Statistics for Engineers: An Introduction with Examples from Practice [1 ed.] 3658323965, 9783658323967

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Hartmut Schiefer Felix Schiefer

Statistics for Engineers An Introduction with Examples from Practice

Statistics for Engineers

Hartmut Schiefer • Felix Schiefer

Statistics for Engineers An Introduction with Examples from Practice

Hartmut Schiefer Mönchweiler, Germany

Felix Schiefer Stuttgart, Germany

ISBN 978-3-658-32396-7 ISBN 978-3-658-32397-4 https://doi.org/10.1007/978-3-658-32397-4

(eBook)

The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). The present version has been revised technically and linguistically by the authors in collaboration with a professional translator. # Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Fachmedien Wiesbaden GmbH part of Springer Nature. The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany

There is at least one solution for every technical task, and for every solution exists a better one.

Foreword

Engineers use statistical methods in many ways for their work. The economic, scientific, and technical requirements of this approach mean that it is necessary to obtain better knowledge of technical systems, such as their cause–effect relationship, a more precise collection and description of data from experiments and observations, and also of the management of technical processes. We are engineers. For us, mathematics and thus also statistics are tools of the trade. We use mathematical methods in many ways, and engineers have made impressive contributions to solving mathematical problems. Examples that spring to mind include the solution of Fourier’s differential equation through numerical discretization by L. Binder (1910) and E. Schmidt (1924), or the elastostatic-element method (ESEM, later called FEM) developed by A. Zimmer in the 1950s, and K. Zuse, whose freely programmable binary calculator (the Z1, 1937) earned him a place among the forefathers of modern computing technology. The present explanations serve as an introduction to the statistical methods used in engineering, where engineers are under constant pressure to save time, money, and materials. However, they can only do this if their knowledge of the sequence ranging from design through to production and the application of the product is as comprehensive as possible. The application of statistics is an important aid to establish such knowledge of interrelationships. Technical development is accompanied by an unprecedented increase in data volumes. This can be seen in all fields, from medicine through to engineering and the natural sciences. Evaluating this wealth of data for a deeper penetration of cause and effect represents both a challenge and an opportunity. The phenomenological description of the relationship between cause and effect enables further theoretical investigation—from the phenomenological model to the physical and technical description. The aim of this textbook is to contribute to the wider application of statistical methods. Applying statistical methods makes statistically founded statements available, reduces the expenditure associated with experiments, and ensures that experiment results are evaluated completely, meaning that more statistically sound information is gained from the

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Foreword

statistically planned experiments or from observations. All in all, the application of statistical methods can lead to more effective and efficient development, more costeffective production with greater process stability, and faster identification of the causes of damage. The contents of statistical methods presented here in seven chapters are intended to facilitate access to the extensive and comprehensive literature that exists in print and online. Examples are used to demonstrate the application of these methods. We hope that the contents of this book will help to bridge the gap between statisticians and engineers. Please also use the opportunities for calculation available online. We are grateful for any suggestions on how to improve the book’s contents and for notification of any errors. We thank Mr. Thomas Zipsner from Springer Vieweg for his constructive cooperation. To satisfy our requirements for the English version of Statistics for Engineers, we would like to thank Mr. James Fixter for his professional cooperation in editing the target text. Mönchweiler, Germany Stuttgart, Germany 2018 (English edition 2021)

Hartmut Schiefer Felix Schiefer

Contents

1

Statistical Design of Experiments (DoE) . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Designing Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Basic Principles of Experiment Design . . . . . . . . . . . . . . . . . 1.1.3 Conducting Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Experiment Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Full Factorial Experiment Designs . . . . . . . . . . . . . . . . . . . . 1.2.2 Latin Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Fractional Factorial Experiment Designs . . . . . . . . . . . . . . . 1.2.4 Factorial Experiment Designs with a Center Point . . . . . . . . . 1.2.5 Central Composite Experiment Designs . . . . . . . . . . . . . . . . Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . .

1 1 2 3 4 7 8 10 12 14 15 20

2

Characterizing the Sample and Population . . . . . . . . . . . . . . . . . . . . . . 2.1 Mean Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Arithmetic Mean (Average) x . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Geometric Mean xG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Median Value xz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Modal Value xD (Mode) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Harmonic Mean (Reciprocal Mean Value) xH . . . . . . . . . . . . . 2.1.6 Relations Between Mean Values . . . . . . . . . . . . . . . . . . . . . . 2.1.7 Robust Arithmetic Mean Values . . . . . . . . . . . . . . . . . . . . . . 2.1.8 Mean Values from Multiple Samples . . . . . . . . . . . . . . . . . . . 2.2 Measures of Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Range R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Variance s2, σ 2 (Dispersion); Standard Deviation (Dispersion of Random Sample) s, σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Coefficient of Variation (Coefficient of Variability) v . . . . . . . 2.2.4 Skewness and Excess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 22 22 24 25 26 26 26 27 27 28 28 29 31 31

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Contents

2.3

Dispersion Range and Confidence Interval . . . . . . . . . . . . . . . . . . . 2.3.1 Dispersion Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Confidence Interval (Confidence Range) . . . . . . . . . . . . . . . Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

33 34 35 38

3

Statistical Measurement Data and Production . . . . . . . . . . . . . . . . . . . 3.1 Statistical Data in Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Machine Capability; Investigating Machine Capability . . . . . . . . . . . 3.3 Process Capability; Process Capability Analysis . . . . . . . . . . . . . . . 3.4 Operating Characteristic Curve; Average Outgoing Quality . . . . . . . Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

39 39 41 42 45 50

4

Error Analysis (Error Calculation) . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Errors in Measured Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Systematic Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Random Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Errors in the Measurement Result . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Errors in the Measurement Result Due to Systematic Errors . . 4.2.2 Errors in the Measurement Result Due to Random Errors . . . 4.2.3 Error Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

51 51 52 53 54 55 59 61 68

5

Statistical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Parameter-Bound Statistical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Hypotheses for Statistical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 t-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 F-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 The Chi-Squared Test (χ 2-Test) . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Conditions for the Chi-Squared Test . . . . . . . . . . . . . . . . . . 5.5.2 Chi-Squared Fit/Distribution Test . . . . . . . . . . . . . . . . . . . . 5.5.3 Chi-Squared Independence Test . . . . . . . . . . . . . . . . . . . . . . Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

69 69 72 75 80 83 83 85 90 93

6

Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Covariance, Empirical Covariance . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Correlation (Sample Correlation), Empirical Correlation Coefficient . 6.3 Partial Correlation Coefficient, Partial Correlation . . . . . . . . . . . . . . Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

95 95 96 97 98

7

Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Cause–Effect Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Nonlinear Regression (Linearization) . . . . . . . . . . . . . . . . . . . . . . .

. . . .

99 99 101 105

Contents

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7.4 Multiple Linear and Nonlinear Regression . . . . . . . . . . . . . . . . . . . . . 7.5 Examples of Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 109 114

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Tables of standard normal distributions . . . . . . . . . . . . . . . . . . . . . . A.2 Tables on the t-Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Tables on the F-Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Chi-Squared Distribution Table . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115 115 118 120 123

Further Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

129

List of Abbreviations

B c cm cmk cp cpk D e F f fx fxi Gym Gys Hj hj H0 H1 K k L( p) N n nx P p Q R r

Coefficient of determination (r2 ¼ B) Number of defective parts Machine controllability Machine capability Process capability index Process capability index, minimum Average outgoing quality Residuum (remainder) Coefficient according to Fisher (F-distribution, F-test) Degree of freedom Confidence region for measured values Measure of dispersion for measured values Error margin, absolute maximum Error margin, absolute statistical Theoretical frequency Empirical frequency, relative frequency Null hypothesis Alternative hypothesis Range coefficient Number of classes Operating characteristic curve Number of values in the basic population Number of samples, number of measured values Number of specific events Basic probability Mean probability, probability of acceptance Basic converse probability Range Correlation coefficient

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S s s2 t ux , u y x xD xG xH xi xz y Z z α α β β γ Δx, Δy δx η λ μ ν σ σ2 Φ(x) φ(x) χ2

List of Abbreviations

Confidence level Standard deviation, dispersion of random sample Sample variance, dispersion Student’s coefficient (t-distribution, t-test) Random error, uncertainty Arithmetic mean of the random sample Most common value, modal value Geometric mean Harmonic mean, reciprocal mean value Measured value, single measured value Median value Mean value of y Relative range z-transformation Probability of error, producer’s risk, supplier’s risk Level of significance, type I error, error of the first kind Type II error, error of the second kind Consumer’s risk Skewness Inaccuracy, systematic error Systematic error Excess Value of normal distribution Mean of basic population Coefficient of variation, coefficient of variability Standard deviation, dispersion of basic population Variance of basic population Distribution function Density function Coefficient according to Helmert/Pearson (χ 2-distribution, χ 2-test)

1

Statistical Design of Experiments (DoE)

In a cause–effect relationship, the design of experiments (DoE) is a means and method of determining the interrelationship in the required accuracy and scope with the lowest possible expenditure in terms of time, material, and other resources. From a given definite cause, an effect necessarily follows; and, on the other hand, if no definite cause be granted, it is impossible that an effect can follow. Baruch de Spinoza (1632–1677), Ethics

1.1

Designing Experiments

Conducting experiments answers the question of what type and level of effect influencing variables (factors, variables) have on the result or target variable(s). To achieve this, the influencing variable(s) must be varied in order to determine the effect on the result. The task can be easily formulated as a “black box” (see Fig. 1.1). The influencing variables (constant, variable), the target variable(s), and the testing range are thus to be defined. An objective evaluation of results is not possible without statistical test procedures. When planning experiments, this requires that the necessary statistical results are available (i.e., the experiment question posed can be answered). To this end, the question must be carefully considered, and the sequence of activities must be determined. In contrast to experiments, observation does not influence the cause–effect relationship. However, observations should also be conducted according to a plan in order to consciously exclude or include certain influences, for example. Despite the differences between experiments and observations, the same methods can be used to evaluate the results. # Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 H. Schiefer, F. Schiefer, Statistics for Engineers, https://doi.org/10.1007/978-3-658-32397-4_1

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1

Statistical Design of Experiments (DoE)

Fig. 1.1 The cause–effect relationship as a “black box”

1.1.1

Basic Concepts

• The testing unit (experiment unit) is the system to be investigated. • The procedure refers to the type of action on the experiment unit. • The influencing variable is an independently adjustable value; it is either varied in the experiment (variable) or is constant. • A variable is either a discrete (discontinuous) variable, which can only assume certain values, or a continuous variable with arbitrary values of real numbers. • The target variable is a variable dependent on the influencing variables. • The experiment range is the predefined or selected range in which the influencing variables are varied. • The levels of the influencing variables are their setting values. • The experiment point is defined by the setting values of all variables and constant influencing variables. • The experiment design is the entire program of experiments to be conducted and is systematically derived in accordance with the research question. In a first-order experiment design, for example, the influencing variables vary on two levels; in a second-order design, they vary on three levels. Influences that interfere with the test are eliminated by covariance analysis with known, qualitatively measurable disturbance variables, through blocking, and through random allocation of the experiment points.

1.1

Designing Experiments

1.1.2

3

Basic Principles of Experiment Design

In general, the following basic principles are generally to be applied when planning, conducting, and evaluating tests or observations: Repeating tests In order to determine the dispersion, several measurements at the experiment point are required, thus enabling statements to be made about the confidence interval (see Sect. 2. 3.2). With optimized test plans, two measurements per experiment point are appropriate. As the number of measurements at the experiment point increases, the mean of the sample x approaches the value of the population (see Chap. 2); the confidence interval becomes smaller. Randomization The allocation within the predefined experiment range must be random. This ensures that trends in the experiment range, such as time- or location-related trends, do not falsify the results. If such trends occur, dispersion increases due to random allocation. If, on the other hand, the trend is known, it can be considered a further influencing variable and thus reduces the dispersion. Changes in temperature and humidity due to seasonal influences are examples of trends, particularly in a comprehensive experiment design. Blocking A block consolidates tests that correspond in terms of an essential characteristic or factor. The evaluation/calculation of the effect of the influencing variables is then conducted within the blocks. If the characteristic that characterizes the blocks is quantifiable, its influence in the testing range is calculable. This, in turn, reduces the dispersion in the description of the cause–effect relationship. If possible, the blocks should be equally extensive. Example of blocking Semi-finished aluminum products are used to produce finished parts exhibiting a strength within an agreed range. The semi-finished product is supplied by two manufacturers. The strength test on random samples of the semi-finished products from both suppliers shows a relatively large variation of values within the agreed strength range. When the values of the two suppliers are evaluated separately, it becomes clear that the characteristic levels of the semi-finished products vary. The mean values and levels of dispersion are different. Through this blocking (whereby each supplier forms a block), it becomes clear which

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Statistical Design of Experiments (DoE)

level of semi-finished product quality is available in each case. Other examples of possible blocking include: • Summer operation/winter operation of a production plant • Blocking upon a change of batch Symmetrical structure of the experiment design A symmetrical structure in the experiment range enables a complete evaluation of the results and avoids a loss of information. For feasibility reasons (e.g., cost-related factors), symmetry in the experiment design can be dispensed with. In the case of unknown result areas within the experiment range, symmetry of the experiment design (i.e., symmetrical distribution of the experiment points) is to be targeted. If the result area is determined by the tests or already known, the symmetry can be foregone. In particular, further experiments can be carried out in a neighboring range in order to follow up the results in this range, such as in the case of optimization.

1.1.3

Conducting Experiments

Procedure A systematic approach to experimentation makes it possible to answer the experiment questions posed under conditions that are economical. Errors are also avoided. The following procedure can be described as a general procedure: Describing the initial situation The experiment unit (i.e., the system to be examined) must be defined. For this purpose, it is worth considering the system as a “black box” (see Sect. 1.1) and recording all physical and technical variables. This concerns the influencing variables to be investigated, the values to be kept constant in the experiment, disturbance variables, and the target variable (s). It is important to record all values and variables before and during the experiment. This will avoid confusion and repetition in the test evaluation at a later point. Since each effect can have one or more causes, it is possible to include other variables, for example, values defined as “constant,” in the calculation (correlation, regression) retroactively. Under certain circumstances, disturbance variables can also be quantified and included in the calculation. In any event, the consideration of the experiment system as a “black box” ensures a concrete question for the experiment. The finding that an “influencing variable” has no influence on a certain target variable is also valuable. Defining the objective of the experiments; forming hypotheses on the cause–effect relationship The aim of the experiments, the variation of the influencing variables on the target variable(s), is to determine the functional influence (i.e., considering the effect of a number

1.1

Designing Experiments

5

of influencing variables xi on the target variable y. This is the hypothesis to be tested in the experiments. The functional dependence is obtained as confirmation of the hypothesis:

y ¼ f ð xi Þ

or with multiple target variables:

y j ¼ f ð xi Þ

As such, the influence of the variables xi on several target variables generally differs qualitatively and quantitatively (see Sect. 7.1). If there is no correlation between the selected variables xi and yj, the hypothesis is wrong. This finding is experimentally justified; an improved hypothesis can be derived. Defining the influencing variables, the target variables, and the values to be kept constant From the hypothesis formation follows the determination of the influencing variables, the definition of the experiment range, and the setting values derived from the test planning. It should be noted that the number of influencing variables xi is generally not the same for all target variables yj. Selecting and preparing the experiment design The dependency that exists, or is to be assumed, between the influencing variables and target variable(s) must be clarified and specified; in other words, whether linear or nonlinear correlations exist. Over larger experiment ranges, many dependencies in science and technology are nonlinear. If, however, smaller experiment ranges are considered, they can often be described as linear dependencies by way of approximation. For example, cooling processes are nonlinear as they take place in accordance with an e-function. However, after a longer cooling time, the temperature changes can be seen to be linear in smaller sections. The degree of nonlinearity to be assumed must be taken into account in the experiment design. It is entirely possible to identify a higher degree of nonlinearity and then, after calculating the relationship, determine that a lower degree exists. In addition, the number of test repetitions per measuring point must be specified. Finally, the sequence of experiment points must be defined. The order of the measurements is usually determined through random assignment (random numbers) in order to avoid gradients in the design. However, this increases the effect of the gradient; the dispersion in the context of cause and effect. Alternatively, the gradient can be recorded and treated as a further influencing variable; for example, recording the ambient temperature for extensive series of measurements (e.g., the problem of summer/winter temperature) or measuring the current room humidity (e.g., the problem of high humidity after rain showers). If a disturbance/influencing variable occurs for certain values, the experiments can be combined in the form of blocks for these constant values. The effect of the influencing variable/disturbance variable is then visible between the blocks.

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Statistical Design of Experiments (DoE)

Conducting tests/experiments Experiments must always be carried out correctly at the experiment point under specified conditions. Time pressure, for example, can be a source of error. Results gained from improperly performed tests cannot be corrected; such tests must be repeated over the appropriate amount of time. Evaluating and interpreting test results Test results are evaluated using statistical methods. To name examples, these include averaging at the experiment point, calculating the degree of dispersion, and the treatment of outlying values. This also includes consideration of correlation and, finally, the calculation of correlations between influencing variables and target variable(s). In general, a linear regression function is started with an unknown functional dependence, which is further developed as a higher-order function while also considering the interaction between the influencing variables. It should be noted that the regression function only represents what has been determined in the experiment: information from the experiment points. If, for example, the experiment points in the experiment range are far apart, local extremes (maxima and minima) between the experiment points are not recorded and therefore cannot be described mathematically. It is also understandable that a regression function is not more accurate than the values from which it was calculated. Neat and correct test execution is thus indispensable for the evaluation. The regression function can only describe what has been recorded in the experiments. It follows from this that, the more extensive the test material is and the more completely the significant (essential) influencing variables have been recorded, the better the expression of a regression function will be. Being polynomial, the regression function is empirical in nature. Alternatively, if the functional dependence of the target variable on the influencing variable is known, this function can be used for regression. The interpretation of correlations between the target variable and variables associated with the influencing variables due to polynomials, such as structural material quantities, must be performed with great caution. It should be remembered that many functions, for example, the e-function or the natural logarithm, can be described using polynomials. A regression function only applies to the investigated range (experiment range). Extrapolation (i.e., calculations with the regression function outside the experiment range) are generally associated with high risk. In the case of known or calculated nonlinear relationships between cause(s) and effect(s), for example, extrapolation beyond the experiment range is problematic. In the experiment range, the given function is adapted to the measuring points (“curve fitting”) in such a way that the sum of the deviation squares becomes a minimum (C. F. Gauß, Sect. 7.2). This means that large deviations between calculated and actual values can occur outside the experiment range, especially in the case of nonlinear relationships.

1.2

1.2

Experiment Designs

7

Experiment Designs

The conventional methods of conducting experiments are as follows: • Random experiment (i.e., random variation of the influencing variables xi and measurement of the target quantity y). Many experiments are required in this case. • Grid line experiment, whereby the influencing variables xi are varied in a grid. In order to obtain good results, a fine grid is required, thus many experiments are required. • Factorial experiment, whereby only one influencing variable xi is changed at a time; the other influencing variables remain constant, thus the interaction of the influencing variables cannot be determined. The effort involved is high. As such, conventional methods of experiment design involve a great deal of effort, both when conducting the experiments and when evaluating results. In practice, the question in the experiment is often limited (e.g., due to a desire to minimize the effort involved). This is achieved through statistical experiment designs. The effort involved in planning experiments, conducting experiments, and evaluating the results is significantly lower. It should be noted that the experiment design is structured/selected in such a way that the question can actually be answered. In the case of extensive tests, a random allocation of the tests must be performed in order to eliminate gradients (e.g., a change in test conditions over time). If the experiment conditions for one parameter cannot be kept constant (e.g., in the case of batch changes), blocking must be applied. The other influences are then calculated for the same batch. With regard to the variation of one influencing variable xi to the target value y, experiment designs have the following advantages: • Significantly less effort through reduction in the number of tests. • Planned distribution of the experiment points and thus no deficits in recording (structured approach). • Simultaneous variation of the influencing variables at the experiment points, thus enabling determination of the interaction between influencing variables. • The more influencing variables there are, the more effective (less effort) statistical experiment designs are. In most cases, experiment designs can be extended unconventionally in one direction or in several directions (parameters); for example, in order to pursue a maximum value at the edge of the previous experiment range. This requires the extended parameters to be varied on two levels in the linear scenario, and on at least three levels in the nonlinear scenario.

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1

Table 1.1 Experiment design with two factors

1.2.1

Statistical Design of Experiments (DoE)

Level combination A – + – +

Experiment no. 1 2 3 4

B – – + +

Full Factorial Experiment Designs

Full factorial experiment designs are designs in which the influencing factors are fully combined in their levels; they are varied simultaneously. The simplest scenario with two influencing variables (factors) A and B that are varied on two levels (plus and minus) results in the design in Table 1.1. In this case, a total of four experiments exist that can be visualized as the corners of a square, see Fig. 1.2. If the experiment design has three influencing variables (factors) A, B, and C, this provides the results in Table 1.2. The level combinations yield the corners of a cube. In general, full factorial experiment designs with two levels and k influencing factors yield the following number of experiments: Number of experiments ¼ LevelsInfluencing variables z ¼ 2k For instance, k ¼ 4: z ¼ 16 experiments or k ¼ 6: z ¼ 64 experiments. In the case of full factorial experiment designs with two levels, the main effects of the influencing variables and the interactions among variables can be calculated. The dependencies are linear since variation only takes place on two levels. In general, this means the following with two influencing variables: y ¼ a0 þ a1 x1 þ a2 x2 þ a12 x1 x2 þ sR |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflffl{zfflfflfflffl} Main linear effect

Reciprocal effect

It also means the following with three influencing variables: y ¼ ao þ a1 x1 þ a2 x2 þ a3 x3 þ a12 x1 x2 þ a13 x1 x3 þ a23 x2 x3 þ a123 x1 x2 x3 þ sR During the calculation, a term is created that cannot be assigned to the influencing factors. This is the residual dispersion sR.

1.2

Experiment Designs

9

Fig. 1.2 Factorial experiment design with four experiments

Table 1.2 Factorial experiment design with three factors

Experiment no. 1 2 3 4 5 6 7 8

Level combination A B – – + – – + + + – – + – – + + +

C – – – – + + + +

For full factorial experiment designs, all effects of the parameters (i.e., main effects and interactions between the parameters with two or more parameters together) can be calculated independently of one another. There is no “mixing” of parameters. The effects of the parameters must be clearly assigned. The test results yield a system of equations of this kind: y1 ¼ a11 x1 þ a12 x2 þ . . . þ a1n xn y2 ¼ a21 x1 þ a22 x2 þ . . . þ a2n xn ⋮ ym ¼ am1 x1 þ am2 x2 þ . . . amn xn The system of equations can be solved for n unknowns with m linearly independent equations where n  m. When m > n, overdetermination occurs. The test dispersion can be determined using this information. Since there is no solution matrix with which all equations are fulfilled, the solution is determined using Gaussian normal equations, whereby the deviation squares are a minimum. Overdetermination can also be reduced or eliminated by further influencing parameters.

10

1

Statistical Design of Experiments (DoE)

Table 1.3 Factorial experiment design with blocking

Experiment no. 1 4 6 7 2 3 5 8

A – + + – + – – +

B – + – + – + – +

Factors C – – + + – – + +

ABC 1 (–) 1 (–) 1 (–) 1 (–) 2 (+) 2 (+) 2 (+) 2 (+)

Block 1

Block 2

General solution methods for the system of equations are the Gaussian algorithm and the replacement procedure; these are implemented in the statistics software. The linear approach contained in a full factorial experiment design can be easily verified through tests at the center point (see Sect. 1.2.4). The full factorial experiment design with the three factors A, B, and C consists of 23 ¼ 8 factor-level combinations. These factor-level combinations are used to calculate the main effects of factors A, B, and C, their two-way interaction (i.e., AB, AC, and BC), as well as their three-way interaction (ABC). Refer to Table 1.3 in this case. If the three-way interaction is omitted (e.g., because this interaction cannot occur due to general findings), then a fractional factorial experiment design is created. If the three-way interaction is used as a blocking factor (see Sect. 1.1.2), this results in the plan shown in Table 1.3 (block 1 where ABC is “”, while block 2 with ABC is “+”).

1.2.2

Latin Squares

As experiment designs, Latin squares enable the main effects of three factors (influencing variables) with the same number of levels to be investigated. Compared to a full factorial experiment design, Latin squares are considerably more economical because they have fewer experiment points. The dependence of the influencing variables relative to the target variable is to be described linearly as well as nonlinearly, depending on the number of levels. However, it is not possible to determine an interaction between the factors since the value of the factor occurs only once for each level. Under certain circumstances, the main effect is thus not clearly interpretable; if interaction occurs, it is assigned to the main effects. The use of Latin squares is therefore only advisable if it is certain that interaction between influencing variables will not play a significant part or can be disregarded. An example of a Latin square with three levels ( p ¼ 3) is shown in Table 1.4.

1.2

Experiment Designs

Table 1.4 Latin square with three levels

11 a2 c2 c3 c1

a1 c1 c2 c3

b1 b2 b3

a3 c3 c1 c2

The level combinations of the influencing variables (factors) are as follows: a1 b1 is combined with c1 a2 b1 is combined with c2 ... The following also applies: a3 b3 is combined with c2 In every row and every column, every c level is thus varied once, see Table 1.5. Since not only the sequence of the levels c1, c2, and c3 is possible, but also permutations (i.e., as shown in Table 1.5), there are a total of 12 different configurations that satisfy the same conditions. For a Latin square where p ¼ 2, thus with two levels for the influencing variables, the experiment designs shown in Table 1.6 are generated. With two levels for the influencing variables, a linear correlation of the following form can be calculated: y ¼ α0 þ α1 a þ α2 b þ α3 c

or generally:

y ¼ a0 þ a1 x 1 þ a2 x 2 þ a3 x 3

In the case of three or more levels, nonlinearities can also be determined, i.e.: y ¼ α0 þ α1 a þ α11 a2 þ α2 b þ α22 b2 þ α3 c þ α33 c2 The calculation of interaction is excluded for well-known reasons. Nine experiments are generated with the aforementioned Latin square with three levels ( p ¼ 3). Compared to a complete experiment design with a complete combination of the levels of the influencing factors (see Sect. 1.2.1) yielding 27 experiments, the Latin Square, therefore, requires only an effort of 1/p. Since the levels of the influencing variables are not completely permuted, the interaction of the influencing variables is missing from the evaluation. Any interaction that occurs is therefore attributed to the main effects of the influencing factors. For example, a Latin square with four levels is written as shown in Table 1.7.

12

1

Table 1.5 Permutations in the Latin square

Table 1.6 Latin square with two levels

Table 1.7 Latin square with four levels

b1 b2 b3 b4

a2 c2 c3 c1

a1 c3 c1 c2

b1 b2 b3

b1 b2

Statistical Design of Experiments (DoE)

a2 c2 c1

a1 c1 c2

a1 c1 c2 c3 c4

a3 c1 c2 c3

or b1 b2

a2 c2 c3 c4 c1

a3 c3 c4 c1 c2

a1 c2 c1

a2 c1 c2

a4 c4 c1 c2 c3

Example The effect of three setting variables for a system/plant on the target variable on three levels is to be investigated, see Table 1.8. On a lathe, for example, the setting values for the speed, feed rate, and depth of cut are varied on three levels. This results in the following combinations of setting values (the values ai, bi, ci correspond to the setting values): a1 b1 c 1 ,

a2 b1 c 2 ,

a3 b1 c 3

a1 b2 c 2 , a1 b3 c 3 ,

a2 b2 c 3 , a2 b3 c 1 ,

a3 b2 c 1 a3 b3 c 2

With these nine experiments, the main effect of the three setting variables a, b, and c can be calculated.

1.2.3

Fractional Factorial Experiment Designs

If the complete effect and interaction of the parameters/influencing variables on the target variable is not of interest for the technical task in question, the experiment design can be reduced. This is associated with a reduction in time and costs. Such reduced experiment designs are fractional factorial experiment designs. With three influencing variables, for example, eight experiments are required in a full factorial experiment design with two levels (settings per influencing variable). In the example, these make up the eight corners of the cube. In the fractional factorial experiment design, the number of experiments is reduced to four.

1.2

Experiment Designs

Table 1.8 Sample setting values for Latin squares

13

Setting variable a Setting variable b Setting variable c

Setting values (levels) a2 a1 b1 b2 c1 c2

a3 b3 c3

Fig. 1.3 Fractional factorial design with three influencing variables in comparison to the full factorial design

In the case of fractional factorial designs, it should generally be noted that each parameter is also varied (changed) in the design, otherwise, its effect cannot be calculated. This results in the two fractional factorial designs. In the example (Fig. 1.3), there are the four experiments with the numbers 2, 3, 5, 8 (red experiment points), or there is alternatively the design with experiments 1, 4, 6, 7 (black experiment points). The fractional factorial experiment design with three influencing variables (A, B, and C) then appears as shown in Fig. 1.3. With three factors (A, B, and C), the fractional factorial design thus only has four experiments. Here, it is necessary that each factor A, B, and C exhibits measured values of the target function y at both the first test limit (plus) and the second test limit (minus).

14

1

Statistical Design of Experiments (DoE)

The calculable function of the effects of the three influencing variables then results in the following: y ¼ a0 þ a1 A þ a2 B þ a3 C As such, no interaction among the influencing variables can be determined. “Mixing” can occur in the case of fractional factorial designs. Since interaction among parameters can no longer be determined through a lack of parameter variations, the main effects contain any potentially existing interactions; therefore, the effect of the parameter A and the interaction of AB add up, for instance.

1.2.4

Factorial Experiment Designs with a Center Point

Full factorial experiment designs and the fractional factorial experiment designs derived from them assume that cause–effect relationships are linear. This can often be assumed as a first approximation. In an initial approach, a linear assumption is also entirely possible when reducing the testing range and thus the distances between the experiment points. However, nonlinearities generally exist in a technical context, in the case of engineering-related questions, and also in natural processes between influencing variables and the effects of these variables. Examples include transient thermal processes, relaxation, and retardation. If such a situation occurs, it must be determined whether nonlinearity occurs in the case under consideration (engineering problem, extent of the experiment range, distance of experiment points). In the case of experiment designs using a linear model (Sects. 1.2.1 and 1.2.3), it is possible to determine whether nonlinearity exists with little effort. A test is performed at the center point, see Fig. 1.4. The center point is an equal distance from the other experiment points. If the mathematical value of the linear model of the regression function at the center point is then compared to the measured value at the center point, it is possible to determine whether the linear approach is justified. If the center point is repeated multiple times (i.e.,

Fig. 1.4 Experiment design with center point

1.2

Experiment Designs

15

weighted), information on the dispersion is obtained. The center point is a suitable experiment point for repetitions. In order to obtain confidence intervals (see Sect. 2.3.2), two or three influencing variables are repeated once per experiment point. With four or more parameters, the center point is weighted through three to ten repetitions.

1.2.5

Central Composite Experiment Designs

Central composite experiment designs have three components, namely: • Factorial core (see Sect. 1.2.1): A, B with levels + and  • Center point (see Sect. 1.2.4): A, B at level “0” • Star points (axial points): A, B with levels +α and α This is illustrated in Fig. 1.5. α is used to refer to the distance between the axial points and the center point. For two influencing variables A and B, the following experiment design with nine experiments is given. The illustration in Fig. 1.5 is intended to clarify this. See also Fig. 1.6 and Table 1.9.

Fig. 1.5 Central composite experiment design

16

1

Statistical Design of Experiments (DoE)

Fig. 1.6 Experiment design with two influencing variables A and B

Table 1.9 Experiment points with two influencing variables A and B

Experiment no. 1 2 3 4 5 6 7 8 9

Influencing variable A – + – + 0 –α +α 0 0

B – – + + 0 0 0 –α +α

The influencing variables A and B are thus varied on five levels. This makes it possible to calculate the nonlinearity of the influencing variables. In the aforementioned example with the influencing variables A and B, the regression function is thus as follows:

The central composite experiment designs can be subdivided as follows: 1. Orthogonal design In the case of a factorial 22 core, α ¼ 1. In other words, the experiment points of the “star” fall in the middle of the connecting lines for the experiments at the core. With a 26

1.2

Experiment Designs

17

Fig. 1.7 Box–Hunter design for two factors (α ¼ 1.414)

Table 1.10 Experiment design according to Box and Hunter Number of independent factors (influencing variables) n Number of corner points nc Number of axial values nα Number of center experiments n0 Total experiments N Center distance α ¼ nc1/4

2

3

4

5

6

4 4 5 13 1.414

8 6 6 20 1.682

16 8 7 31 2.000

32 10 10 52 2.378

64 12 15 91 2.828

core, then α ¼ 1.761, see Fig. 1.5. The regression coefficients are calculated independently of one another; there is no mixing of effects. 2. Rotatable design; Box–Hunter design [1], Fig. 1.7. With this design, and with a 22 core, α has the value α ¼ 1.414, and with a 26 core, α ¼ 2.828 (see Table 1.10). The center point is weighted. With two factors, five tests are carried out at the center point; with five factors, ten tests are carried out. The experiment points of the core and the star points are the same distance from the center point; they lie on the same spherical surface. For n influence quantities, the experiment points (measuring points) are located on the surface of an n-dimensional sphere and in the weighted center of the sphere. With two influencing variables (factors), the experiment points lie on a circle around the weighted center. The experiment points on the circle are four corner points and four axial values, see Fig. 1.7. In the threedimensional case (three influencing variables), the experiment points lie on the sphere surface, with an axis distance of α ¼ 1.682 and in the weighted center. The weighting of the center ensures that the same confidence level exists throughout the entire experiment range.

18

1

Statistical Design of Experiments (DoE)

3. Pseudo-orthogonal and rotatable design This design combines the advantages of the orthogonal and the rotatable design. No mixing occurs, although the number of experiment points increases compared to the rotatable plan. The α values are the same as for the rotatable plan; the center point has a higher weighting. For example, there are 8 experiments at the center point with a 22 core and 24 experiments with a 26 core.

Example: Ceramic injection molding with gas injection technology [2] The parameters injection volume (A), delay time for the GIT technique (B), gas pressure (C), and gas pressure duration (D) on a ceramic injection-molded part produced with gas injection technology (GIT) were examined for the following dependent parameters (target variables): bubble length, wall thickness, weight, and crack formation (crack length). The four-factor experiment design according to Box and Hunter was used for this purpose. With four independent factors (influencing variables), there are a total of 31 experiments (see Table 1.10). The test values for these conditions are shown in (α ¼ 2) Table 1.11. The individual values of the parameters result from the experiment range (i.e., from the respective minimum and maximum parameter values). These limits are determined by the product volume, and the mechanical and technological conditions of the injection-molding machine. In the example for the injection volume [ccm], values range from 26.3 (minimum value) to 26.9 (maximum value). These values correspond to the standardized values 2 and +2. The values for 1, 0, +1 result from the limits of the parameter range. The standardized values and the experiment values for all 31 experiments are given in Table 1.12. The test results were statistically evaluated through regression; sample results for these tests are listed in Sect. 7.5. In addition to product optimization with regard to specified quality criteria, technical production decisions can also be substantiated. The optimal injection-molding machine (in terms of quality and cost) can be determined from the knowledge of the essential (significant) influencing factors during production.

Table 1.11 Experiment values for GIT technology Parameter Injection volume [ccm] Delay [s] Gas pressure [bar] Gas pressure duration [s]

Designation A B C D

2 26.30 0.52 140 0.50

1 26.45 0.89 165 0.88

0 26.60 1.26 190 1.25

+1 26.75 1.63 215 1.63

+2 26.90 2.00 240 2.00

1.2

Experiment Designs

19

Table 1.12 Experiment design: experiment values and standardized values No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

Experiment values A B 26.60 1.26 26.60 1.26 26.60 1.26 26.60 1.26 26.60 1.26 26.60 1.26 26.60 1.26 26.75 1.63 26.75 1.63 26.75 1.63 26.75 1.63 26.75 0.89 26.75 0.89 26.75 0.89 26.75 0.89 26.45 1.63 26.45 1.63 26.45 1.63 26.45 1.63 26.45 0.89 26.45 0.89 26.45 0.89 26.45 0.89 26.90 1.26 26.30 1.26 26.60 2.00 26.60 0.52 26.60 1.26 26.60 1.26 26.60 1.26 26.60 1.26

C 190 190 190 190 190 190 190 215 215 165 165 215 215 165 165 215 215 165 165 215 215 165 165 190 190 190 190 240 140 190 190

D 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.63 0.88 1.63 0.88 1.63 0.88 1.63 0.88 1.63 0.88 1.63 0.88 1.63 0.88 1.63 0.88 1.25 1.25 1.25 1.25 1.25 1.25 2.00 0.50

Standardized values A B 0 0 0 0 0 0 0 0 0 0 0 0 0 0 +1 +1 +1 +1 +1 +1 +1 +1 +1 1 +1 1 +1 1 +1 1 1 +1 1 +1 1 +1 1 +1 1 1 1 1 1 1 1 1 +2 0 2 0 0 +2 0 2 0 0 0 0 0 0 0 0

C 0 0 0 0 0 0 0 +1 +1 1 1 +1 +1 1 1 +1 +1 1 1 +1 +1 1 1 0 0 0 0 +2 2 0 0

D 0 0 0 0 0 0 0 +1 1 +1 1 +1 1 +1 1 +1 1 +1 1 +1 1 +1 1 0 0 0 0 0 0 +2 2

Example: Investigation of a cardiac support system [3] The performance parameters of a cardiac support system based on the two-chamber system of the human heart were investigated. The system is driven by an electrohydraulic energy converter. An intraoral balloon pump is used to relieve the heart. For this purpose, a balloon catheter is inserted into the aorta. Filling and emptying the specially developed balloon provides the support function for the heart.

20

1

Statistical Design of Experiments (DoE)

Table 1.13 Default and experiment values of the Box–Hunter design Air–vessel pressure [mmHg] Experiment value Default 60 +2 70 +1 80 0 90 1 100 2

Speed [rpm] Experiment value 10,000 9000 8000 7000 6000

Default +2 +1 0 1 2

Trigger frequency [Hz] Experiment value Default 2 +2 1.75 +1 1.5 0 1.25 1 1 2

Symmetry, diastole/ systole Experiment value Default 65/35 +2 60/40 +1 55/45 0 50/50 1 45/55 2

The following four factors (input parameters) were investigated: • • • •

Frequency Systole/diastole ratio Speed of the pump Air–vessel pressure

The target value is the volumetric flow within the cardiac support system. In order to save time and costs, and to process the task in a given time, a Box–Hunter experiment design was used. With this design, the factors were varied on five levels. This also enables nonlinearities to be calculated (number of levels greater than 2). There is a total of 31 experiments with this design. With conventional experiment procedures and variation on five levels, 54 ¼ 625 experiments are required. The savings in terms of experiments are therefore considerable. Table 1.13 shows the default values and the experiment values in the Box– Hunter design.

Literature 1. Box, G.E., Hunter, J.S.: Ann. Math. Stat. 28(3), 195–241 (1957) 2. Schiefer, H.: Spritzgießen von keramischen Massen mit der Gas-Innendruck-Technik. Lecture: IHK Pforzheim, 11/24/1998 (1998) 3. Noack, C.: Leistungsmessungen eines elektrohydraulischen Antriebes in zwei Anwendungsfällen. Thesis: FH Furtwangen, 22 June 2004

2

Characterizing the Sample and Population

Measured values or observed values are only fully characterized by the mean value and measure of dispersion or indication of the error. Experiments or observations initially yield single values; repetitions under the same conditions give totals of single values. For infinite repetitions (n ! 1), infinite totals are created, which are referred to as the basic population. If the single values xi are finite, then the basic population is N. In practice, the repetitions are finite (i.e., there is a sample from the population N ). The sample size n is the number of repetitions. The degree of freedom f is the number of supernumerary measurements/observations required for their characterization (i.e., f ¼ n  1). See also DIN ISO 3534-1 [1] and DIN ISO 3534-2 [2]. The sample is characterized by the relative frequency distribution of the characteristics. As the sample size increases, the distribution of characteristics approaches the probability distribution of the population. Provided that the sample is taken randomly, the parameters of the sample (e.g., mean and dispersion) can be used to deduce the corresponding population. The randomness of the sample taken requires equal conditions and mutual independence. Parameters with a constant probability distribution are described by the relative frequency; discrete characteristics are described by the corresponding probability. In science and technology, the measured variables (continuous random variables) usually have a population described by the normal distribution—see also Fig. 2.1. While the variables of the basic population, such as the mean value μ and variance σ 2, represent unknown parameters, the values of a concrete sample (random sample) including the mean value x and the variance s2 from sample to sample are each realizations of a random variable. In addition to the mean value μ resp. x and the variance σ 2 resp. s2, other measures for average values (mean values) and measures of dispersion are formed. Both the various mean values and the measures of dispersion have different properties.

# Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 H. Schiefer, F. Schiefer, Statistics for Engineers, https://doi.org/10.1007/978-3-658-32397-4_2

21

22

2

Characterizing the Sample and Population

Fig. 2.1 Illustration of distributions. (a) Bar chart with symmetrical distribution. (b) Modal value xD and median value xZ on a numerical scale

Among the measures of dispersion, variance/dispersion has the smallest deviation and should therefore preferably be used. If there is an asymmetrical distribution (non-Gaussian distribution), the asymmetry is described by the skewness. The deviation of a symmetrical distribution from the normal distribution as a flatter or more pointed distribution is recorded by the excess. The dispersion range is specified with a given confidence level to characterize the dispersion of a sample. The confidence interval is the range of the mean value of the population to be estimated with a sample.

2.1

Mean Values

Various characterizing values/mean values can be calculated from a discrete number of measurement or observation values. However, the most important mean value is the arithmetic mean. Under certain conditions, other mean values are nonetheless also effective, such as when the values are widely dispersed.

2.1.1

Arithmetic Mean (Average) x

The arithmetic mean of a sample x is the mean value of the measured values xi. The arithmetic mean of the basic population is μ.

2.1

Mean Values

23

x¼

n 1 1 X ð x1 þ x2 þ . . . þ xn Þ ¼ x n n i¼1 i

n Number of values in the sample In the case of cumulative values (weighted average), the following applies: x¼

n 1 X xh n j¼1 j j

where h j ¼

nj n

hj Relative frequency of jth class with class width b, see Fig. 2.1 nj Number of values (population density) in the jth class k X

nj ¼ n

j¼1

For classified values, the frequency of a class is calculated using the mean of this class to give the arithmetic mean. The following applies for the classification: Number of classes k ¼

Range of the values xmax  xmin ¼ Δx Δx

Here, there is a Δx class width. Guideline values for the number of classes: k¼

pffiffiffi n or k  10 for n  100 k  20 for n  105

Properties of x: X X

ðx  xi Þ2 ¼ Min

ð x  xi Þ ¼ 0 x!μ

where

n!N

or

1

N Number of values in the population. The arithmetic mean of the sample x is a faithful estimate of the mean value of the population.

24

2

Characterizing the Sample and Population

For counter values, the following correspondence applies: x ≙ Mean probability p p¼

nx Number of specific events ¼ Total number of events n

Basic population for counter values μ ≙ P (Basic probability) Q ¼ 1  P (Basic converse probability) The arithmetic mean is the first sample moment. Example of the arithmetic mean value The following individual thickness values in mm were measured on steel test plates: 0.54; 0.49; 0.47; 0.50; 0.50

x¼

1X 2:50 mm xi ¼ ¼ 0:50 mm n 5

Example of the population density The frequency of the class is the number of measured values relative to the total number of measured values in the sample. With n7 ¼ 17 values in the class width of the 7th class and the total number of values in the sample n ¼ 112, the following is given:

h7 ¼ 17=112 ¼ 0:15

2.1.2

Geometric Mean x G

The geometric mean xG of a sample with the number of measured values n is the nth root of the product of their measured values. If at least one measured value is equal to zero or negative, the calculation of the geometric mean is not possible. xG ¼

ffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p n x1 ∙ x2 ∙ . . . xn ¼ n Πxi

where

xi > 0

2.1

Mean Values

25

Example of the geometric mean As an example, the following values represent the development of a production line’s productivity over the last 4 years: 104.3%; 107.1%; 98.7%; 103.3%. The geometric mean is calculated as follows: xG ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 4 104:3 ∙ 107:1 ∙ 98:7 ∙ 103:3 ¼ 103:3%

The average increase in productivity thus amounts to 3.3%.

2.1.3

Median Value xz

The middle value xz or median is the value that halves a series of values ordered by size. If the number of measured values is odd (2k+1), the median is the (k+1)th value from the beginning or end of the series. If a series consists of an even number (2k) of values, then the median of the two values is the kth value from the start or the end of the series. Properties of xz: • It remains unaffected by extreme values, such as outliers. P • ðxz  xi Þ ! Min • In the case of extensive value series, xz is negligibly different from x. The median is also referred to as the 0.5 quantile (50th percentile) of an ordered series of values. The value of the pth percentile has the ordinal number p (n+1) of the value series where 0 < p < 1. The 0.25 quantile is also called the lower quartile, while the 0.75 quantile is called the upper quartile. Example The 25th percentile (0.25 quantile) of an ascending series of 50 measured values results in the following value:

p ðn þ 1Þ ¼ 0:25 ð50 þ 1Þ ¼ 12:75, thus the 13th value

26

2

2.1.4

Characterizing the Sample and Population

Modal Value xD (Mode)

The most common value xD, modal value, or mode is the value that occurs most frequently in a series of values. xD lies below the peak of the frequency distribution. If the value series is normally distributed, the most common value xD is negligibly different from x . The modal value remains unaffected by extreme values (outliers).

2.1.5

Harmonic Mean (Reciprocal Mean Value) x H

The harmonic mean xH of the values xi is defined as follows: xH ¼

1 x1

n n ¼ þ x12 þ . . . x1n ∑ni¼1 x1i

where xi 6¼ 0

The reciprocal of the harmonic mean value n 1 1X 1 ¼ xH n i¼1 xi

is the arithmetic mean of the reciprocal values x1i . If the values xi are assigned positive weightings gi, the weighted harmonic mean xHg is obtained: xHg ¼

∑ni¼1 gi ∑ni¼1 gxii

The harmonic mean xH is used to calculate mean values for quotients.

2.1.6

Relations Between Mean Values

The different mean values x, xG , xz , xD , and x H calculated from the single values xi lie between xmax and xmin in an ordered series of values. Outliers (extreme measured values above and below) influence the arithmetic mean value x relatively strongly. On the other hand, the median xz and the most common value xD are not changed. This insensitivity of the median and mode xD is called “robustness.” With a sample size of n ¼ 2, the sample median and the range center are identical.

2.1

Mean Values

2.1.7

27

Robust Arithmetic Mean Values

Robust arithmetic means are obtained when the measured values are truncated, such as the α-truncated mean and the α-winsorized mean; here, the following preferably applies α ¼ 0.05; α ¼ 0.1; α ¼ 0.2. The truncation depends on the number of suspected outliers among the measured values. The arithmetic mean truncated by 10% (α ¼ 0.1) is obtained by shortening the ordered series of measured values on both sides by 10% and calculating the arithmetic mean from the remaining values. For the winsorized mean value, the shortened values are replaced by the adjacent value at the beginning and end of the series after a percentage reduction at the beginning and end of the series of measured values, and the arithmetic mean is calculated from this. Example • 20% truncated arithmetic mean x g0.2 of a series of values arranged in ascending order x1. . .x20

xg 0:2 ¼

16 1 X x 12 i¼5 i

• 10% winsorized arithmetic mean xw0.1 of the measured values arranged in ascending order x1. . .x10 xw 0:1 ¼

x2 þ

9 X

! xi þ x9

i¼2

1 10

The “processing” of primary data, including the exclusion of certain values (truncated means, winsorized means) always results in a loss of information.

2.1.8

Mean Values from Multiple Samples

If samples n1, n2. . . ni where ∑ni ¼ n and their averages x1, x2, . . . xi are present, the overall ̳ mean x (weighted mean) is calculated as follows:

28

2

Characterizing the Sample and Population

• With equal variances si2: ̳

x¼

∑ki¼1 ni xi ∑ki¼1 ni

• With unequal variances: ̳

x¼

2.2

∑ki¼1 ni xi =s2i ∑ki¼1 ni =s2i

Measures of Dispersion

Measured values or observed values are subject to dispersion. Recording the dispersion requires measurements that describe this variation. The most important descriptive variable for dispersion in the technical field is the standard deviation.

2.2.1

Range R

The range or variation width is the simplest value used to describe dispersion. This is the difference between the largest and smallest value in a measurement series. R ¼ xmax  xmin The range is suitable for small series of measured values as a measure of dispersion. In the case of large samples, however, the range provides only limited information about dispersion. The range is used in quality control. The quotient between the largest and smallest value of a sample is the range coefficient K. K ¼ xmax =xmin The range of variation R can be placed in relation to the mean value of the sample x in order to obtain the relative range Z. Z ¼ R=x ¼

Variation width ðrangeÞ Arithmetic mean of the sample

2.2

Measures of Dispersion

29

The limits for the standard deviation s of the sample can be estimated using the variation range R in the following way: R R pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  s  2 2 ð n  1Þ

2.2.2

rffiffiffiffiffiffiffiffiffiffiffi n n1

Variance s2, s2 (Dispersion); Standard Deviation (Dispersion of Random Sample) s, s

Variance and the standard deviation of the sample (s2, s) and basic population (σ 2, σ) are the most important descriptive variables for random deviations from the mean value. The variance—or standard deviation—is suitable for describing single-peak distributions, not for asymmetrical distributions. The variables s2, σ 2, s, and σ are sensitive to outliers. s2 ¼ 0 applies for x1 ¼ x2 ¼ . . . ¼ xn. The variance of a sample is defined as follows: s2 ¼

n 1 X ð x  xÞ 2 n  1 i¼1 i

Or, in another form:  2 ∑ni¼1 x2i  ∑ni¼1 xi s ¼ n ð n  1Þ 2

The positive root of s2 is the standard deviation s, also known as dispersion or mean deviation. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 1 X s¼ ðx  xÞ2 n  1 i¼1 i The standard deviation has the same unit of measurement as the variable to be characterized. The variance among counter values is a measure of the deviations of the individual values around the mean probability p of the sample.

30

2

p ð100  pÞ s2 ¼ n1

p¼

and

Characterizing the Sample and Population

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ð100  pÞ s¼ n1

Number of specific events Total number of events

By contrast to the sample, the variance σ 2 and standard deviation σ of the basic population N (total population) are calculated as follows: σ2 ¼

∑Ni¼1 ðxi  μÞ2 N

For the standard deviation (dispersion), the following applies: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∑Ni¼1 ðxi  μÞ2 σ¼ N The following then applies for counter values: P∙Q σ ¼ N 2

rffiffiffiffiffiffiffiffiffiffi P∙Q and σ ¼ N

Here, p is the basic probability: P¼

Number of events to be characterized Population of events ðN Þ

Q is the basic converse probability: Q¼

N  ðNumber of events to be characterizedÞ Population of events

Q¼1P

2.2

Measures of Dispersion

2.2.3

31

Coefficient of Variation (Coefficient of Variability) v

The coefficient of variation is a measure for comparing the dispersions of samples of different measures. For this purpose, the ratio of the dispersion s to the arithmetic mean value x is formed. v¼

s x

x>0

The coefficient of variation expressed as a percentage is then calculated as follows: v ½% ¼

2.2.4

s ∙ 100 x

Skewness and Excess

If the frequency distribution deviates from the normal distribution (Gaussian distribution), the form of the distribution is described as follows: • By the position and height of the peaks for multimodal frequency distribution. • By the skewness and excess for unimodal distributions. Skewness and excess are the third and fourth central moments of a distribution function (frequency distribution). Here, the moment is the frequency multiplied by the distance from the middle of the distribution, see also Fig. 2.2. The first moment is zero (mean absolute deviation). The second central moment corresponds to the variance.

2.2.4.1 Skewness The sums of the odd power of the difference (xi  x) can be used to describe the asymmetry of a distribution. The skewness γ is described as follows: γ¼

The following also applies:

n   1 X xi  x 3 n i¼1 s

32

2

Characterizing the Sample and Population

Fig. 2.2 Skewness and excess

γ¼

n 1 X 3 z n i¼1 i

where zi ¼

xi  x s

Since the third moment can be positive or negative, a positive or negative skewness is defined; see Fig. 2.2. It must be clarified whether the skewness is due to metrological or mathematical causes. This might be because of an insufficient number of measured values, class division, or the scale of the X-axis. The aforementioned description of the skewness for the sample is not an unbiased estimator of the skewness of the population. To estimate the skewness of the basic population γ G, the following correlation is used: γG ¼

n   X n xi  x 3 s ðn  1Þðn  2Þ i¼1

This corresponds to a correction of the systematic deviation. In sets of statistics, the skewness is also described by the aforementioned variable.

2.2.4.2 Excess, Curvature, and Kurtosis As the fourth moment of the distribution function, excess is defined as follows:

2.3

Dispersion Range and Confidence Interval

η¼

33

∑ni¼1 ðxi  xÞ4 3 n ∙ s4

The following also applies: n   1 X xi  x 4 η¼ 3 n i¼1 s

or

η¼

n 1 X 4 z n i¼1 i

! 3

where zi ¼ ðxi  xÞ=s

This describes the deviation from the normal distribution in such a way that a more pointed or broader distribution exists. In a normal distribution, the following variable has a value of 3: η¼

n   1 X xi  x 4 n i¼1 s

Due to the aforementioned definition of the excess, the normal distribution has an excess of zero. If the distribution is wider and flatter, the excess is less than zero, while the latter is greater than zero if the distribution is narrower and higher than the normal distribution. In sets of statistics, the following variable where zi ¼ ðxi  xÞ=s is used: η¼

n X n ð n þ 1Þ 3 ð n  1Þ 2 z4i  ð n  2Þ ð n  3Þ ðn  1Þðn  2Þðn  3Þ i¼1

For large samples n, the following applies: 3 ð n  1Þ 2 3 ð n  2Þ ð n  3Þ If the skewness and excess of a frequency distribution are substantially (significantly) different from “zero” (i.e., greater than 2), it is to be assumed that the distribution of the basic population differs significantly from the basic population.

2.3

Dispersion Range and Confidence Interval

Dispersion values (measured values) lie within a range defined by the arithmetic mean with a certain confidence interval, the dispersion, and Student’s factor t. This range is called the dispersion range. The confidence interval with a chosen confidence level is the estimated mean value μ of the population.

34

2

2.3.1

Characterizing the Sample and Population

Dispersion Range

For a homogeneous measurement series with the description variables x , s, and n, the dispersion range can be specified as the best possible estimate with the following: x  t∙s  x  x þ t∙s The dispersion range is symmetrical around the following arithmetic mean value: x  t∙s Or with the measure of dispersion fxi: x  f xi Here, fxi ¼ t ∙ s. The factor t is Student’s factor as defined by W. S. Gosset. This is dependent on the degree of freedom f ¼ n  1 (see Fig. 2.3), the confidence interval as a percentage, and on whether a one-sided test or a two-sided test is considered. In Tables A2.1 and A2.2, in Appendix A.2, a selection of t-values is given. Any given value of the basic population falls as follows: • With a 68.3% probability within the range of x  s and with a 31.7% probability outside of x  s • With a 95.5% probability within the range of x  2s and with a 4.5% probability outside of x  2s • With a 99.7% probability within the range of x  3s and with a 0.3% probability outside of x  3s • With a 15.85% probability below x  s and with the same probability (symmetrically) above x þ s In a Gaussian normal distribution with a two-sided confidence interval, the following applies: • 95% of all measured values fall within the range μ + 1.96σ • 99% of all measured values fall within the range μ + 2.58σ • 99.9% of all measured values fall within the range μ + 3.29σ In the case of counter values, the dispersion range is determined as follows: x  fz

where f z ¼ λ  s

2.3

Dispersion Range and Confidence Interval

35

Fig. 2.3 Normal distribution and t-distribution

fz is the measure of dispersion for counter values, while λ is the Gaussian parameter of the normal distribution. The parameter λ also applies to the basic population of the measured values. Table A1.3 in Appendix A.1 provides a selection for parameter x depending on the confidence level. Note on the confidence level: Results with a 95% confidence level are considered likely; those with a 99% confidence level are considered significant, while those with a 99.9% confidence level are considered highly significant.

2.3.2

Confidence Interval (Confidence Range)

With a specified confidence level, the confidence interval denotes the limits (confidence limits) within which the parameter to be estimated (mean value) of the basic population μ defined on the basis of the sample falls. Within the confidence limits, the arithmetic mean value x has a certain confidence. The unknown mean value of the basic population μ is also located in this area with a stated confidence level. μ ¼ x  fx f x Confidence region The following also applies: x  fx < μ < x þ fx The following applies in this case: xu ¼ x  f x is the lower confidence limit of the arithmetical mean

36

2

Characterizing the Sample and Population

xo ¼ x þ f x is the upper confidence limit According to the rule of dispersion transfer (propagation), the following is obtained for f x: ts f x ¼  pffiffiffi n This results in the following: ts μ ¼ x  pffiffiffi n For counter values, the confidence interval is: P¼pf Here, the following applies: f Measure of dispersion as a percentage P Basic probability p Probability of the random sample In addition: p  f < P < p + f. p  f and p + f are the confidence limits of the mean probability. Example: Dispersion range and confidence interval for measured values The following characteristics of steel 38MnVS6 (1.1303) are determined in the incoming inspection of three deliveries A, B, and C, see Table 2.1. The producer provides the following values (reference of the population): Yield strength Re: Tensile strength Rm: Elongation at break A5:

min. 520 MPa 800–950 MPa min. 12%

Calculation of mean values and dispersion The mean value is calculated as follows:

2.3

Dispersion Range and Confidence Interval

37

Table 2.1 Properties of steel 38MnVS6 (1.1303) A B C

Yield strength Re [MPa] 655 648 623

x¼

Tensile strength Rm [MPa] 853 852 867

Elongation A5 [%] 18 18 17

n 1 1 X ð x1 þ x2 þ . . . þ xn Þ ¼ x n n i¼1 i

The dispersion is calculated as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 1 X s¼ ð x  xÞ 2 n  1 i¼1 i This results in the following values for the sample: xRe ¼ 642 MPa xRm ¼ 857:33 MPa

sRe ¼ 16:82 MPa sRm ¼ 8:39 MPa

xA5 ¼ 17:67%

sA5 ¼ 0:58%

Calculation of the dispersion range x t  s With a confidence level of S ¼ 95% (α ¼ 0.05), the following is obtained: Yield strength [MPa]: Tensile strength [MPa]: Elongation at break [%]:

642  49.11 857.33  24.50 17.67  1.69

The t-values are presented in Tables A2.1 and A2.2 in the Appendix A.2; selected values for the example are given in Table 2.2. If the confidence level is increased to S ¼ 99%, the following values are obtained: Yield strength [MPa]: Tensile strength [MPa]: Elongation at break [%]:

642  117.15 857.33  58.44 17.67  4.04

The producer’s specifications are no longer adhered to for the tensile strength. With a confidence level of 99.9%, the following values are obtained: Yield strength [MPa]: Tensile strength [MPa]:

642  375.54 857.33  187.32

38

2

Table 2.2 t-Value as a function of the degree of freedom f and confidence level S

Elongation at break [%]:

Characterizing the Sample and Population

Confidence level S ¼ 95% (α ¼ 0.05) S ¼ 99% (α ¼ 0.01) S ¼ 99.9% (α ¼ 0.001)

t-Value where f ¼ n  1 f¼2 f¼9 2.920 1.833 6.965 2.821 22.327 4.300

17.67  12.95

None of the manufacturer’s specifications are still adhered to. If the sample size were to be based on n ¼ 10, and the mean values and dispersions remained approximately the same, the result would be a confidence level of 99.9%: Yield strength [MPa]: Tensile strength [MPa]: Elongation at break [%]:

642  72.33 857.33  36.08 17.67  2.49

Under these conditions, all producer specifications would be met, even with a confidence level of 99.9%. Confidence interval ts μ ¼ x  pffiffiffi n For the three material deliveries, a confidence level of S ¼ 95% is obtained: • For the yield strength μRe [MPa]: • For the tensile strength μRm [MPa]: • For the elongation at break μA5 [%]:

642  28.36 857.33  14.14 17.67  0.98

In the aforementioned ranges, the mean value of the basic population is to be expected with the stated confidence level.

Literature 1. DIN ISO 3534-1: Statistik – Begriffe und Formelzeichen – Teil 1 (Statistics – Vocabulary and symbols – Part 1: General statistical terms and terms used in probability) 2. DIN ISO 3534-2: Statistik – Begriffe und Formelzeichen – Teil 2: Angewandte Statistik. (Statistics – Vocabulary and symbols – Part 2: Applied statistics)

3

Statistical Measurement Data and Production

Statistical methods are an integral part of quality assurance. Quality management and quality assurance are distinct and extensive fields of knowledge. Therefore, only the relationships between statistical data and production are presented here. With regard to more detailed content, reference is made to the extensive literature on quality management and quality assurance.

3.1

Statistical Data in Production

In engineering production, the product properties required for its function are determined (measured) and monitored. Further properties of the product can also be recorded. The property profile/property matrix (column matrix) is agreed between the consumer of the part or product and the producer. On the part of the consumer, the tolerances of the function-related property or properties are determined by the design. The values described below are used to comply with these variables. The statistical distribution of the characteristics of products (continuous random variable) is recorded in production through division into classes. Arithmetic mean value x x¼

n X

xi =n

i¼1

xi Single measured value n Number of measured values # Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 H. Schiefer, F. Schiefer, Statistics for Engineers, https://doi.org/10.1007/978-3-658-32397-4_3

39

40

3

Statistical Measurement Data and Production

Fig. 3.1 Frequency distribution for classified values

pffiffiffi For class formation with classes of the same class width b, where b ¼ 3:5 s= 3 n, and the class number k, where k  1 + 3.3 log n, and with the frequency hj of the class, the following is given:

x¼

n 1X x h n j¼1 j j

where

hj ¼

nj n

and x j represents the mean values of the classes:

The number of classes k is also determined with the rule of thumb k ¼ 7 < k < 20, whereby the class width is b ¼ R/(k1). See also Fig. 3.1. Range R R ¼ xmax  xmin

xmax Largest measured value xmin Smallest measured value Standard deviation s For single measured values xi:

n 1 X s2 ¼ ðx  xi Þ2 ; n  1 i¼1

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∑ni ðx  xi Þ2 s¼ n1

pffiffiffi n , where

3.2

Machine Capability; Investigating Machine Capability

41

For classified values: 2 n 1 4X 1 s ¼ h  x2  n  1 j¼1 j j n 2

n X

!2 3 hj  xj 5

j¼1

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2ffi u n n u 1 X X 1 s¼t h  x2  hj  xj n  1 j¼1 j j n j¼1 Here, xj represents the class mean values among the classified measured values. The sixfold value of the dispersion (6s) is the production variance T. The range x  3  s contains 99.73% of all individual measured values, while the range x  3:29  s contains 99.9%. With the conventional formulation, the production equipment is unsuitable for maintaining the required tolerance where 6s  T. Where 6s  2=3 T, and therefore 9s ¼ T, production is free of errors. Where 6s  T, production is free of errors, although the production equipment is not used. With more stringent manufacturing requirements in the sense of zero-defect production (Gaussian distribution/normal distribution only begins when 1 approaches zero), the relation between tolerance and production dispersion has been increased.

3.2

Machine Capability; Investigating Machine Capability

The investigation of the machine capability is used to evaluate a machine and/or plant during acceptance (i.e., upon purchasing a machine/system or upon commissioning at the consumer’s facility). An investigation of machine capability is conducted under defined conditions over a short time. The influences of the factors man, material, measuring method, machine temperature, and manufacturing method (5M) on the manufacturing conditions should essentially be constant. In general, the function-determining characteristic value/measured value is determined on approximately 50 parts. This characterizes the short-term capability. The measured values are used to calculate the mean value x and the dispersion s. These variables are used to formulate two characteristic values: Machine controllability cm: cm ¼ Tolerance T=machine dispersion ¼ T=ð6  sÞ Machine capability cmk:

42

3

cmk ¼

Statistical Measurement Data and Production

ðSmallest distance from x to the upper or lower tolerance limitÞ Half of machine dispersion

Here, Zcrit is the smallest distance from x to the upper or lower tolerance limit. cmk ¼ Z crit =ð3  sÞ If the mean value x is in the middle of the tolerance field, then cm  cmk. The cm value describes the ability to manufacture within a specific tolerance field; the cmk value takes into account the position within the tolerance zone. As a guideline, a cm value greater than 1.67 and a cmk value greater than 1.33 applies.

3.3

Process Capability; Process Capability Analysis

Process capability describes the stability and reproducibility of a production process. Here, process stability is a prerequisite for process capability testing. The investigation of process capability (PFU) contains two factors: The preliminary PFU with the characteristic values pp and ppk investigates the process before the start of series production; the lower and upper intervention limits are determined. The long-term PFU evaluates the production process after the start of series production. All practical influences (real process conditions) are taken into account. The characteristic values cp and cpk are determined. Figure 3.2 shows the values for machine capability and process capability within the tolerance zone. The manufacturing process is stable if it is reproducible, quantifiable, and traceable, which means that it is also irrespective of personnel, and can be planned and scheduled. For the long-term PFU, at least 25 samples with three or five individual measurements each (n ¼ 3; n ¼ 5) are examined. The system calculates the following from the samples: • Mean value of the sample mean values: ̳

x¼

x j Mean value of the sample m Number of samples

m 1 X x m j¼1 j

3.3

Process Capability; Process Capability Analysis

43

Fig. 3.2 Machine capability and process capability within the tolerance range

• Estimated value of standard deviation b σ: From

s¼

m 1 X s m j¼1 j

where b σ¼

pffiffiffiffi s2

• Mean range of the sample: R¼

m 1 X R m j¼1 j

Under normal conditions, b σ ¼ 0.4 R in accordance with [1].

44

3

Statistical Measurement Data and Production

These variables are used to calculate the following: • Process controllability (process potential) σÞ cp ¼ Tolerance T=Process dispersion ¼ T=ð6  b

• Process capability cpk ¼

Smallest distance from x to upper or lower tolerance limit, Z crit Half of process dispersion cpk ¼ Z crit =ð3  b σÞ

If the process is centered, the following applies: cpk ¼ cp. Otherwise, cpk < cp. The following also applies: cm  cp. Evaluation of process capability: • • • • •

cp ¼ 1; therefore, tolerance T ¼ 6  b σ . The proportion of reject parts is roughly 0.3% cp ¼ 1.33; T ¼ 8  b σ ; general requirement cm, cp, cpk  1.33 cp ¼ 1.67; T ¼ 10  b σ cp ¼ 2; T ¼ 12  b σ (requirement cp ¼ 2; cpk ¼ 1.67) cp values between 3 and 5 result in very safe processes (“TAGUCHI philosophy”). As cp values increase, so too do the process costs in most cases. The process is as follows:

• • • •

Capable and controlled where cp  1.33 or cpk  1.33 Capable and conditionally controlled where cp  1.33 or 1.00 < cpk 1.33 Capable and not controlled where cp  1.33 and cpk < 1.00 Not capable and not controlled where cp < 1.33 or cpk < 1.00

The limit of process capability where cp ¼ 1.33 means T ¼ 8 σ the limit for cpk ¼ 1.00, Zcrit ¼ 3 σ. If the center position of production is within the tolerance range, 99.73% of all values are in the range of x  3  s , 99.994% of all values are in the range x  4  s , while 99.99994% of all measured values are in the range of x  5  s. The normal distribution does not tend toward “zero” until 1.

3.4

Operating Characteristic Curve; Average Outgoing Quality

3.4

45

Operating Characteristic Curve; Average Outgoing Quality

Operating Characteristic Curve Monitoring the characteristics of a product is important for both the manufacturer and the customer. Here, a definition is made for whether a 100% inspection or a sample inspection is to be conducted. The 100% inspection gives a high degree of certainty that only parts fulfilling the requirements will reach the consumer. This inspection is time-consuming, and there is also not absolute certainty that no faulty parts will reach the consumer. Sampling is the alternative [2, 3]. A sample is used to determine the population and the lot size. This raises the fundamental question of how many missing parts a sample can contain in order for the lot size to be accepted or rejected. In other words: Lot size N with sample n

Number of defective parts is smaller than or equal to c

Number of defective parts is greater than c

Accept delivery of lot size N

Reject delivery of lot size N

This approach implies the notion that the proportion of defects in the sample is the same as in the lot size. However, this is not the case; the defective proportion of the sample is close to the defective proportion of the lot size. As a result, the lot size can be accepted or rejected due to the defect in the sample. Risks, therefore, arise for both the producer and the consumer. The risks taken by both are described by the operating characteristic curve, see Fig. 3.3 and DIN ISO 2859 [2]. This represents the probability of acceptance L( p) for lot size N depending on the percentage of scrap (defective parts) p. The course and shape of the (operating) characteristic curve depends on the sample size n and the number of faulty parts c. Operating characteristic curves are generally calculated using a binomial, hypergeometric, or Poisson distribution. Where n ! 1, a binomial distribution converges with the normal distribution. The hypergeometric distribution and the binomial distribution merge with a large basic population N and a small sample size n (n/N 0.05). The Poisson distribution is the boundary distribution of the binomial distribution for a large n and a small p (n  50, p 0.05). The probability function of the binomial distribution where n/N < 1/10 is the probability of acceptance.  c  X n L ð pÞ ¼ k ¼ 0:1, . . . c  pk  ð1  pÞnk k k¼0 This includes the following: n N

Sample size Basic population (lot size)

46

3

Statistical Measurement Data and Production

Fig. 3.3 Operating characteristic (OC) curve

p q¼1p c

Probability of acceptance (proportion of defects in the population) Converse probability Number of defective parts (acceptance number)

The function L( p) is the acceptance characteristic or operating characteristic (OC); α is the producer risk/supplier risk, and β is the consumer risk.

3.4

Operating Characteristic Curve; Average Outgoing Quality

47

The value p1α is the acceptable quality level (AQL) value; the value pβ is the rejectable quality level (RQL). The larger the sample size is, the steeper the acceptance characteristic will be. The smaller the number of defective parts c with the same sample size n, the steeper the characteristic curve will be. This means that the selectivity of the sampling instruction becomes greater (smaller β-error) as the steepness of the operating characteristic increases. α

β

With this probability, a lot is rejected even though the maximum percentage of defects is adhered to as per the agreement (AQL). The α risk is the supplier risk; α ¼ 1  L (AQL). With this probability, a lot is accepted even though the proportion of defects is exceeded as per the agreement. The β risk is the consumer risk; β ¼ L (LQ).

AQL (acceptance quality limit):

LQ (limiting quality):

Acceptable quality level (DIN ISO 2859 [2]); for the AQL, a probability of acceptance of the lot greater than 90% is generally required. Quality limit to be rejected; the usual requirement is that the acceptance probability for a lot is less than 10% for LQ.

Average Outgoing Quality The average outgoing quality is the average proportion of defective parts in the lot that is subjected to testing but not sifted out (i.e., the proportion that slips through undetected). Defective parts can only slip through if a lot is accepted (risk for the consumer). The average outgoing quality is also known as the AOQ value. The rate to be expected must be calculated using the operating characteristics. The average outgoing quality (as a percentage) is the quotient of the defective parts relative to the lot size (population). The result also depends on how the sample is handled. The possibilities are as follows: • The proportion of the sample is omitted. • The defective parts are sifted out from the proportion of the sample. • The defective parts from the proportion of the sample are replaced with good parts. The solutions for average outgoing quality under the above conditions for the sample and the lot result in a 3 3 matrix [4]. If the defective parts of the sample are sifted out, the average outgoing quality is calculated as follows:

D¼

Nn ∙ LðpÞ ∙ p; N

where n  N,

D  LðpÞ ∙ p follows

48

3

Statistical Measurement Data and Production

This contains the following: n p L( p) N

Sample size Proportion of bad parts in lot Probability that the lot will be accepted Lot size

The average outgoing quality is zero if the lot is free of defects. As the proportion of defects increases, the AOQ increases to a maximum value and then decreases to zero because the probability that lots will be rejected increases with bad lots, see Fig. 3.4. The maximum for the average outgoing quality (i.e., Dmax) is also referred to as the AOQL (average outgoing quality limit) value; the proportion of defects at this maximum is pAOQL. Example: Operating characteristic curves and average outgoing quality Calculation of the operating characteristic curve and calculation of the average outgoing quality using the algorithm according to Günther/TU Clausthal, Institute of Mathematics [5]. Calculation with the binomial distribution; specified values: α ¼ 0.05; β ¼ 0.1

“Good” limit: 1  α ¼ 0.95 p1α ¼ 0.01 “Bad” limit: β ¼ 0.1 pβ ¼ 0.1% Sampling plan: n ¼ 51; c ¼ 2 As a result of the calculation, 51 parts (random sample) are removed from the lot size (N ) and subjected to inspection. The delivery is accepted if a maximum of two parts of the sample do not meet the test conditions. The sampling plan adheres to the “good” and “bad” limits. Figure 3.5 shows the result of the calculation. The average outgoing quality calculated in the example is shown in Fig. 3.6. The conditions for the “good” and “bad” limits correspond to those for the operating characteristic curve (Fig. 3.5). The average outgoing quality is as follows: AOQ ¼ p ∙ LN,n,c ðpÞ The maximum average outgoing quality limit is calculated as follows: AOQL ¼ max ðp ∙ LðpÞÞ  0:027:

3.4

Operating Characteristic Curve; Average Outgoing Quality

Fig. 3.4 Operating characteristic curves and average outgoing quality in accordance with [5]

Fig. 3.5 Sample calculation example for operating characteristic curve

49

50

3

Statistical Measurement Data and Production

Fig. 3.6 Sample calculation for average outgoing quality

Literature 1. Hering, E., et al.: Qualitätsmanagement für Ingenieure. Springer, Berlin (2013) 2. DIN ISO 2859: Annahmestichprobenprüfung anhand der Anzahl fehlerhafter Einheiten oder Fehler (Attributprüfung) (Sampling procedures for inspection by attributes) 3. DIN ISO 3951: Verfahren für die Stichprobenprüfung anhand quantitativer Merkmale (Variablenprüfung) (Sampling procedures for inspection by variables) 4. Thümmel, A.l.: www.thuemmel.co/FBMN-HP/download/QM/Skript.pdf. Accessed 19 Sept 2017 5. Algorithm according to Günther: TU Clausthal. www.mathematik.tu-clausthal.de/interaktiv/ qualitätssicherung/qualitätssicherung. Accessed 19 Sept 2017

4

Error Analysis (Error Calculation)

Measured or observed values without an indication of the error are incomplete. The quality of measured or observed values is described by the errors among those values, whereby a distinction is made between random and systematic errors. Random errors are dispersed, while systematic errors are essentially identifiable.

4.1

Errors in Measured Values

All measurements are subject to errors, even under ideal measurement conditions. Every metrological process is influenced by a multitude of influences. This means that the process of data acquisition is not strictly determined: The readings are dispersed and are characterized by probability. Such errors are random errors and are referred to as “uncertainty.” Furthermore, there are still systematic errors; here, a systematic influence generates a change in the measured value (e.g., through a defined temperature change). Systematic errors, also known as methodological errors or controllable errors, are detected in the form of inaccuracy. The sign of the inaccuracy is fixed by defining the following systematic error: Error ¼ Incorrect  Correct In other words, Error ¼ Actual value  Setpoint. The full specification of a measurement result then includes the following: # Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 H. Schiefer, F. Schiefer, Statistics for Engineers, https://doi.org/10.1007/978-3-658-32397-4_4

51

52

4

Error Analysis (Error Calculation)

y ¼ f ðxi Þ  Inaccuracy  Uncertainty Systematic errors consequently make the measurement result inaccurate, while random errors make it uncertain. Random errors are sometimes referred to as “random deviations” and systematic errors as “systematic deviations.” In the following section, the term “error” is retained.

4.1.1

Systematic Errors

Systematic errors, also known as “methodical” or “controllable” errors, are constant in terms of value and sign under identical measurement and test conditions. The causes of systematic errors are manifold; ultimately, it is always assumptions about the measurement system that are not satisfied in this way. Examples include the following: • Calibration conditions of the measuring standard (e.g., gage block) do not correspond to the test conditions (different temperature, pressure, humidity, etc.). • Errors in the measuring standard due to manufacturing tolerance and wear (systematic error of a gage block). • Behavior of the measured object under the test load. Example If different temperatures between the calibration temperature and the measuring temperature exist during the length measurement, then the systematic error of the measured length Δlz is calculated as follows: 
  Δlz ¼ l αp T p  T 0  αM ðT M  T 0 Þ Here, the following applies: l Tp TM T0 αp αM

Measuring length Temperature of the test specimen Temperature of the measuring standard Calibration temperature Coefficient of expansion of the test specimen Coefficient of expansion of the measuring standard

The detected systematic error is termed inaccuracy. The measured value can be corrected to take account of the systematic error.

4.1

Errors in Measured Values

53

A single measurement yields the error Δxi as follows: Δxi ¼ xis  xi Here, the following applies: xis Incorrect measured value due to systematic error xi Correct measured value Systematic errors can be determined by changing the measuring arrangement or measuring conditions. If systematic errors are not determined, they can be estimated and included in the measurement uncertainty (!) (see Sect. 4.1.2). Systematic errors cannot be detected through repeated single measurements.

4.1.2

Random Errors

Random errors differ in terms of value and sign under identical measurement conditions (i.e., they are dispersed). The deviations of the values are generally statistically distributed around a mean value. Dispersion of the measurement result always occurs, which makes the result uncertain. The causes of random errors are fluctuations in the measuring conditions, in addition to dispersed instrument and load characteristics. A change in the observation approach also generates random errors. The random error detected is the uncertainty uxi: uxi ¼ xiz  xi Here, the following applies: xiz Incorrect measured value due to random error xi Correct measured value The random deviations of the individual measured values from the mean value are characterized by the dispersion or standard deviation. The distribution of probabilities W (xiz) of the random error is described by the Gaussian bell curve (normal distribution). The curve reaches its apex at x (cf. Fig. 2.3).   ðxiz  xÞ2 1 pffiffiffiffiffi exp  W ðxiz Þ ¼ 2σ 2 σ  2π Here, the following applies:

54

4

Error Analysis (Error Calculation)

xiz Measured value σ Standard deviation σ 2 Variance x Most likely value (i.e., x or arithmetic mean) The uncertainty of a measurement result uxi is determined by the confidence interval of the random errors f x and the estimated (technically undetected) systematic errors and uncertainty of the detected systematic errors δx. δx therefore contains an estimated variable. Since there is a low probability that random errors, estimated systematic errors, and as the uncertainty of the detected systematic errors occur with the same sign and with the respective maximum value, they are summarized quadratically: uxi ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 2x þ δ2x

The following is also true: ts f x ¼  pffiffiffi n Here, the following applies: f x Confidence range of the random error δx Estimated systematic errors and uncertainty of the detected systematic errors The uncertainty of the measurement result uxi is the probable maximum value. If the estimated quantities are not available, only the confidence interval of the random error is used as the uncertainty of a measurement result.

4.2

Errors in the Measurement Result

In general, a measurement result y is a function of multiple influencing variables xi. y ¼ f ð x1 , x2 , . . . x i . . . xk Þ Only in exceptional cases is the value recorded in a measurement procedure actually the measurement result. With the inaccuracy Δy and uncertainty uy, the measurement result is described as follows: y ¼ f ðxi Þ  Δy  uy

4.2

Errors in the Measurement Result

4.2.1

55

Errors in the Measurement Result Due to Systematic Errors

If a result y is determined by the influencing variables/measured values xi and these values are incorrect due to Δxi, the result will be as follows: xi ¼ xi þ Δxi Here, the following applies: xi Single measured value xi Arithmetic mean of xi Δxi Systematic error in the single measurement Assuming small Δxi values, the systematic error of the measurement result Δy is calculated through series expansion (Taylor series) and termination after the first term as follows, using an example with two influencing variables: y ¼ f ðx1 þ Δx1 ; x2 þ Δx2 Þ This results in: y ¼ f ðx1 , x2 Þ þ Δx1 f ðx1 , x2 Þ þ Δx2 f ðx1 , x2 Þ where y ¼ f ðx1 , x2 Þ, the systematic error is obtained for Δy ¼ y  y ¼ Δx1 f x1 ðx1 , x2 Þ þ Δx2 f x2 ðx1 , x2 Þ and Δy ¼

∂f ∂f  Δx1 þ  Δx2 ∂x1 ∂x2

For a general scenario (Taylor series), this therefore gives the following: f ðx þ hÞ ¼ f ðxÞ þ Here, R is a remainder and

f 0 ð xÞ 1 f 00 ðxÞ 2 f n ð xÞ n ∙h þ ∙h þ ... þ ∙h þ R 1! 2! n!

56

4

Δy ¼

Error Analysis (Error Calculation)

∂f ∂f ∂f ∂f  Δx1 þ  Δx2 þ . . .  Δxi þ . . .  Δxk ∂x1 ∂x2 ∂xi ∂xk

or Δy ¼

k X ∂f  Δxi ∂x i i¼1

Examples

1. The measured values add up to the measured result: y ¼ x1 þ x2

• Absolute error Δy ¼

∂f ∂f  Δx1 þ  Δx2 ∂x1 ∂x2

because

∂f ¼1 ∂x1

and

∂f ¼1 ∂x2

Δy ¼ Δx1 þ Δx2

• Relative error Δy Δx1 þ Δx2 x1 Δx x2 Δx  1þ  2 ¼ ¼ y y x1 þ x2 x1 x1 þ x2 x2 Δx2 1 Here, Δx x1 and x2 are relative errors among the measured values.

2. The measured values are subtracted to yield the measurement result. • Absolute error y ¼ x1  x2 ;

Δy ¼ Δx1  Δx2

• Relative error Δy Δx1  Δx2 x1 Δx x2 Δx ¼  1  2 ¼ y y x1  x2 x1 x1  x2 x2

4.2

Errors in the Measurement Result

57

3. The measured values are multiplied to yield the measurement result. y ¼ x1  x2

• Absolute error Δy ¼

∂f ∂f  Δx1 þ  Δx2 ; ∂x1 ∂x2

∂f ¼ x2 ; ∂x1

∂f ¼ x1 ∂x2

Δy ¼ x2  Δx1 þ x1  Δx2

• Relative error Δy x2  Δx1 þ x1  Δx2 Δx1 Δx2 ¼ þ ¼ y x1  x2 x1 x2

4. The measured values are divided to yield the measurement result. y¼

x1 x2

• Absolute error Δy ¼

1 x  Δx1  12  Δx2 x2 x2

• Relative error Δy 1 x Δx Δx ¼  Δx1  1 2  Δx2 ¼ 1  2 y y  x2 x1 x2 y  x2

5. The measured value is exponentiated to yield the measurement result. y ¼ xn

58

4

Error Analysis (Error Calculation)

• Absolute error Δy ¼ n  xn1  Δx

• Relative error Δy Δx ¼n y x

6. The square root of the measured value is calculated to yield the measurement result. y¼

p ffiffiffi n x

• Absolute error Δy ¼

1 1n1  x  Δx n

• Relative error Δy 1 Δx ¼  y n x It should be noted that the partial derivatives are generally more complicated functions! If the measurement result y is obtained in a function F( y), i.e.: F ð yÞ ¼ f ð x1 , x2 . . . xi . . . xk Þ The following is given with the exterior derivative: F 0 ð yÞ ¼

∂F ∂y

The following is given with the interior derivative: k X ∂f  Δxi ¼ Δy ∂x i i¼1

As such, the following applies:

4.2

Errors in the Measurement Result

F 0 ðyÞ  Δy ¼

59

∂f ∂f ∂f  Δx1 þ . . .  Δxi þ . . .  Δxk ∂x1 ∂xi ∂xk

Therefore, the inaccuracy Δy is calculated as follows: Δy ¼

k 1 X ∂f  Δxi F ðyÞ i¼1 ∂xi 0

◄

4.2.2

Errors in the Measurement Result Due to Random Errors

The uncertainty of the measured values of the influencing variables uxi is used to calculate the uncertainty of the measurement result uy as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2  2 ffi ∂f ∂f ∂f uy ¼ ∙ ux1 þ . . . þ ∙ uxi þ . . . þ ∙ uxk ∂x1 ∂xi ∂xk Alternatively: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u k  2 uX ∂f t uy ¼  uxi ∂xi i¼1 If the measurement result y in a function F(y) is included, the uncertainty uy is accordingly calculated as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u k  2 1 uX ∂f uy ¼ 0 t  uxi F ðyÞ i¼1 ∂xi The maximum value of uxi is determined from the random errors recorded by the confidence interval f x and the estimated, not calculated, systematic error ϑxi as follows:

60

4

Error Analysis (Error Calculation)

t∙s f x ¼  pffiffiffi n

uxi ¼ j f x j þ jϑxi j;

Alternatively, the uncertainty components can be summarized according to the quadratic error propagation: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uxi ¼ j f x j2 þ jϑxi j2 Since the error component in the case of f x is a dispersion value and ϑxi is an estimated value, a probable value for uxi is obtained. Example Determination of the random error in the calculation of tensile strength Rm of steel from the uncertainty of the measuring equipment. Material :

E295 ðSt 50  2Þ : Rm ¼ 490 N=mm2 ðas per standardÞ:

Random errors identified during diameter measurement using calipers: u∅ ¼ ur ¼ 0:5 ∙ Scale value ¼ 0:05 mm; Diameter ¼ 10 mm Maximum force when tearing the specimen F ¼ 39.191 kN; uncertainty of force measurement uF ¼ 5 N. Rm ðsampleÞ ¼ f ðF, r Þ;

uRm

uRm

Rm ðsampleÞ ¼

F π r2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2 ∂f ∂f ∙ uF þ ∙ ur ¼ ∂F ∂r

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

2 1 2F ¼ ∙ u þ  ∙ u F r π ∙ r2 πr 3 uRm ¼ þ9:98

Rm ðsampleÞ ¼ 499:00

N mm2

N N þ 9:98 mm2 mm2

If a micrometer is used to determine the diameter instead of a caliper gage, the random error is calculated as uRm ¼ 1.00 N/mm2.

4.2

Errors in the Measurement Result

Rm ðsampleÞ ¼ 499:00

61

N N þ 1:00 mm2 mm2

This also demonstrates that optimal selection of the measuring equipment is enabled by error calculation. Example for optimal selection of measuring equipment The errors (uncertainty) in a measuring task should be determined using the error components from the measured variables A, B, and C, thus: uy ¼ f ð uA , uB , uC Þ Taking the values for the individual error components (derivation of the functional dependence multiplied by the uncertainty), e.g., uB  uA > uC, the need for metrological action can be read out at B if the uncertainty of the result error is to be considerably reduced.

4.2.3

Error Limits

Error limits are limit values that are not exceeded. A distinction can be made between calibration error limits and guaranteed error limits.

4.2.3.1 Error Characteristic, Error Limits [1, 2] The relationship between input variable xE and output variable xA of a measuring system is referred to as the characteristic curve, see Fig. 4.1. As an example, Fig. 4.1 shows the linearity error F1 at xE1 and F2 at xE2 . The error characteristic can be derived from the characteristic curve. It is the representation of the error Fi as a function of the output variable xA of the measuring system (shown in tabular form or as a diagram). The relative error is used in addition to the absolute error. The error Fi is related to the input or output variable; therefore, the reference variable must always be specified. The error curve is also called the “linearity error” and is determined through various methods (see Fig. 4.2). These are as follows: • Fixed-point method The start and end of the measuring range are adjusted so that they coincide with the correct value. The straight line through the start and end point is the nominal characteristic; the linearity error is the deviation from the measured characteristic. • Least-squares method The measured characteristic curve is placed relative to the nominal characteristic curve (passing through the zero point) in such a way that the sum of the quadratic deviations is a minimum (Gaussian error-square minimization).

62

4

Error Analysis (Error Calculation)

Fig. 4.1 Characteristic curve of a measuring device

Fig. 4.2 Error curve with various methods of adjustment

• Tolerance-band method The measured characteristic curve is positioned relative to the nominal characteristic curve in such a way that the sum of the deviation squares is a minimum, ∑(Fi)2 ! Min. The tolerance-band method yields the smallest error, but should not be measured in the lower range (approx. 20% of the measuring range), because the measured characteristic curve does not pass through the zero point of the nominal curve. For calibrated instruments, error limits are referred to as calibration error limits. These are the limit values for the errors prescribed by the calibration authorities and must not be exceeded. The guaranteed error limits of a measuring instrument are the error limits guaranteed by the manufacturer of the instrument (that are not exceeded). Measuring instruments with multiple measuring ranges can have various error limits. Calibrated measuring instruments are often assigned class designations (accuracy classes).

4.2

Errors in the Measurement Result

63

Examples • Accuracy classes for electrical measuring instruments (VDE 0410) Error classes in [%]: 0.1; 0.2; 0.5; 1; 1.5; 2.5; 5 • Gage blocks for dimension metrology Accuracy grades (ISO/TC 3/SC 3) 1–4 • Guaranteed error limits for measuring instruments For example, 1.5% of measuring range for analog devices or 0.1% of measuring value for digital devices +2 digits, 1 digit. For digital devices, the error therefore consists of an error dependent on the display value and a constant digitization error. • Reading error: Error ¼ 0.5 ∙ Scale unit (also scale graduation value). Example: With an accuracy class of 1.5% in a 30 V measuring range, 1.5% of 30 V equals 0.45 V.

4.2.3.2 Result Error Limits Error limits in practical measurement technology are the agreed or guaranteed, permissible extreme deviations upward or downward from the target display value or from the nominal value. Error limits can be one-sided or two-sided. Maximum result error limits If the errors Δxi among the measured values are unknown, but the error limits Gi are known, the maximum error limit of the result is calculated by adding up the values of the individual error limits. The values are added up because the errors among the measured values can be both positive and negative within the error limits. The maximum result error limits are the maximum (safe) limits of the result error. The absolute maximum error limit Gym of the result y ¼ f(xi) with G1 is calculated as follows: Gym

         ∂y   ∂y   ∂y   ∂y         ¼  ∙ G1  þ  ∙ G2  þ . . . þ  ∙ Gi  þ . . . þ  ∙ Gk  ∂x1 ∂x2 ∂xi ∂xk

The relative maximum error limit of the result

Gym y

is:

         ∂y G1   ∂y G2   ∂y Gi   ∂y Gk  Gym        ∙ ∙ ∙ ∙  ¼  þ þ ... þ  þ ... þ  y ∂x1 y  ∂x2 y  ∂xi y  ∂xk y 

64

4

Error Analysis (Error Calculation)

The following also applies:          ∂y x1 G1   ∂y x2 G2   ∂y xi Gi   ∂y xk Gk  Gym        þ þ ... þ  þ ... þ  ¼  ∙ ∙ ∙ ∙ ∙ ∙ ∙  y ∂x1 y x1  ∂x2 y x2  ∂xi y xi  ∂xk y xk  Here, Gxii denotes the relative error limits of the measured variables. Examples • y ¼ x1 + x2 and y ¼ x1  x2 Gym ¼ ðjG1 j þ jG2 jÞ • y ¼ x1 ∙ x2 and y ¼ xx12     G  G  Gym ¼   1  þ  2  y x1 x2 In the above example, the absolute error limits add up when the measured values are added or subtracted. With multiplication and division, the relative error limits add up. Statistical result error limits Since it is unlikely that the errors Δxi among all measured values are only at the positive or negative error limits, it is equally unlikely that these safe result error limits will be used. Therefore, the statistical (probable) result error limit is determined through quadratic addition of the single error limits. The absolute statistical result error limit Gys is calculated as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2  2  2ffi ∂y ∂y ∂y ∂y Gys ¼  ∙ G1 þ ∙ G2 þ . . . þ ∙ Gi þ . . . þ ∙ Gk ∂x1 ∂x2 ∂xi ∂xk And for the relative statistical result error limit

Gys y :

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 2 2 u ∂y G1 2 Gys ∂y G2 ∂y Gi ∂y Gk t ∙ ∙ ∙ ∙ ¼ þ þ ... þ þ ... þ y ∂x1 y ∂x2 y ∂xi y ∂xk y

4.2

Errors in the Measurement Result

65

Alternatively: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 2 u ∂y x1 G1 2 Gys ∂y xi Gi ∂y xk Gk ∙ ∙ ∙ ∙ ∙ ¼ t ∙ þ ... þ þ ... þ y ∂x1 y x1 ∂xi y xi ∂xk y xk

Examples • y ¼ x1 + x2 or y ¼ x1  x2 Gys ¼ 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G21 þ G22

• y ¼ x1 ∙ x2 or y ¼ xx12 Gys ¼ y

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  2ffi G1 G2 þ x1 x2

In the above examples, the statistical result error limits are calculated by the root of the squared error limits when adding and subtracting the measured values. When multiplying and dividing the measured values, the relative error limits are determined by the root of the squares of the relative error limits. Example The absolute maximum error limit and the absolute statistical error limit of the total conductance is to be calculated for four parallel resistors. R1 ¼ R2 ¼ 120 Ω R3 ¼ R4 ¼ 150 Ω The color code for the resistance tolerance means that a golden band is present on all resistors (denoting 5%). Absolute maximum error limit of the total conductance GRm: 1 1 1 1 1 ¼ þ þ þ ; RG R1 R2 R3 R4

1 2 2 ¼ þ RG R1 R3

66

4

Error Analysis (Error Calculation)

     ∂G   ∂G  1 2 ∙ R3 þ 2 ∙ R1    G¼ ¼ ; GRm ¼   ∙ G1  þ  ∙ G3  RG R1 ∙ R3 ∂R1 ∂R3

GRm

     2   2     ¼   2 ∙ G1  þ  2 ∙ G3  ¼ 0:0015 Ω1 R R 1

3

Absolute statistical result error limit for the total conductance:

GRs

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 2 u ∂G ∂G ¼ t ∙ G1 þ ∙ G3 ∂R1 ∂R3 GRs ¼ 0:00107 Ω1

4.2.3.3 Errors and Error Limits of Electrodes In a measuring chain, each individual measured value is converted into several measuring links connected in series. The output signal of the first measuring element is the input signal for the second measuring element, and so on. If the ratio of the output signal of a measuring element to the input signal is determined as the transmission factor K, then the following applies: y Output value of the electrode x Input signal of the electrode y ¼ x ∙ f ðK 1 , K 2 , . . . , K i Þ For a measuring chain with linear (!) links, the following applies: y ¼ x ∙ K1 ∙ K2 ∙ . . . ∙ Ki In the case of nonlinear relationships between the input and output signal of the measuring element, the respective derivative must be used (see Fig. 4.3). The relative error of a combination electrode is then determined to be as follows: Δy ΔK 1 ΔK 2 ΔK i ¼ þ þ ... þ y K1 K2 Ki

4.2

Errors in the Measurement Result

67

Fig. 4.3 Linear and nonlinear relationship between input and output signal

The relative maximum error limit with the relative error limits of the measuring G elements where KK11 results in the following:       GK 1  GK 2  GK  Gym     þ þ . . . þ  i  ¼    y K1 K2 Ki Finally, the relative statistical error limit is calculated as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 2 u GK 2 Gys GK i GK 2 1 t ¼ þ þ ... þ y K1 K2 Ki

Example: Thermoelectric temperature measurement Input signal: x ¼ Temperature Thermocouple

Thermocouple transmission factor

Voltage K 1 ¼ Temperature

Transducer (measuring amplifier)

Transmission factor

Current K 2 ¼ Voltage

Moving-coil device (indicator)

Transmission factor of the display unit

angle K 3 ¼ Deflection Current

68

4

Error Analysis (Error Calculation)

Output signal: y ¼ Deflection angle Electrode: y ¼ x ∙ K1 ∙ K2 ∙ K3

y ¼ Temperature ∙

Voltage Current Deflection angle ∙ ∙ Temperature Voltage Current

Literature 1. VDI/VDE 2620: Unsichere Messungen und ihre Wirkung auf das Messergebnis (Dokument zurückgezogen) (Propagation of error limits in measurements; examples on the propagation of errors and error limits (withdrawn)) 2. DIN 1319: Grundlagen der Messtechnik (Fundamentals of metrology) (1995–2005)

5

Statistical Tests

Statistical tests enable comparisons across a very wide range of applications. The results are quantitative statements with a given confidence level. Examples of statistical tests in technical fields include comparisons of material deliveries in incoming inspections, machine comparisons with regard to production quality, aging tests, and dynamic damage. The statistical tests discussed here are parameter-bound tests. These include the t-test, the F-test, and the chi-squared test. These tests use different parameters to distribute a size or characteristic; therefore, the quality of the statement is also different. The t-test uses the arithmetic mean and dispersion, while the F-test uses dispersion and the chi-squared test the frequency of a characteristic.

5.1

Parameter-Bound Statistical Tests

Parameter-bound or parametric tests refer to the characterizing parameters of the distribution and the normal distribution, the mean value, the dispersion, and the frequency. In contrast, nonparametric or distribution-free tests use other parameters [1]. The test statistics for distribution-free tests do not depend on the distribution of the sample variables and thus not on the normal distribution (Gaussian distribution), for example. The following variables are among those used: median, quantiles, quartiles, and rank correlation coefficient. Distribution-free tests are weaker than parametric tests. The following explanations refer exclusively to parameter-bound tests. Normal distribution/t-distribution is usually guaranteed or approximately fulfilled for technical and scientific results, measurements, or observations. The value distribution should nevertheless be mapped in order to detect whether other distributions are present (e.g., skewed distribution; see also Sect. 2.2.4). # Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 H. Schiefer, F. Schiefer, Statistics for Engineers, https://doi.org/10.1007/978-3-658-32397-4_5

69

70

5

Statistical Tests

Table 5.1 Examples of transformation Application Distribution skewed to the right

Transformation yx ¼ y3 yx ¼ y2

No transformation where yx ¼ y1 Distribution skewed to the left

yx ¼ ln y 1 yx ¼ y0:5 x y ¼ y1 yx ¼ y12

Retransformation 1

y ¼ ðyx Þ3 1

y ¼ ðyx Þ2 y ¼ e yx y ¼ (yx)2 y ¼ (yx)1 y ¼ ðyx Þ2 1

If the distribution is skewed, the data must be transformed. For example, with a distribution skewed to the right (i.e., the maximum falls to the right of the center of the frequency distribution), this would involve a cubic transformation, while distribution by a root function would be used for a distribution skewed to the left. Therefore, the relative cumulative frequency lies along a straight line (normal distribution). After evaluation with the transformed data, the transformation is reversed (see Table 5.1). The statistical test is performed on the basis of a sample from the population, whereby the sample is taken at random for this purpose. The method for this might be assignment using random numbers, for example. Normal distribution, also called Gaussian distribution, has the shape of a bell curve. The function (density function) has its maximum at μ and does not tend toward 0 until
1 (e-function). 1 xμ 2 1 f ðx, μ, σ Þ ¼ pffiffiffiffiffi ∙ e2 ð σ Þ σ 2π

Here, the following applies: σ Standard deviation (σ 2 variance) μ Expected value For μ ¼ 0 and σ 2 ¼ 1, the standard normal distribution will be retained. The density function of the standard normal distribution φ(x) is as follows: 1 2 1 φðxÞ ¼ pffiffiffiffiffi ∙ e2 x 2π

The distribution function of standard normal distribution Φ(x) is the Gaussian integral. See Fig. 5.1 in this case.

5.1

Parameter-Bound Statistical Tests

71

Fig. 5.1 Distribution function of the standard normal distribution

Within the bounds of 1 to x, the following applies: 1 ΦðxÞ ¼ pffiffiffiffiffi 2π

Z

x

e2 t dt 1 2

1

The integral from 1 to +1 results in a value of exactly one with a prefactor of 1=pffiffiffiffi 2π. ¼ 0:3989, and the inflection Where x ¼ 0, the function φ(x) has the maximum value 1=pffiffiffiffi 2π points of the functions are at x ¼ 1 and x ¼ 1 (second derivative equal to 0). Below are several values for the standard normal distribution: • 68.27% of all values are in the range μ
σ (0.6827); therefore, approximately 32% are outside this range. • 95.45% of all values fall within the range μ
2 ∙σ • 99.73% of all values fall within the range μ
3 ∙σ • 99.994% of all values fall within the range μ
4 ∙σ • 99.9999% of all values fall within the range μ
5 ∙σ • 99.999999% of all values fall within the range μ
6 ∙σ See also Chapter 3 “Statistical Measurement Data and Production.” Tables on standard normal distribution are provided in Tables A1.1 and A1.2, Appendix A.1.

72

5.2

5

Statistical Tests

Hypotheses for Statistical Tests

Performing a statistical test involves establishing hypotheses that are answered with a chosen confidence level. Statistical tests serve to verify assumptions about the population using the results (parameters) of the sample. This test includes two hypotheses: the null hypothesis H0 and the alternative hypothesis H1. The alternative hypothesis contradicts the null hypothesis. Only one of the two hypotheses (H0 or H1) is valid. The null hypothesis is always tested. The more serious error is chosen as the first-type error, or α-error, which is the decision in favor of H1 where H0 actually applies. The probability of a first-type error is lower than the significance level (the probability of error) α. The β-error (second-type error) is the decision in favor of H0 where H1 actually applies. Table 5.2 shows the decision-making options. Determining the level of significance α also defines the probability of a second-type error (β-error). A reduction in α causes an increase in β. The α-error should be kept as small as possible so that the correct hypothesis is not rejected. If the distance between the parameter values of the sample (e.g., x) and the true value of the basic population, the result will be a small β-value (second-type error). However, if the distance between the parameter values becomes ever smaller, the β-value will increase; see Fig. 5.2. Both error probabilities cannot be reduced at the same time. Since the sample size reduces the dispersion, a larger sample will result in a more significant result. The test becomes more sensitive as the sample size increases. When testing null and alternative hypotheses, a distinction must be made between one-sided and two-sided tests; see also Fig. 5.3. In the case of two-sided tests, the null hypothesis is a point hypothesis (i.e., it refers to a permissible value), for example, in the case of the t-test for the mean value of the population μ. The null hypothesis is then H0: x ¼ μ The alternative hypothesis is H1: x 6¼ μ The one-sided test [i.e., the consideration of the distribution function from the righthand side (right-sided test) or from the left-hand side (left-sided test)] has the following hypotheses in the above example:

Right-sided test Left-sided test

Null hypothesis H0 : x  μ H0 : x  μ

Alternative hypothesis H1 : x > μ H1 : x < μ

Whether a one- or two-sided test is performed depends on the problem at hand. One-sided tests are more significant than two-sided tests: A one-sided test more frequently reveals the incorrectness of the hypothesis to be tested.

5.2

Hypotheses for Statistical Tests

73

Table 5.2 Errors in decision options for null and alternative hypothesis

Decision in favor of H0

Decision in favor of H1; H0 is rejected

Null hypothesis H0 also applies State of basic population ≙ H0 Correct decision (1  α)-error

Incorrect decision α-error (First-type error) Level of significance α

Alternative hypothesis H1 also applies State of basic population ≙ H1 Incorrect decision β-error (second-type error) Level of significance β Correct decision (1  β) error

Fig. 5.2 Hypothesis of the sample; α- and β-error

Fig. 5.3 One-sided and two-sided test

In a statistical test, decisions are made on the basis of the sample with a certain probability: that is to say, even if a hypothesis is assumed to be preferable to other hypotheses with the given probability.

74

5

Statistical Tests

The α-error (or first-type error) is referred to as the probability of error, statistical uncertainty, risk measure, or safety threshold. The probability of error α and confidence level S add up to a value of 1. The following applies: S þ α ¼ 1 or alternatively S þ α≙100% For a normal distribution, S + α corresponds to the area below the curve of the Gaussian distribution. The α-values are much smaller than a value of 1. In general, the following values are agreed, for example: 0.05; 0.01; 0.001. For the β-values, for example, 0.1 is specified. General procedure for statistical tests: 1. 2. 3. 4. 5. 6.

Problem formulation; what is the serious error? Definition of null hypothesis and alternative hypothesis One-sided or two-sided test Determination of the α- and the β-error Calculation of the test size from the sample Comparison of the test size with the table value; acceptance or rejection of the null hypothesis 7. Test decision and statement on the problem With the one-sided test, the table value zr is read out with the significance level (probability of error) taken from the table of the test statistic (t-, F-, and χ 2-function for the parameter-bound tests). On the other hand, the check function is used to determine the pffiffiffi pffiffiffi check value zp. For the t-test, the value is thus: t p ¼ xμ n. s2

If the value is zp  zr, the sample belongs to the basic population ðx ¼ μÞ. Where zp > zr, ðx 6¼ μÞ applies (alternative hypothesis). Since the significance level of the two-sided test lies on both sides of the distribution, it is prudent to choose αl ¼ αr ¼ α/2. The following applies for the two one-sided tests with the probability of error a, and for the left-sided and the right-sided test: • Where zp  zr, the sample belongs to the basic population; alternatively, where zp > zr in the example ðx > μÞ • Where zp  zr, then the sample also belongs to the basic population; alternatively, where zp < zl in the example ðx < μÞ

5.3

5.3

t-Test

75

t-Test

The t-test or “Student’s test” according to W. S. Gosset (Student is the pseudonym of W. S. Gosset) is a statistical method for testing hypotheses (significance test). The t-test uses the mean and the dispersion. A check is conducted for whether the mean values of two samples, or the mean value of one sample and a standard value/reference value (population), differ significantly or whether the difference is random. The t-test checks whether the statistical hypothesis is correct. To apply the t-test, the following requirements must be met: • The sample is taken randomly. • The samples are independent of each other. • The variable to be examined (the characteristic) is interval-scaled (evenly divided unit along a scale; parametric statistics). • The characteristic is normally distributed. • The variances of the characteristics to be compared are the same (variance homogeneity). Note: If variance heterogeneity is present, the degrees of freedom must be adapted to the test distribution. For this purpose, it is necessary to check whether the variances of the sample are actually different. The test used for this is Levene’s test or the modification of this test according to Brown–Forsythe, an extension of the F-tests; the authors refer the reader to the listed literature sources in this case. For the variables used in the relationships, see also Chap. 2. The t-test uses the statistical characteristic values “arithmetic mean” and “dispersion”: • Difference in mean values (sample characteristic value) x1  x2 • Standard error of the mean-value differences s 1 pffiffiffi ¼ pffiffiffi ∙ n n

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∑ni¼1 ðxi  xÞ2 n1

The standard deviation of the sample and the population differ. With a small sample size, the correcting influence is more pronounced.

76

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Statistical Tests

Standard deviation of the basic population: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∑ni¼1 ðxi  xÞ2 σ¼ N Standard deviation of the sample: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∑ni¼1 ðxi  xÞ2 s¼ n1 For large samples where n > 30, the t-distribution converges with the standard normal distribution. This causes the t-test to become a Gauss test or z-test. Gauss’s test uses the standard normal distribution (population) and is therefore not suitable for small samples with an unknown dispersion. See Fig. 5.4 in this case. The dispersion of the t-distribution is greater than a value of 1. Where n ! 1 (1 ≙ population), the standard normal distribution has a value of 1. For the t-test, a distinction is made between a one-sample t-test and a two-sample t-test, whereby the one-sample t-test compares a sample with a standard value or reference value. Example Comparison of the tensile strength of a sample with the tensile-strength reference value defined in the applicable norm. The t-value, test value tp is calculated as follows for this test:

Fig. 5.4 Standard normal distribution and t-distribution

5.3

t-Test

77

tp ¼

x  μ pffiffiffi ∙ n s

x , s, and n are the values of the sample. μ is the value used as a reference for testing (normative value). For the two-sample t-test with independent samples (unpaired t-test), the following applies: j x1  x2 j j x  x2 j ffiffiffiffiffiffiffiffiffiffiffiffiffiffi or t p ¼ 1 tp ¼ q ∙ sd 1 1 sd n1 þ n2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 ∙ n2 n1 þ n2

Here, the following also applies: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn1  1Þs21 þ ðn2  1Þ s22 sd ¼ n1 þ n2  2 n1 + n2  2 is the degree of freedom f. The value of jx1  x2 j ensures that no negative t-values occur. When the t-values for the samples (test values) are present, it must also be determined whether the null hypothesis or alternative hypothesis is applicable (see Sect. 5.2). Two-sided test (point hypothesis): Null hypothesis H0: ðx ¼ μÞ Alternative hypothesis H1: ðx 6¼ μÞ One-sided test: • Left-sided test (a is on the left-hand side of the distribution) H0: ðx  μÞ H1: ðx < μÞ • Right-sided test (a is on the right-hand side) H0: ðx  μÞ H1: ðx > μÞ For the t-test, the empirical mean-value difference is significant if the value of the empirical t-value (test value) is greater than the critical t-value (table value), which means that the null hypothesis is rejected. If a test is performed to determine whether the mean values x1 , x2 from two samples differ, the tp-value for the two-sample t-test is to be calculated. This value is compared to the table value of the t-distribution.

78

5

Statistical Tests

The following generally applies: • If the tp-value is smaller than the table value with a confidence level of 95%, the mean values x1 and x2 do not differ. • If the tp-value  t-table value at 95%, the mean values x1 and x2 are probably different. • If the confidence level is increased to 99% and tp  t-table value, then x1 and x2 are significantly different. • If the confidence level is further increased to 99.9% and tp  t-value in the table, then x1 and x2 are highly significant. Instead of the comparison of the critical t-value (table value) with the empirical t-value (test value), the α-level can also be compared with the p-value. The p-value is a probability between 0 and 1. The smaller the p-value, the less likely the null hypothesis is. Selected tables for the t-test are given in the Appendix Tables A2.1 and A2.2. Example: Thermooxidative damage on polypropylene [2] In order to test the influence of thermal oxidation on the property profile of polypropylene films, film strips were tested after thermooxidative stress. This stress on the films lasted up to 60 days at 150  C. Using the example of determining the elastic modulus in a tensile test (based on DIN EN ISO 527 [3]), Fig. 5.5 shows the chronological progression of the change. The module initially increases due to tempering as a result of post-crystallization. This is followed by a reduction in the modulus due to the accumulation of stabilizers and low-molecular components in the amorphous phase, because this is situated in the high-elasticity range. Testing in the form of a t-test is performed to determine whether the changes are statistically certain (95% and 99% confidence level; one-sided test). Calculation example Reference 30 days’ thermooxidative sample stress tested on tempered sample. Measured values: • Elastic modulus (tempered) [MPa]: x1 ¼2506 s1¼ 62 n1 ¼ 10 • Elastic modulus (after 30 days of thermooxidative stress) [MPa]: x2 ¼2435 s2¼ 95 n2 ¼ 10

5.3

t-Test

79

Fig. 5.5 Dependence of the elastic modulus on the storage period

The test value is calculated as follows: tp ¼

j x1  x2 j qffiffiffiffiffiffiffiffiffiffiffiffiffiffi or sd n11 þ n12

tp ¼

j x1  x2 j ∙ sd

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 ∙ n2 n1 þ n2

The following is obtained for the dispersion sd from the two samples: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn1  1Þ s21 þ ðn2  1Þ s22 9  622 þ 9  952 sd ¼ ¼ ¼ 80:21 n1 þ n2  2 10 þ 10  2 The following is obtained for the check value tp: j2506  2435j tp ¼ ∙ 80:21

rffiffiffiffiffiffiffiffiffiffiffiffiffi 10 ∙ 10 20

Therefore, tp ¼ 1.979 According to the table in the Appendix A.2, the critical t-value for f ¼ 18 is as follows: • • • •

With a one-sided confidence interval and 95% confidence level: tcrit ¼ 1.734 With a one-sided confidence interval and 99% confidence level: tcrit ¼ 2.552 With a two-sided confidence interval and 95% confidence level: tcrit ¼ 2.101 With a two-sided confidence interval and 99% confidence level: tcrit ¼ 2.878

80

5

Statistical Tests

Table 5.3 Statistical investigation of the elastic modulus Elastic modulus in relation to the tempered sample Untempered to tempered 10 days after tempering to tempered 20 days after tempering to tempered 30 days after tempering to tempered 40 days after tempering to tempered 50 days after tempering to tempered 60 days after tempering to tempered

Significant change, two-sided confidence range 95% 99% Yes Yes Yes Yes Yes Yes No No No No Yes Yes No No

tp-Value 20.67 9.50 8.87 1.98 0.31 12.08 0.07

Therefore, with a 95% confidence level and a one-sided confidence interval, the mean values probably differ. With a one-sided confidence interval and 99% confidence level, there is no significant change in the elastic modulus. The same applies to the two-sided test with a 95% and 99% confidence level. The test results are listed in Table 5.3. It can be seen—as can also be confirmed in Fig. 5.5—that different statements on significance exist for both a 95% and 99% confidence level. Overall, this demonstrates the complexity of the investigation.

5.4

F-Test

The F-Test according to Fisher, R. A. is a statistical test for checking whether the dispersions (variances) of two samples are identical or different. The prerequisite for applying F-tests is that normally distributed populations N are present and that the populations must be independent (prerequisite for all parameter tests). The F-test only uses the dispersions and sample sizes of the two samples to be checked. F¼

σ 21 σ 22

If both dispersions are the same, then F ¼ 1. The F-values for verifying the hypotheses are derived from the F-distribution, also known as the Fisher–Snedecor distribution. The F-distribution results from all possible dispersion ratios between the sample distributions in the two populations. The F-distribution has the following as special

5.4

F-Test

81

Fig. 5.6 Distribution function of the F-distribution with the degrees of freedom (20,20) and (5,10)

cases: normal distribution F(1, 1); t-distribution F(1, n2), and chi-squared distribution F(n1, 1). Figure 5.6 shows that the distribution function is asymmetrical. Furthermore, it is dependent on the degrees of freedom of the two samples, namely: f 1 ¼ n1  1 and f 2 ¼ n2  1 In the table of the F-values, the degrees of freedom f1 are the degrees of freedom for the numerators, while f2 refers to the degrees of freedom for the denominators. Selected table values for the F-distribution are given in Tables A3.1, A3.2, and A3.3 of the Appendix A.3. Values between the degrees of freedom given in the tables can be obtained through harmonic interpolation, whereby the confidence level remains the same. The calculation is carried out for the required F-value for f1 and f2 with the table values for f2 and the values f11 and f1+1, between which the value f1 lies: ( f11 < f1 < f1+1). Ff 11 , f 2  Ff 1 , f 2 ¼ Ff 11 , f 2  Ff 1þ1 , f 2

1 f 11 1

f 11





1 f1 1

f 1þ1

The F-test for the samples is determined as the quotient of the estimated variances s21 and s22 (larger variance to smaller variance). F samples ¼

s21 s22

where s21 > s22

82

5

Statistical Tests

The test distinguishes between one-sided and two-sided tests, whereby the one-sided test has the greater test power. There are three possibilities: One-sided test a) Hypotheses H0: σ 21  σ 22 H1: σ 21 < σ 22 Test statistic: F ¼ s21=s22 H0 is rejected if F > FTab (for the statistical uncertainty α) H0 is accepted if F < FTab (for α) b) Hypotheses H0: σ 21  σ 22 H1: σ 21 > σ 22 Test statistic: F ¼ s21=s22 H0 is rejected if F > FTab (for α). Two-sided test Hypotheses H0: σ 21 ¼ σ 22 H1: σ 21 6¼ σ 22 Test variable F ¼ s21=s22 where s21 > s22 has been chosen. H0 is rejected if F > FTab (for α=2). The decision as to whether the test is one-sided or two-sided does not depend on the asymmetry and one-sidedness of the F-function. The following can be determined concerning the influence of the confidence level S or the statistical uncertainty α (probability of error): • If F is smaller than the table value of F at 95%, the tested dispersions s1 and s2 do not differ. • If F is greater than or equal to the table value of F at 95%, the dispersions s1 and s2 are probably different. • If F with a confidence level of 99% is greater than the table value, the dispersions s1 and s2 are significantly different. • If F where S ¼ 99.9% is greater than the table value, the dispersions s1 and s2 are highly significant.

The Chi-Squared Test (w2-Test)

5.5

83

Example for the F-test: Adherence to target dispersion The distribution of weights is determined to monitor the quality of injection-molded parts. The agreed target dispersions svb is svb ¼ 0.024; fvb in this case is fvb ¼ 1 (≙ f2). Using a sample size comprising np ¼ 101, the following dispersion is determined: sp ¼ 0.0253 where fp ¼ np  1 ¼ 100 (≙ f1). Calculation of F-value where F  1: F¼

s2p s2vb

¼

0:02532 ¼ 1:11 0:0242

The comparison with the table value where S ¼ 99% results in FTab ¼ 1.36. Since F < FTab, the agreed target dispersion is adhered to.

5.5

The Chi-Squared Test (x2-Test)

The chi-squared test is used to investigate frequencies and is a simple statistical test.

5.5.1

Conditions for the Chi-Squared Test

The following conditions must be met: • Normal distribution; application of the data is sensible. • Random sampling. • The random variables are independent of each other and subject to the same normal distribution. • Application of the chi-squared test also requires the following: – Sample size not too small: n > 30. – For division into frequency classes (cell frequency): n  10 or at least 80% of all classes where n > 5. – For the four-field test: The expected value of the four fields (expected value ¼ (Row total  Column total)/Total number) must be at least 5. With an expected value of < 5, Fisher’s test is to be applied. In addition to using this test for dichotomous characteristics (binary characteristic; only two expressions possible, for example, smaller or larger than a certain size), ordinal- and interval-scaled values (variables) can also be used to investigate the frequencies of these characteristics.

84

5

Statistical Tests

Note: The interval scale is a metric scale; for an ordinal scale, the ordinal variables can be sequentially ordered. All chi-squared methods are comparisons of observed and expected frequencies. In other words, they always concern the following variable: ðDetermined frequency  Expected frequencyÞ2 Expected frequency The chi-squared value is then calculated as the sum across all of these values. χ2 ¼

X ðDetermined frequency  Expected frequencyÞ2 Expected frequency χ2 ¼

X ðhif  hie Þ2 hie

hif is the determined frequency; hie is the expected frequency Remark: R. Helmert investigated the square sums of normally distributed variables. The resulting distribution function was then named “K. Pearson’s chi-squared distribution.” The chi-squared test is carried out as an independence test (testing two characteristics for stochastic independence) and as an adaptation test (also a distribution test; testing data for distribution). The chi-squared distribution function cannot be written in elementary form, rather by means of the gamma function. Here, the authors refer the reader to the listed literature sources. The chi-squared distribution is asymmetrical: It starts at the coordinate origin, see Fig. 5.7. The distribution is continuous, and the parameter f is the degree of freedom. Table values for the chi-squared distribution are provided in Table A4.1 of the Appendix A.4. With a confidence level of 95%, the χ 2 value ( f ¼ 3) is determined as 7.81; the c2-value ¼ 0.3518. If the confidence level is increased to 99%, then χ 2 ¼ 11.34 and c2 ¼ 0.1148. The density function is a falling curve for the degree of freedom f ¼ 1 and f ¼ 2. Where f is greater than 2, this is a skewed bell curve (to the right-hand side).

5.5

The Chi-Squared Test (w2-Test)

85

Fig. 5.7 Chi-squared distribution, f ¼ 3, f(x) density function

The asymmetry of the chi-squared curve depends on the degree of freedom. The chi-squared curve becomes more symmetrical with an increasing degree of freedom and is thus also more similar to the normal distribution.

5.5.2

Chi-Squared Fit/Distribution Test

The chi-squared distribution test, also known as the fit test, concerns a statistical characteristic. A check is conducted into whether the distribution of a sample corresponds to an assumed or predefined distribution (e.g., distribution of the population). The test can be used for categorical characteristics (limited number of characteristics/ categories) and for continuous characteristics that have been previously classified. The chi-squared fit test squares the difference between an observed (sample) and expected (for example, population) frequency and uses the sum of the resulting density function to accept or reject the null hypothesis, therefore with frequency hi of the sample and frequency hE for the given (expected) distribution normalized with hE. The following therefore applies to the chi-squared test statistic: χ 2p ¼

n X ðhi  hE Þ2 hE i¼1

n is the number of values/classified characteristics. The null hypothesis H0 (i.e., the equality of the frequency distributions) is rejected if the following is true: χ 2p > χ 21α,ðk1Þ ðtable valueÞ

• Degree of freedom f ¼ k1 • Confidence level 1  α

86

5

Statistical Tests

• Level of significance α If the distribution of the sample and that of the specified distribution match, the following χ 2p ¼ 0, since ðhi  he Þ2  0. The clearer the differences between the distributions to be compared are, the greater the 2 χ p will be; therefore, it is more likely that the null hypothesis will be rejected. The rejection area for H0 is on the right-hand side of the distribution function; testing is always conducted on the right-hand side. If it is to be checked whether a given distribution satisfies a normal distribution, the measured distribution must first be transformed. This transformation is called the ztransformation and converts any normal distribution into a standard normal distribution. The transformed distribution then has the mean value x ¼ 0 and the standard deviation or dispersion s ¼ 1. The z-transformation for any xi value of the distribution is calculated as follows: zi ¼ ðxi  xÞ=s Here, there are the following values for the sample: xi Mean s Standard deviation And for the basic population: zi ¼ ðxi  μÞ=σ μ Mean value of the population σ Standard deviation Example: Checking for normal distribution (goodness-of-fit test) Diameters of steel shafts should be checked to see whether they are subject to normal distribution. Normal distribution is an important prerequisite in quality assurance. A sample of 50 shafts is taken from production in immediate succession (machine capability analysis; MCA). The diameter of these is determined. Table 5.4 shows the results of an ordered series of diameters. The chi-squared distribution test (“goodness-of-fit test”) is used to check for normal distribution. For this purpose, the sample is divided into classes: in this case, six classes with a class width of five shafts. Since the distribution is to be checked, the more classes are created, the more accurate the check will be. Six classes are regarded as being the

15.0872 15.0880 15.0887 15.0895 15.0903 15.0915

15.0873 15.0880 15.0888 15.0896 15.0904 15.0916

15.0874 15.0881 15.0889 15.0898 15.0905 15.0916

15.0874 15.0881 15.0890 15.0901 15.0906 15.0918

Table 5.4 Diameter of steel shafts, ordered value series 15.0876 15.0882 15.0890 15.0901 15.0906 15.0927

15.0876 15.0883 15.0890 15.0901 15.0909

15.0876 15.0886 15.0890 15.0903 15.0909

15.0878 15.0886 15.0894 15.0903 15.0912

15.0880 15.0887 15.0895 15.0903 15.0913

5.5 The Chi-Squared Test (w2-Test) 87

88

5

Statistical Tests

minimum, while a class width of at least five is required, see also Sect. 5.5.1. These conditions are satisfied. The chi-squared distribution test examines the deviations of the observed and expected frequencies. The expected frequencies correspond to the normal distribution. This is the distribution against which the check is performed. It is understandable that, in the case of little or no deviation of the sample from the normal distribution, the sum of the deviation squares as present in the test are small or equal to 0. χ2 ¼

X ðhif  hie Þ2 hie

In the classes of the sample, the values that limit the class are entered for the same class width. In order to compare the measured values with the normal distribution, the measured values must be transformed. Here, the z-transformation is applied, which normalizes values to a normal distribution. For the limitation values of the classes, the z-values are obtained using the following relationship: zi ¼

ðxi  xÞ s

Here, x is the mean value and s the dispersion of the sample. In the example, all 50 single values for the diameters result in a mean value of x ¼ 15:0894 mm and the dispersion of s ¼ 0.001393 mm calculated from this. The zi-values are calculated with the limit values of the classes and are listed in Table 5.5. Class 1 starts at 1, and class 6 ends the sample at +1. This takes account of the normal distribution. Column A lists the frequency of the measured values, while column B shows the calculated frequency after the z-transformation. For this purpose, the area of the normal distribution in the class boundaries is initially determined. To this end, it is necessary to determine the areas using the table of the normal distribution for the class boundaries. The area within the limits of the normal distribution is obtained by subtracting the area value for the lower class limit from the area value for the upper class limit. See also Fig. 5.8 in this case. Alternatively, this area proportion can also be determined using software. Since the frequency corresponds to the area proportion, multiplication by the sample size yields the calculated frequency. The difference between the values under A and B is already a measure for the deviation of the measured frequency relative to the calculated frequency. The smaller the difference, the clearer the convergence with the normal distribution will be.

The Chi-Squared Test (w2-Test)

5.5

89

Table 5.5 Values for chi-squared distribution test

Diameter classes No. 1 2 3 4 5 6

From 1 15.0879 15.0887 15.0895 15.0903 15.0911

To 15.0879 15.0887 15.0895 15.0903 15.0911 +1

zTransformed 1.0768 0.5025 0.0718 0.6461 1.2204 2.3690

A Frequency of measured values 8 11 9 9 6 7 Σ ¼ 50

B Calculated frequency (after ztransformation) Area Φ (x) Number 0.1408 7.04 0.1669 8.34 0.2210 11.05 0.2123 10.62 0.1480 7.40 0.1112 5.56 Σ¼ Σ¼ 1.0002 50.01

Diff. A–B 0.96 2.66 2.05 1.62 1.4 1.44

χ2 0.131 0.848 0.380 0.247 0.265 0.373 Σ χ2 ¼ 2.244

Fig. 5.8 Area proportion after z-transformation

The chi-squared value for each class is obtained by dividing the square of the deviations (A–B) by the calculated frequency. Finally, the test value is determined as the sum of the chi-squared values. Result: ∑ χ 2¼ 2.244 Whether this value is significant is shown by comparison with the chi-squared table, see Table A4.1 in Appendix A.4. As can be seen from the table, the number of degrees of freedom is also required. Degree of freedom f:

90

5

Statistical Tests

f ¼ Number of addends  1 degree of freedom  2 degrees of freedom ¼ 6  1  2 ¼3 The additional 2 degrees of freedom that have to be considered result from the estimation of the mean and dispersion for calculating the expected frequencies using the ztransformation. When determining a confidence level or degree of freedom, it should be noted that a check is conducted to determine whether or not the frequencies differ. In the present case, the following values are obtained from Table A4.1 in the Appendix A.4: α ¼ 5%ð0:95Þ;

χ 2 ¼ 7:81

α ¼ 1%ð0:99Þ;

χ 2 ¼ 11:34

α ¼ 0:1%ð0:999Þ;

χ 2 ¼ 16:27

Since the test value ∑χ 2¼ 2.244 is smaller than the table values with a varying confidence level, it must be assumed that the diameters measured in the sample originate from a normal distribution.

5.5.3

Chi-Squared Independence Test

In the chi-squared independence test, the statistical relationship between two characteristics X and Y is examined. The system checks whether these characteristics are statistically interdependent or independent. The empirical frequencies are entered in a cross table; see Table 5.6. The prerequisite for this is that no frequency is less than 5. hΣΣ ¼ Column total ¼ Row total The test statistic for the independence test is calculated as follows:

χ2 ¼

 2 hn,m XX hn,m  b n

Here, the following applies: hn, m Observed (empirical) frequency

m

b hn,m

5.5

The Chi-Squared Test (w2-Test)

91

Table 5.6 Chi-squared test Row 1 Row 2 Column total

Column 1 h11 h21 hΣ1

Column 2 h12 h22 hΣ2

Column 3 h13 h23 hΣ3

Row total h1Σ h2Σ hΣΣ

Table 5.7 Four-field table Row y1 y2 Column total

Column x1 h11 h21 hΣ1

x2 h12 h22 hΣ2

h ∙h b hn,m ¼ n,s s,m hs,s

Row total h1Σ h2Σ hΣΣ

expected=theoretical frequency

n ¼ columns; m ¼ rows; hn,s ¼ column frequency; hs,m ¼ row frequency Degree of freedom f ¼ ðn  1Þ ðm  1Þ If there are two values for each of the characteristics X and Y, the result is a four-field table (2 2 panel) and thus a four-field χ 2-test; see Table 5.7. hΣΣ ¼ Column total ¼ Row total The chi-square test variable is calculated from the four addends to give the following: χ2 ¼

ðh11  h1Σ ∙ hΣ1 Þ2 ðh12  h1Σ ∙ hΣ2 Þ2 ðh21  h2Σ ∙ hΣ1 Þ2 ðh22  h2Σ ∙ hΣ2 Þ2 þ þ þ h1Σ ∙ hΣ1 h1Σ ∙ hΣ2 h2Σ ∙ hΣ1 h2Σ ∙ hΣ2

Through arithmetic transformation, the following results are obtained: χ2 ¼

ðh11 ∙ h22  h12 ∙ h21 Þ2 ∙ hΣΣ h1Σ ∙ hΣ1 ∙ h2Σ ∙ hΣ2

The expected frequencies correspond to the null hypothesis.

92

5

Statistical Tests

In general, the following is true: H0: The expected distribution and the empirical distribution are the same. H1: Empirical distribution and expected distribution are not equal. In a one-sided test, the null hypothesis H0 is rejected if the test value χ p2 is greater than the theoretical χ 2-value (table value from Appendix A.4). χ 2p > χ 2f ,1α ;

f ¼ k  1 ðdegree of freedomÞ

In a two-sided test, the null hypothesis H0 is rejected if the test value is greater or less than the theoretical frequency. χ 2p > χ 2f ,1α=2 and χ 2p < χ 2f ,α=2 The general procedure for the chi-squared independence test is as follows: 1. 2. 3. 4. 5. 6.

Formulate null and alternative hypotheses Calculate degree of freedom Determine significance level Calculate chi-squared test statistic Comparison of the test statistic with the table value Interpretation of the result

When determining the significance level, a general probability of error of α ¼ 0.05 was chosen. This is the probability of rejecting the null hypothesis despite it being true. The null hypothesis is rejected if the table value (critical value) is exceeded. The sample size has an influence since the chi-squared distribution is a function of the error probability α and the degree of freedom f. For large samples, the differences become significant at an earlier stage. Example: Comparison of cold-forming systems Cold forming is used to produce identical parts for fasteners (screws) on two machines. For a property (characteristic) that is not defined in more detail, sorting is performed manually on one machine with a lot size of 4 million and is not carried out on the other machine with a lot size of 2 million.

Literature

93

Table 5.8 Example for the four-field test Characteristic A Machine with manual sorting Machine without sorting

Characteristic B Parts OK 3,999,400 1,999,580

Parts defective 600 420

The examination of the error rates revealed the following: Machine 1: Lot size of 4 million, defective proportion of 150 ppm Machine 2: Lot size of 2 million, defective proportion of 210 ppm A test must be conducted to determine whether there are statistical differences between the two machines. An example of using the chi-squared test (four-field test) is shown in Table 5.8. The null hypothesis is tested: There is no difference in the frequency of defective parts. With a statistical certainty of 95% (probability of error α ¼ 0.05), the following table value (critical value) is obtained for the one-sided test: χ2

f ,1α

¼ 3:84:

The test statistic is calculated as being χ 2p ¼ 28.24. Since χ 2p > χ 2f,1α, the null hypothesis is rejected. There is a statistical difference in the proportion of defects between the two installations.

Literature 1. Bortz, J., Lienert, G.A., Boehnke, K.: Verteilungsfreie Methoden in der Biostatistik. Springer, Berlin (2008) 2. Schiefer, F.: Evaluation von Prüfverfahren zur Bestimmung der künstlichen Alterung an Polypropylen-Folien. Bachelor thesis: University of Stuttgart, 27 Feb 2012 3. DIN EN ISO 527: Kunststoffe – Bestimmung der Zugeigenschaften (Plastics – Determination of tensile properties)

6

Correlation

Correlation (or interrelation) characterizes a relationship or a connection between variables (usually two). For the “black box” model, this describes the relationship between a causative variable and an effect. Correlations between causes and effects can also be determined.

6.1

Covariance, Empirical Covariance

If there is a relationship between two values/variables (xi, yi), then there is a descriptive measure, namely the empirical covariance: s2xy ¼

n 1 X ðx  xÞðyi  yÞ n  1 i¼1 i

Here, the following are mean values of the samples in the population n: x¼

n 1 X x n i¼1 i

y¼

n 1 X y n i¼1 i

and

As can be seen, empirical covariance depends on the dimension of the variable and is therefore dimensional.

# Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 H. Schiefer, F. Schiefer, Statistics for Engineers, https://doi.org/10.1007/978-3-658-32397-4_6

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Correlation

This disadvantage is eliminated by the correlation coefficient (empirical correlation coefficient) by dividing the empirical covariance by the variances of the samples. This results in a dimensionless key figure.

6.2

Correlation (Sample Correlation), Empirical Correlation Coefficient

Correlation answers the question of whether or not there is a relationship between two quantities. No distinction is made as to which value/variable is considered dependent or independent. The variables should be distributed normally. The degree of correlation is quantitatively specified by the correlation coefficient rxy. The correlation coefficient, also known as the correlation value or product-moment correlation (Pearson correlation), is a dimensionless variable that describes the linear context. Depending on the strength of the connection, the value rxy can range between 1 and +1. Where rxy ¼ +1 or rxy ¼ 1, there is a functional interrelationship: directly linear in the case of +1 and indirectly linear for 1. With these values, all values/points in the scatterplot lie along a straight line. If rxy is zero, there is no linear correlation. Yet a nonlinear connection can still exist. The sign of the correlation coefficient is determined by the sign of the covariance; the variances in the denominator are always greater than “zero.” The correlation coefficient (empirical correlation coefficient) is calculated from the value pairs (xi, yi) where i ¼ 1 . . . n as follows: ∑ni¼1 ðxi  xÞðyi  yÞ ∑ni¼1 xi yi  nx y ffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r xy ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ∑ni¼1 x2i  nx2 ∑ni¼1 y2i  ny2 ∑ni¼1 ðxi  xÞ2 ∙ ∑ni¼1 ðyi  yÞ2 Here, x, y are the arithmetic mean values. The strength of the linear relationship is interpreted as follows: 0 0–0.5 0.5–0.8 0.8–1.0 1.0

No linear relationship Weak linear relationship Mean linear relationship Strong linear relationship Perfect (functional) relationship

The limit values +1 or 1 can never be greater than “1” because the variable in the numerator (covariance) cannot be greater than the denominator—the product of the standard deviations. The square of the correlation coefficient rxy is the coefficient of determination B:

6.3

Partial Correlation Coefficient, Partial Correlation

97

r 2xy ¼ B In a first approximation, the coefficient of determination indicates the percentage of total variance (dispersion) determined with regard to a statistical correlation. For example, where r ¼ 0.6, and thus B ¼ 0.36, 36% of the variance is statistically expressed. Whether or not a correlation coefficient is significant (substantially different from zero) is determined through the t-test. The correlation between two variables is a necessary, albeit insufficient, condition for a causal relationship.

6.3

Partial Correlation Coefficient, Partial Correlation

If the correlation between two variables A and B is influenced by a third variable C, the partial correlation provides the correlation coefficient between A and B without the influence of C (disturbance variable). r AB  r AC ∙ r BC r AB,C ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi 1  r 2AC 1  r 2BC Here, rAB, C is the correlation coefficient for the linear relationship between the variables (influencing variables) of A and B without the effect of variable C. rAB, rAC, and rBC are the correlation coefficients between the variables AB, AC, and BC. Example: Factors influencing the tensile strength of reinforced polyamide (nylon) 66 The tensile strength of plastics is an important factor for their application. For example, the influence of temperature on the strength of plastics is much greater than that of metals or ceramics. However, other influencing variables are also significant for plastics. This applies to both influencing variables such as the material composition and to influences present during application. Materials that absorb moisture have a considerable influence on the property profile. Such materials include polyamides (PA). These can be modified by additives (e.g., using fillers or reinforcing materials; i.e., fibers). In the example, the correlation coefficients are used to show the influence of density and moisture on the tensile strength of reinforced PA66, see Table 6.1. Using the data, the correlation coefficients are calculated for PA 66 as follows: • Correlation between tensile strength and density rAB ¼ 0.9776 • Correlation between tensile strength and moisture absorption rAC ¼ 0.9708 • Correlation between density and moisture absorption rBC ¼ 0.9828

98

6

Correlation

Table 6.1 Tensile strength, density, and moisture absorption of PA66 [1]

PA66 Without glass fiber 15% GF 25% GF 30% GF 35% GF 50% GF

A Tensile strength (lf) [MPa] ISO 527 60 80 120 140 160 180

B Density [g/cm3] ISO 1183 1.13 1.23 1.32 1.36 1.41 1.55

C Moisture absorption (lf) DIN 50014 2.8 2.2 1.9 1.7 1.6 1.2

lf ¼ Humid after storage in standard climate 23/50 in accordance with DIN 50014 GF ¼ Glass fiber content

All three correlation coefficients exhibit a strong linear correlation. This also determines the partial correlation coefficient rAB, C (i.e., the linear correlation between tensile strength and density) without the influence of moisture absorption. A result for rAB, C ¼ 0.5304 is obtained using the following formula: r AB  r AC ∙ r BC r AB,C ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi 1  r 2AC 1  r 2BC From this, it can be seen that moisture absorption has a considerable influence on tensile strength.

Literature 1. Bottenbruch, L., Binsack, R. (eds.): Technische Thermoplaste Polyamide – Kunststoff Handbuch 3/4. Hanser, Munich (1998)

7

Regression

The quantitative relationship between two or more values/variables is described by regression. This establishes a mathematical relationship—a functional description—between these variables. The question of what causes and effects are is generally defined for technical tasks.

7.1

Cause–Effect Relationship

“The effect must correspond to its cause.” Leibniz, G. W., 1646–1716 Before calculating the regression model, a scatterplot must be created. From this graph (the representation of the value pairs for the regression), the following facts can be ascertained: • • • •

The dispersion of the values The degree of correlation The possible functional relationship The homogeneity of the result function or result area

Note on the result area: The experiment must be designed such that a consistent quality prevails in the experiment space; only the quantity may change. In other words, no jumps (inhomogeneities) may occur in the experiment space. Examples of inhomogeneities include phase transition of a metal from solid to liquid and glass transition in plastics. The cause–effect relationship in the form of a “black box” is a simple hypothesis. Namely, it is the representation of all influencing variables (xi) affecting the target variable(s) yj. See Fig. 7.1 in this case. # Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 H. Schiefer, F. Schiefer, Statistics for Engineers, https://doi.org/10.1007/978-3-658-32397-4_7

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100

7

Regression

Fig. 7.1 Representation of the cause–effect relationship as a “black box”

The influencing variables are then varied in the experiment design in order for their effect to be determined. When testing in a laboratory or on a production line, care must be taken to ensure that the “constant influencing variables” actually remain constant. If this is not possible, for example, due to a change in the outdoor temperature (summer vs. winter) or humidity, these influences can be assigned to the causes. Put simply, the target variable y is dependent on one or more influencing variables xi. With an influencing variable x, the following therefore applies: y ¼ f ð xÞ Alternatively, with several influencing variables, for example, y ¼ f (x1, x2, x3). If there are reciprocal effects between the influencing variables in terms of their effect on the target variable, the result might be the following simplified phenomenological relationship with two influencing variables x1 and x2, for example: y ¼ a þ b1 ∙ x1 þ b2 ∙ x2 þ b12 ∙ x1 ∙ x2 þ e: Here, b12∙x1∙x2 is the term that describes the interaction. If the effects of the influencing variables on several target variables are investigated, the following system of equations might be given (cf. Fig. 7.2). y1 ¼ f ðx1 , x2 , x5 Þ y2 ¼ f ð x1 , x3 Þ y3 ¼ f ðx2 , x4 , x5 Þ Here, it can be seen that not all influencing variables xi influence the target value yj (magnitude of effect) in the same way; the quantitative influence is also generally varied.

7.2

Linear Regression

101

Fig. 7.2 Selective effect of causes

Note: “Identical causes [...] produce identical effects.” On the Nature of Things, Lucretius (Titus Lucretius Carus), 93–99 BCE to 53–55 BCE This must be taken into account when formulating an optimization function. Under certain circumstances, certain effects can be optimized independently of each other.

7.2

Linear Regression

Regression is a method of statistical analysis. If it can be observed from the scatterplot that the dependence of the target variable on the influencing variable is linear or roughly linear, the correlation (linear regression) is calculated. A function of the following type is the result: y¼aþbxþe Dependence is thus described quantitatively. Here, a is the intersection with the y-axis, b is the slope, and e the residuum (the remainder) of the mathematical model. These circumstances are shown in Fig. 7.3.

102

7

Regression

Fig. 7.3 Linear regression

The straight line runs through the points on the graph in such a way that the sum of the squares of the distances to the linear equation is a minimum (C. F. Gauss): y ¼ f ð x1 Þ n X

ðyi  ða þ b ∙ xi ÞÞ2 ⟹Minimum

i¼1

i Number of measuring points yi Measured y value at point i a + bxi Function value y at point i Note on the least-squares method: “[...] by showing in this new treatment of the object that the method of the least squares delivers the best of all combinations, not approximately but necessarily [...].” Theoria Combinationis Observationum Erroribus Minimis Obnoxiae, Carl Friedrich Gauss, Göttingen, 1821–1823 If the distances from the measuring points to the linear equation are greater than the dispersion of the measuring points, a systematic influence exists. At least one other influencing factor is therefore responsible for this. The potential influencing variable is determined through the correlation analysis (see Sect. 6.2). In Fig. 7.4, this variable is designated by x2. The deviations of the measured values from the straight line equation are Δy.

7.2

Linear Regression

103

Fig. 7.4 Dependence of the variable Δy on x2

Where Δy ¼ f (x2), the following applies: y ¼ f ðx1 Þ þ Δyðx2 Þ ¼ f ðx1 Þ þ f ðx2 Þ ¼ f ðx1 , x2 Þ If Δy is also linearly dependent on x2, it follows that: y ¼ a þ b1 x 1 þ b2 x 2 þ e From this representation, it can also be seen that it is important to keep the dispersion small so that potential influencing variables can be identified. The coefficient of determination B accordingly increases in this case. When using a regression function, always specify r or r2 ¼ B as well. This shows how well the function used maps the measured values (reality). To calculate the variables a and b (regression coefficients), the sum of the distance squares is initially mapped; the first derivation in accordance with a and b yields the extremum (in this case the minimum; refer to the second derivation). Additionally, the regression coefficients are obtained by setting the derivatives to zero: b¼

∑ni¼1 xi yi  nx y ∑ni¼1 x2i  nx2

or alternatively

b¼

∑ni¼1 ðxi  xÞðyi  yÞ ∑ni¼1 ðxi  xÞ2

And for a: a ¼ y  b∙x

or alternatively

x and y are mean values of the data xi and yi

a¼

∑ni¼1 yi  b ∙ ∑ni¼1 xi n

104

7

x¼

Regression

n n 1 X 1 X xi ; y ¼ y n i¼1 n i¼1 i

The regression function is the phenomenological model of the relationship between causes and effects. The function can be physically and technically consolidated if the influencing variables/parameters are formulated; for example, in terms of their dependence on physical variables. Example y ¼ a þ b∙x here, x ¼ k ∙ e

Ea T

Ea—Activation energy For certain technical tasks, theoretical derivations might also be available as alternative influencing variables on the target variable. However, if no direct calculation is possible due to insufficient knowledge of material values, for example, the effect of these can initially be evaluated through correlation calculation with defined influencing variables. From the strength of the influencing variables in the regression, it can be deduced which variables are significant for the circumstances in question. The model is incrementally refined, resulting in the completeness of the model. The reformulation of the measured variable into structural variables or other theoretical influencing variables also results in a consolidation of the description of the physical and technical connection. If the linear approach of regression does not achieve the required coefficient of determination (correlation coefficient), further steps need to be formulated and calculated, namely: • The nonlinear behavior of the influencing variables. • The interaction between the influencing variables. If these measures do not yield the necessary improvements in the coefficient of determination, the description model is incomplete. In addition, an investigation must be conducted to determine the additional influencing variables/parameters that have a significant influence on the cause–effect relationship. A linear approach can also be used initially for these other influencing variables.

7.3

Nonlinear Regression (Linearization)

7.3

105

Nonlinear Regression (Linearization)

Since linear relationships are assumed for the calculation of the regression function, a linearization must first be performed for a nonlinear relationship between the influencing variable and target variable. If this is not possible, other methods should be used (see Sect. 7.4). Many nonlinear correlations (i.e., dependencies described by functions that describe this nonlinearity as practically as possible) are to be transformed mathematically in such a way that facilitates simple linear regression. Examples The nonlinear function y ¼ a ∙ eb ∙ x is transformed to ln y ¼ ln a þ b ∙ x and thus Y ¼ A þ b∙x where ln y≙Y; ln a≙A The function y¼

a∙x bþx

can also be linearized through transformation. 1 bþx x b ¼ ¼ þ y a∙x a∙x a∙x

and therefore

1 1 b 1 ¼ þ ∙ , i:e:, where y a a x

106 Table 7.1 Transformation of functions into linear functions

7

Function y ¼ a ∙ xb y ¼ a ∙ eb ∙ x y ¼ a ∙ eb=x y ¼ aþb1 ∙ x y ¼ aþb1∙ ex a > 0; b > 0 a∙x y ¼ bþx a > 0; b > 0 y ¼ a + b ∙ x + c ∙ x2

Regression

Transformed (linearized) function ln y ¼ ln a + b ∙ ln x ln y ¼ ln a + b ∙ x ln y ¼ ln a þ b=x 1 y ¼ a þ b∙x 1 x y ¼ a þ b∙e 1 y

¼ 1a þ ba ∙

1 x

y¼a+b∙x+c∙z where x2 ¼ z

1^ 1^ b^ 1^ ¼Y; ¼A; ¼B and ¼X y a a x The result is the linear function Y ¼ A + BX. Further examples for the transformation of functions are listed in Table 7.1. If the available data cannot be described by functions that can be linearized, the description range might need to be divided in such a way that approximately linear relationships exist within these ranges. Small ranges are often approximately linear. A nonlinear relationship can also be described with polynomials. The first degree of a polynomial is the linear equation. A second degree polynomial describes a curvature, namely: y ¼ a þ b1 x þ b2 x 2 If the degree of the polynomial is further increased (e.g., a third-degree polynomial), a function with an inflection point is created: y ¼ a þ b1 ∙ x þ b2 ∙ x 2 þ b3 ∙ x 3 However, a polynomial can no longer be adjusted on the basis of the data (curve fitting) with the linearization method and subsequent closed calculation. However, the polynomial is also calculated according to the method of the smallest square sum (minimization of the square sum, C. F. Gauss). Numerical analysis programs are used to calculate these polynomials. Statistical packages (software) contain these numerical methods. These methods are not discussed in further detail at this juncture. As the polynomial degree increases, so too does the quality of the description increases. The possibility of interpretation generally decreases unless there is a physical/technical background in addition to the polynomial to be interpreted.

7.4

Multiple Linear and Nonlinear Regression

7.4

107

Multiple Linear and Nonlinear Regression

Multiple regression for linear cases (first-order polynomial) where f ðx1 . . . xn , ao , a1 . . . an12 Þ ¼ ao þ a1 ∙ x1 þ . . . an ∙ xn has the residuals e e1 ¼ ao þ a1 x11 þ . . . þ an xn,1  y1 ⋮ ei ¼ ao þ a1 x1,i þ . . . þ an xn,i  yi ⋮ e j ¼ ao þ a1 x1,j þ . . . þ an xn,j  y j This system of equations is usually presented in a simplified form as follows: e ¼ Aa  y In this form, the column matrix e contains the residuals ei, the matrix A contains the basic function values of the influencing variables (x1,i . . . xn,j); a is the parameter matrix across all ai, and y is the column matrix of the individual yi values. If the random errors (see Sect. 4.1.2) are small compared to the residuals, it can be assumed that at least one further influencing variable (parameter) is present. The solution of this system of equations takes place across all residuals from e1 to ej with the proviso of minimizing the sum of the distance squares:

Min

j X i¼1

e2i

The system of equations is considered determined when it has as many equations as unknowns (i +1 ¼ j). It is considered underdetermined when there are fewer equations than unknowns, and overdetermined if it has more equations (i +1 > j). A definite system has at least one solution, while an overdetermined system might have one solution, and an underdetermined system might have multiple solutions. The solution of the system of equations is generally numerical. Statistical software packages contain solution programs/solution strategies (e.g., MATLAB, EXCEL).

108

7

Regression

The classical approach is the solution of the inhomogeneous equation system (application of Cramer’s rule). Certain systems of equations (one solution) with two or three unknowns, (i.e., x1; x2) or (x1; x2; x3) can be solved with the following: • The addition method (method of identical coefficients) • The substitution method • The combination method The prerequisite is that the equations are independent of and do not contradict one another. If the equations contradict one another, the system has no solution. If equations in the system are interdependent, an infinite number of solutions emerges. Certain systems of equations with two second-degree unknowns can also be solved with the same methods as for linear systems. Simple polynomials of the form y ¼ ao þ a1 x1 þ a11 x21 þ e can be transferred into the linear compensation function by redefining a11 x21 to a2 x2 ; with two variables in this example: y ¼ ao þ a1 x 1 þ a2 x 2 þ e If the equations cannot be linearized, then (real) multiple nonlinear systems of equations exist. e i ¼ ao þ a1 f 1 ð x 1 Þ þ . . . þ an f n ð x n Þ  y i f1(x1) . . . fn(xn) are nonlinear functions. In certain cases, a series expansion of nonlinear functions might be of use. It makes sense to create a picture of the nonlinearity in advance in order to specify the mathematical model. The starting points for this constitute physical/technical considerations (theory) or logical derivations. The solution of the equation system requires an interactive method to realize the adaptation of the nonlinear equations to the data points/measuring points with a minimum for the distance square sum. As such, there is no direct method for solving the system of equations that produces a unique solution, such as the ones that exist for linear fitting (adaptation). With regard to the interactive solution method, it is generally necessary to specify starting values. These influence the convergence behavior.

7.5

Examples of Regression

109

Table 7.2 Tensile strength, Brinell hardness, and Vickers hardness of steel; extract from [1] Tensile strength [MPa] 1485 1630 1810 1995 2180

7.5

Brinell hardness HB Factor 437 3.40 475 3.43 523 3.46 570 3.50 618 3.52 x ¼ 3.46

Vickers hardness HV 10 Factor 460 3.23 500 3.26 550 3.29 600 3.33 650 3.35 x ¼ 3.29

Examples of Regression

Example: Correlations of steel properties It has long been known in mechanical engineering that there is a correlation between tensile strength Rm and hardness values for unalloyed and low-alloy steels. Table 7.2 shows values for the tensile strength Rm, Brinell hardness HB, and Vickers hardness HV, in addition to the factors calculated from these for the conversion. With a certain tolerance, the correlation between tensile strength Rm and Brinell hardness HBW (EN ISO 6506-1, W-widia ball) can thus be defined:

Rm ½MPa  3:5 HBW And, in a similar way, the relationship between tensile strength Rm and Vickers hardness HV (according to DIN 501500) can be calculated: Rm ½MPa  ð3:2 . . . 3:3Þ HV This proportionality applies in spite of the unequal stress states, which also means that the composition and microstructure under the test load have (approximately) the same influence in this case. It is therefore possible to express one property with another. Instead of destructive strength testing, a test can also be carried out on components in ongoing production. The significant influencing variables affect both properties in the same way; the relationship is linear. Example: Influence of physical structural variables on the tensile strength of polystyrene [2] Determination of the influence of the following physical structural variables: orientation of the macromolecules, short-range order of the molecules, and energy-elastic residual stress on the tensile strength of amorphous polystyrene.

110

7

Regression

Experimental measures ensured that the influencing variables (structural variables) could be varied independently of one another. The calculation of the functional relationship was performed with a second-order polynomial (multiple nonlinear regression).   Rm N=mm2 ¼ 34:127 þ 23:123 ∙ ln ð1 þ εx Þ þ 0:106 ð ln ð1 þ εx ÞÞ2 þ 0:039 ∙ σ EZ  0:0015 σ 2EZ  6146:620 ∙ Δρ=ρ2  27, 594, 410 ðΔρ=ρ2 Þ2 This includes the following: ln(1 + εx) Δ ρ/ρ2 σ EZ

Measurement parameter for the orientation of macromolecules where εx is the degree of stretching Change in density due to the change in the short-range order Energy-elastic residual stress due to the thermal gradient (cooling); the tensile residual stress in the example of tensile strength

Reference density for calculation ρ2 ¼ 1.04465 g/cm3 Coefficient of determination B ¼ 0.9915. Figure 7.5 shows the results. The linear relationship between the degree of orientation and the tensile strength is notable. Although the regression was performed with a second-degree polynomial, a linear correlation was identified. Since ln(1+εx) c ¼ ΔNx/N applies for the measured variable (where N is the total number of binding vectors and Nx the number of binding vectors in the x direction), the physically meaningful relationship follows that the strength increases more linearly with the number of bond vectors in the loading direction. In formal terms, the influence of the change in density and the tensile residual stresses is also correctly represented. Since the number of bond vectors increases in line with the density, the strength also increases. The residual tensile stress is a preload: As it increases, the measured tensile strength decreases. The non-linearities in the change in density and the change in tensile residual stress result from this because these quantities are distributed over the cross-section of the tensile samples. In addition, the calculation is adapted to the measured values as per C. F. Gauss. Example: Setting parameters and molded-part properties for ceramic injection molding [3] Calculation of the relationship between the setting parameters for ceramic injection molding and the properties of the molded part. In Chap. 1, the statistical experimental design was presented for the aforementioned task. A Box–Hunter plan was used with the following influencing variables: injection

7.5

Examples of Regression

111

Fig. 7.5 Change in the tensile strength of polystyrene as a function of the following structural variables: orientation, energy-elastic residual stress, and short-range order

112

7

Regression

volume x1 [cm3], delay time for GIT technology x2 [s], gas pressure x3 [bar], and gas– pressure duration x4 [s]. The target values are: bubble length [mm], wall thickness [mm], mass [g], and crack formation (crack length) [mm]. For a selected material mixture A, the following linear relationships were calculated through regression without considering the interaction: Bubble length yB yB ½mm ¼ 171, 727:3 þ 13, 741:8 x1  7293:5 x2  4:04 x3  238:851 x4 B ¼ 0:92 Wall thickness yW yW ½mm ¼ 2390:8 þ 186:3 x1  2:409 x2 þ 0:0641 x3 þ 3:7995 x4 B ¼ 0:56 Mass yM yM ½g ¼ 2159:0  170:2 x1 þ 111:73 x2 þ 0:0015 x3  4:996 x4 B ¼ 0:84 Crack length yR yR ½mm ¼ 68, 783:9 þ 5299:8 x1 þ 1269:03 x2 þ 2:7120 x3 þ 181:127 x4 B ¼ 0:71 If the material mixture is changed (mixture B), the following correlation coefficients/ coefficients of determination are obtained: Bubble length: Wall thickness: Weight: Crack length:

B ¼ 0.88; r ¼ 0.938 B ¼ 0.63; r ¼ 0.794 B ¼ 0.59; r ¼ 0.768 B ¼ 0.61; r ¼ 0.781

From this, it can be seen that other parameters have an influence on the regression of the four selected influencing variables with the target variables. In the case above, these are at least parameters describing the composition. Furthermore, it must be investigated whether nonlinear dependencies and/or interactions are present.

7.5

Examples of Regression

113

Example: Dependence on molded-part properties [4] Precision injection molding, the production of molded plastic parts within tight tolerances, is a necessity in many areas of application. Under the conditions of zero-defect production, accompanying production monitoring is imperative. The molding dimensions of a polyoxymethylene (POM) circuit board with dimensions of approx. 32 mm  approx. 15 mm  max. approx. 4 mm were investigated. The following injection-molding parameters were varied: • • • • •

Tool temperature TW 80–120  C Melt temperature TM 180–210  C Injection pressure pE 1000–1400 bar Injection time tE 7–13 s Injection speed vE 20–60% of the max. injection speed (Arburg Allrounder injectionmolding machine)

A Box–Hunter plan with the above parameters was used. With five parameters, a total of 52 experiments result from 25 key points, 2∙5 axis values, and 2∙5 center tests of the experiment design (see also Sect. 1.2.4). The standardized values for the injection-molding parameters are then as follows: 0;  1; þ 1;  2:4; þ 2:4 This constitutes a variation in five steps. Compared to a conventional experiment design with a total of 55 (basis: steps; exponent: influencing variables), and thus 3125 tests, considerable savings are achieved. Through variation in five steps, nonlinearities and interactions between influencing variables and target variables can also be calculated. Results The dependence of the total length L on the injection pressure pE under the conditions TW ¼ 100  C, TM ¼ 195  C, tE ¼ 10 s, and vE ¼ 40% is calculated from the test results to give the following: L ½mm ¼ 32:1056 þ 0:002 ∙ pE with a correlation coefficient of r ¼ 0.9412. Under the conditions TW ¼ 100  C and tE ¼ 10 s, the following dependence is obtained for the mass m: m ½g ¼ 1:3298 þ 0:0001 ∙ pE with the correlation coefficient r ¼ 0.9032.

114

7

Regression

Fig. 7.6 Molded-part mass and length in the examined processing range

Mass temperature and injection speed had no influence. The relationship between the total length and weight is shown in Fig. 7.6. The regression calculation results in the following function: L ½mm ¼ 27:5605 þ 3:4067 ∙ m There is a very high statistical correlation, as can be seen in Fig. 7.6: r ¼ 0.9303. This confirms that the gravimetric production monitoring practiced in plastics technology is an efficient method. This particularly applies to small and precision-manufactured parts.

Literature 1. Innorat: Umrechnung Zugfestigkeit – Härte. http://www.innorat.ch/Umrechnung+Zugfestigkeit++Härte_u2_72.html (2017). Accessed 19 Sept 2017 2. Schiefer, H.: Beitrag zur Beschreibung des Zusammenhanges zwischen physikalischer Struktur und technischer Eigenschaft bei amorphem Polystyrol. Dissertation: Technische Hochschule Carl Schorlemmer, Leuna-Merseburg (1976) 3. Schiefer, H.: Spritzgießen von keramischen Massen mit der Gas-Innendruck-Technik. Lecture: IHK Pforzheim, 24 November 1998 (1998) 4. Beiter, N.: Präzistionsspritzgießen – Bedingungen und Formteilcharakteristik. Thesis: Fachhochschule Furtwangen, 31 March 1994

Appendix

A.1. Tables of standard normal distributions Table A1.1 Values of the standard normal distribution, one-sided confidence interval Φ(x) Area x Distance from the maximum of the standard normal distribution

x 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0 0.50000 0.53983 0.57926 0.61791 0.65542 0.69146 0.72575 0.75804 0.78814 0.81594 0.84134

0.01 0.50399 0.54380 0.58317 0.62172 0.65910 0.69497 0.72907 0.76115 0.79103 0.81859 0.84375

0.02 0.50798 0.54776 0.58706 0.62552 0.66276 0.69847 0.73237 0.76424 0.79389 0.82121 0.84614

0.03 0.51197 0.55172 0.59095 0.62930 0.66640 0.70194 0.73565 0.76730 0.79673 0.82381 0.84849

0.04 0.51595 0.55567 0.59483 0.63307 0.67003 0.70540 0.73891 0.77035 0.79955 0.82639 0.85083

0.05 0.51994 0.55962 0.59871 0.63683 0.67364 0.70884 0.74215 0.77337 0.80234 0.82894 0.85314

0.06 0.52392 0.56356 0.60257 0.64058 0.67224 0.71226 0.74537 0.77637 0.80511 0.83147 0.85543

0.07 0.527908 0.56749 0.60642 0.64431 0.68082 0.71566 0.74857 0.77935 0.80785 0.83398 0.85769

# Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 H. Schiefer, F. Schiefer, Statistics for Engineers, https://doi.org/10.1007/978-3-658-32397-4

0.08 0.53188 0.57142 0.61026 0.64803 0.48439 0.71904 0.75175 0.78230 0.81057 0.83646 0.85993

0.09 0.53586 0.57535 0.61409 0.65173 0.68793 0.72240 0.75490 0.78524 0.81327 0.83891 0.86214

115

116

Appendix

Table A1.2 Values of the standard normal distribution, one-sided confidence interval Φ(x) Area x 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

0 0.86433 0.88493 0.90320 0.91924 0.93319 0.94520 0.95543 0.96407 0.97128 0.97725 0.98214 0.98610 0.98928 0.99180 0.99379 0.99534 0.99653 0.99744 0.99813 0.99865 0.99903 0.99931 0.99952 0.99966 0.99977 0.99984 0.99989 0.99993 0.99995 0.99997

0.01 0.86650 0.88686 0.90490 0.92073 0.93448 0.94630 0.95637 0.96485 0.97193 0.97778 0.98257 0.98645 0.98956 0.99202 0.99396 0.99547 0.99664 0.99752 0.99819 0.99869 0.99906 0.99934 0.99953 0.99968 0.99978 0.99985 0.99990 0.99993 0.99995 0.99997

0.02 0.86864 0.88877 0.90658 0.92220 0.93574 0.94738 0.95728 0.96562 0.97257 0.97831 0.98300 0.98679 0.98983 0.99224 0.99413 0.99560 0.99674 0.99760 0.99825 0.99874 0.99910 0.99936 0.99955 0.99969 0.99978 0.99985 0.99990 0.99993 0.99996 0.99997

0.03 0.87076 0.89065 0.90824 0.92364 0.93699 0.94845 0.95818 0.96638 0.97320 0.97882 0.98341 0.98713 0.99010 0.99245 0.99430 0.99573 0.99683 0.99767 0.99831 0.99878 0.99913 0.99938 0.99957 0.99970 0.99979 0.99986 0.99990 0.99994 0.99996 0.99997

Example: Φ(1.3) ¼ 0.90320 (90.32%) For all x > 4.9, Φ(x)  1.0 (100%)

0.04 0.87286 0.89251 0.90988 0.92507 0.93822 0.94950 0.95907 0.96712 0.97381 0.97932 0.98382 0.98745 0.99036 0.99266 0.99446 0.99585 0.99693 0.99774 0.99836 0.99882 0.99916 0.99940 0.99958 0.99971 0.99980 0.99986 0.99991 0.99994 0.99996 0.99997

0.05 0.87493 0.89435 0.91149 0.92647 0.93943 0.95053 0.95994 0.96784 0.97441 0.97982 0.98422 0.98778 0.99061 0.99286 0.99461 0.99598 0.99702 0.99781 0.99841 0.99886 0.99918 0.99942 0.99960 0.99972 0.99981 0.99987 0.99991 0.99994 0.99996 0.99997

0.06 0.87698 0.89617 0.91309 0.92785 0.94062 0.95154 0.96080 0.96856 0.97500 0.98030 0.98461 0.98809 0.99086 0.99305 0.99477 0.99609 0.99711 0.99788 0.99846 0.99889 0.99921 0.99944 0.99961 0.99973 0.99981 0.99987 0.99992 0.99994 0.99996 0.99998

0.07 0.87900 0.89796 0.91466 0.92922 0.94179 0.95254 0.96164 0.96926 0.97558 0.98077 0.98500 0.98840 0.99111 0.99324 0.99492 0.99621 0.99720 0.99795 0.99851 0.99893 0.99924 0.99946 0.99962 0.99974 0.99982 0.99988 0.99992 0.99995 0.99996 0.99998

0.08 0.88100 0.89973 0.91621 0.93056 0.94295 0.95352 0.96246 0.96995 0.97615 0.98124 0.98537 0.98870 0.99134 0.99343 0.99506 0.99632 0.99728 0.99801 0.99856 0.99896 0.99926 0.99948 0.99964 0.99975 0.99983 0.99988 0.99992 0.99995 0.99997 0.99998

0.09 0.88298 0.90147 0.91774 0.93189 0.94408 0.95449 0.96327 0.97062 0.97670 0.98169 0.98574 0.98899 0.99158 0.99361 0.99520 0.99643 0.99736 0.99807 0.99861 0.99900 0.99929 0.99950 0.99965 0.99976 0.99983 0.99989 0.99992 0.99995 0.99997 0.99998

Appendix

117

Table A1.3 The values of the standard normal distribution as a function of x and S Examples:

x 1.000 1.500 1.960 2.000 2.500 2.576 3.000 3.500 3.891 4.000

One-sided confidence interval (S ¼ 1  α/2) 84.1345% 93.3193% 97.5000% 97.7250% 99.3790% 99.5000% 99.8650% 99.9767% 99.9950% 99.9968%

The x values are also displayed as λ values for counter values

Two-sided confidence interval (S ¼ 1  α) 68.2689% 86.6386% 95.0000% 95.4500% 98.7581% 99.0000% 99.7300% 99.9535% 99.9900% 99.9937%

118

Appendix

A.2. Tables on the t-Distribution Table A2.1 Values of the t-distribution

f 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Two-sided confidence interval 0.8 0.9 0.95 One-sided confidence interval 0.90 0.95 0.975 3.078 6.314 12.706 1.886 2.920 4.303 1.638 2.353 3.182 1.533 2.132 2.776 1.476 2.015 2.571 1.440 1.943 2.447 1.415 1.895 2.365 1.397 1.860 2.306 1.383 1.833 2.262 1.372 1.812 2.228 1.363 1.796 2.201 1.356 1.782 2.179 1.350 1.771 2.160 1.345 1.761 2.145 1.341 1.753 2.131 1.337 1.746 2.120 1.333 1.740 2.110 1.330 1.734 2.101 1.328 1.729 2.093 1.325 1.725 2.086

Degree of freedomf ¼ n  1

0.98

0.99

0.998

0.999

0.99 31.821 6.965 4.541 3.747 3.365 3.143 2.998 2.896 2.821 2.764 2.718 2.681 2.650 2.624 2.602 2.583 2.567 2.552 2.539 2.528

0.995 63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355 3.250 3.169 3.106 3.055 3.012 2.977 2.947 2.921 2.898 2.878 2.861 2.845

0.999 318.309 22.327 10.215 7.173 5.893 5.208 4.785 4.501 4.297 4.144 4.025 3.930 3.852 3.787 3.733 3.686 3.646 3.610 3.579 3.552

0.9995 636.578 31.600 12.924 8.610 6.869 5.959 5.408 5.041 4.781 4.587 4.437 4.318 4.221 4.140 4.073 4.015 3.965 3.922 3.883 3.850

Appendix

119

Table A2.2 Values of the t-distribution

f 21 22 23 24 25 26 27 28 29 30 40 50 60 80 100 200 300 500 1

Two-sided confidence interval 0.8 0.9 0.95 One-sided confidence interval 0.90 0.95 0.975 1.323 1.721 2.080 1.321 1.717 2.074 1.319 1.714 2.069 1.318 1.711 2.064 1.316 1.708 2.060 1.315 1.706 2.056 1.314 1.703 2.052 1.313 1.701 2.048 1.311 1.699 2.045 1.310 1.697 2.042 1.303 1.684 2.021 1.299 1.676 2.009 1.296 1.671 2.000 1.292 1.664 1.990 1.290 1.660 1.984 1.286 1.653 1.972 1.284 1.650 1.968 1.283 1.648 1.965 1.282 1.645 1.960

0.98

0.99

0.998

0.999

0.99 2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.457 2.423 2.403 2.390 2.374 2.364 2.345 2.339 2.334 2.326

0.995 2.831 2.819 2.807 2.797 2.787 2.779 2.771 2.763 2.756 2.750 2.704 2.678 2.660 2.639 2.626 2.601 2.592 2.586 2.576

0.999 3.527 3.505 3.485 3.467 3.450 3.435 3.421 3.408 3.396 3.385 3.307 3.261 3.232 3.195 3.174 3.131 3.118 3.107 3.090

0.9995 3.819 3.792 3.768 3.745 3.725 3.707 3.689 3.674 3.660 3.646 3.551 3.496 3.460 3.416 3.390 3.340 3.323 3.310 3.290

Degree of freedomf ¼ n  1 where f ! 1, the t-distribution changes to the standard normal distribution. Example: f ! 1: (1  α) ¼ 0.9995: t ¼ 3.29

1 2 3 4 5 6 7 8 9 10 15 20 30 40 50 100 ∞

1 161 18.50 10.10 7.71 6.61 5.99 5.59 5.32 5.12 4.96 4.54 4.35 4.17 4.08 4.03 3.94 3.84

2 200 19.00 9.55 6.94 5.79 5.14 4.74 4.46 4.26 4.10 3.68 3.49 3.32 3.23 3.18 3.09 3.00

3 216 19.20 9.28 6.59 5.41 4.76 4.35 4.07 3.86 3.71 3.29 3.10 2.92 2.84 2.79 2.70 2.60

4 225 19.30 9.12 6.39 5.19 4.53 4.12 3.84 3.63 3.48 3.06 2.87 2.69 2.61 2.56 2.46 2.37

5 230 19.30 9.01 6.26 5.05 4.39 3.97 3.69 3.48 3.33 2.90 2.71 2.53 2.45 2.40 2.31 2.21

6 234 19.30 8.94 6.16 4.95 4.28 3.87 3.58 3.37 3.22 2.79 2.60 2.42 2.34 2.29 2.19 2.10

7 237 19.40 8.89 6.09 4.88 4.21 3.79 3.50 3.29 3.13 2.71 2.51 2.33 2.25 2.20 2.10 2.01

8 239 19.40 8.85 6.04 4.82 4.15 3.73 3.44 3.23 3.07 2.64 2.45 2.27 2.18 2.13 2.03 1.94

9 241 19.40 8.81 6.00 4.77 4.10 3.68 3.39 3.18 3.02 2.59 2.39 2.21 2.12 2.07 1.98 1.88

10 242 19.40 8.79 5.96 4.74 4.06 3.64 3.35 3.14 2.98 2.54 2.35 2.16 2.08 2.03 1.93 1.83

Numerator degree of freedom

Table A3.1 Values of the F-distribution 1  α ¼ 0.95

Denominator degree of freedom

15 246 19.40 8.70 5.86 4.62 3.94 3.51 3.22 3.01 2.85 2.40 2.20 2.02 1.92 1.87 1.77 1.67

20 248 19.50 8.66 5.80 4.56 3.87 3.44 3.15 2.94 2.77 2.33 2.12 1.93 1.84 1.78 1.68 1.57

30 250 19.50 8.62 5.75 4.50 3.81 3.38 3.08 2.86 2.70 2.25 2.04 1.84 1.74 1.69 1.57 1.46

40 251 19.50 8.59 5.72 4.46 3.77 3.34 3.04 2.83 2.66 2.20 1.99 1.79 1.69 1.63 1.51 1.39

50 252 19.50 8.58 5.70 4.44 3.75 3.32 3.02 2.80 2.64 2.18 1.97 1.76 1.66 1.60 1.48 1.35

100 253 19.50 8.55 5.66 4.41 3.71 3.27 2.98 2.76 2.59 2.12 1.91 1.70 1.59 1.52 1.39 1.24

∞ 254 19.50 8.53 5.63 4.37 3.67 3.23 2.93 2.71 2.54 2.07 1.84 1.62 1.51 1.44 1.28 1.01

120 Appendix

A.3. Tables on the F-Distribution

Denominator degree of freedom

1 2 3 4 5 6 7 8 9 10 15 20 30 40 50 100 ∞

1 647.8 38.51 17.44 12.22 10.01 8.81 8.07 7.57 7.21 6.94 6.20 5.87 5.57 5.42 5.34 5.18 5.02

2 799.5 39.00 16.04 10.65 8.43 7.25 6.54 6.06 5.72 5.46 4.77 4.46 4.18 4.05 3.98 3.83 3.69

3 564.2 39.17 15.44 9.98 7.76 6.60 5.59 5.42 5.08 4.83 4.15 3.86 3.56 3.46 3.39 2.25 3.12

4 599.6 39.25 15.10 9.61 7.39 6.23 5.52 5.05 4.72 4.47 3.80 3.52 3.25 3.13 3.05 2.92 2.79

5 921.8 39.30 14.88 9.36 7.42 5.99 5.29 4.82 4.48 4.24 3.58 3.29 3.03 2.90 2.83 2.70 2.37

6 937.1 39.33 14.73 9.20 6.98 5.82 5.12 4.65 4.32 4.07 3.42 3.13 2.87 2.74 2.67 2.54 2.41

Table A3.2 Values of the F-distribution 1  α ¼ 0.975

7 948.2 39.36 14.62 9.07 6.85 5.70 5.00 4.53 4.20 3.95 3.29 3.01 2.75 2.62 2.55 2.42 2.29

8 956.7 39.37 14.54 8.93 6.76 5.60 4.90 4.43 4.10 3.86 3.20 2.91 2.65 2.53 2.46 2.32 2.19

9 963.3 39.39 14.47 8.91 6.68 5.52 4.82 4.36 4.03 3.78 3.12 2.84 2.28 2.45 2.38 2.24 2.11

10 968.6 39.40 14.42 8.84 6.62 5.46 4.76 4.30 3.96 3.72 3.06 2.77 2.51 2.39 2.32 2.18 2.05

15 984.9 39.43 14.25 8.66 6.43 5.27 4.57 4.10 3.77 3.52 2.86 2.57 2.31 2.18 2.11 1.97 1.83

Numerator degree of freedom 20 993.1 39.45 14.17 8.56 6.33 5.17 4.47 4.00 3.67 3.42 2.76 2.46 2.20 2.07 1.99 1.85 1.71

30 1001 39.46 14.08 8.46 6.23 5.07 4.36 3.89 3.56 3.31 2.64 2.35 2.07 1.94 1.87 1.72 1.59

40 1006 39.47 14.04 8.41 6.18 5.01 4.31 3.84 3.51 3.26 2.59 2.29 2.01 1.88 1.80 1.64 1.48

50 1009 39.48 14.01 8.38 6.14 4.98 4.28 3.81 3.47 3.22 2.55 2.25 1.97 1.83 1.75 1.59 1.43

100 1013 39.49 13.96 8.32 6.03 4.92 4.21 3.74 3.40 3.15 2.47 2.17 1.88 1.74 1.66 1.48 1.30

∞ 1018 39.50 13.90 8.26 6.02 4.85 4.14 3.67 3.33 3.08 2.40 2.09 1.79 1.64 1.55 1.35 1.00

Appendix 121

Denominator degree of freedom

1 2 3 4 5 6 7 8 9 10 15 20 30 40 50 100 ∞

1 4052 98.50 34.12 21.20 16.26 13.75 12.25 11.26 10.56 10.04 8.68 8.10 7.56 7.31 7.17 6.90 6.64

2 4999 99.00 30.82 18.00 13.27 10.92 9.55 8.65 8.02 7.56 6.36 5.85 5.39 5.18 5.06 4.82 4.61

3 5403 99.17 29.46 16.69 12.06 9.78 8.54 7.59 6.99 6.55 5.42 4.94 4.51 4.31 4.20 3.98 3.78

4 5625 99.25 28.71 15.98 11.39 9.15 7.85 7.01 6.42 5.99 4.89 4.43 4.02 3.83 3.72 3.51 3.32

5 5764 99.30 28.24 15.32 10.97 8.75 7.46 6.63 6.06 5.64 4.56 4.10 3.70 3.51 3.05 3.21 3.02

6 5859 99.33 27.91 15.21 10.67 8.47 7.19 6.37 5.80 5.39 4.32 3.87 3.47 3.29 3.19 2.99 2.80

Table A3.3 Values of the F-distribution 1  α ¼ 0.99 7 5928 99.36 27.67 14.98 10.46 8.26 6.99 6.18 5.61 5.20 4.14 3.70 3.30 3.12 3.02 2.82 2.64

8 5981 99.37 27.49 14.80 10.29 8.10 6.84 6.03 5.47 5.06 4.00 3.56 3.17 2.99 2.89 2.69 2.51

9 6022 99.39 27.35 14.66 10.16 7.98 6.72 5.91 5.35 4.94 3.90 3.46 3.07 2.89 2.79 2.59 2.41

10 6056 99.40 27.23 14.55 10.05 7.87 6.62 5.81 5.26 4.85 3.81 3.37 2.98 2.80 2.70 2.50 2.32

15 6157 99.43 26.87 14.20 9.72 7.54 6.31 5.52 4.96 4.56 3.52 3.09 2.70 2.52 2.42 2.22 2.04

Numerator degree of freedom 20 6209 99.45 26.69 14.02 9.55 7.40 6.16 5.36 4.81 4.41 3.37 2.94 2.55 2.37 2.27 2.07 1.88

30 6261 99.47 26.50 13.84 9.38 7.23 5.99 5.20 4.69 4.25 3.21 2.78 2.39 2.20 2.10 1.89 1.70

40 6287 99.47 26.41 13.75 9.29 7.14 5.91 5.12 4.57 4.17 3.13 2.70 2.30 2.11 2.01 1.80 1.59

50 6303 99.48 26.35 13.69 9.24 7.09 5.86 5.07 4.52 4.16 3.08 2.64 2.25 2.06 1.95 1.74 1.52

100 6334 99.49 26.24 13.58 9.13 6.94 5.76 4.96 4.42 4.01 2.98 2.54 2.13 1.94 1.83 1.60 1.36

∞ 6366 99.50 26.13 13.46 9.02 6.88 5.65 4.86 4.31 3.91 2.87 2.42 2.01 1.81 1.69 1.43 1.00

122 Appendix

Appendix

123

A.4. Chi-Squared Distribution Table

Table A4.1 Chi-squared distribution values 1–α f 1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 22 24 26 28 30 40 50 60 70 80 90 100 200 300 400 500

0.900 2.71 4.61 6.25 7.78 9.24 10.64 12.02 13.36 14.68 15.99 18.55 21.06 23.54 25.99 28.41 30.81 33.20 35.56 37.92 40.26 51.81 63.17 74.40 85.53 96.58 107.57 118.50 226.02 331.79 436.65 540.93

0.950 3.84 5.99 7.81 9.49 11.07 12.59 14.07 15.51 16.92 18.31 21.03 23.68 26.30 28.87 31.41 33.92 36.42 38.89 41.34 43.77 55.76 67.50 79.08 90.53 101.88 113.15 124.34 233.99 341.40 447.63 553.13

0.975 5.02 7.38 9.35 11.14 12.83 14.45 16.01 17.53 19.02 20.48 23.34 26.12 28.85 31.53 34.17 36.78 39.36 41.92 44.46 46.98 59.34 71.42 83.30 95.02 106.63 118.14 129.56 241.06 349.87 457.31 563.85

0.990 6.63 9.21 11.34 13.28 15.09 16.81 18.48 20.09 21.67 23.21 26.22 29.14 32.00 34.81 37.57 40.29 42.98 45.64 48.28 50.89 63.69 76.15 88.38 100.43 112.33 124.12 135.81 249.45 359.91 468.72 576.49

0.995 7.88 10.60 12.84 14.86 16.75 18.55 20.28 21.95 23.59 25.19 28.30 31.32 34.27 37.16 40.00 42.80 45.56 48.29 50.99 53.67 66.77 79.49 91.95 104.21 116.32 128.30 140.17 255.26 366.84 476.61 585.21

0.999 10.83 13.82 16.27 18.47 20.52 22.46 24.32 26.12 27.88 29.59 32.91 36.12 39.25 42.31 45.31 48.27 51.18 54.05 56.89 59.70 73.40 86.66 99.61 112.32 124.84 137.21 149.45 267.54 381.43 493.13 603.45

Further Literature

Books 1. Bandemer, H. (ed.): Theorie und Anwendung der optimalen Versuchsplanung. Akademie Verlag, Berlin (1977) 2. Bandemer, H., Bellmann, A.: Statistische Versuchsplanung, 4th edn. Teubner-Verlag, Leipzig (1994) 3. Bleymüller, J., Gehlert, G.: Statistische Formeln, Tabellen und Statistik-Software, 11th edn. Vahlen, Munich (2011) 4. Bortz, J., Lienert, G.A., Boehnke, K.: Verteilungsfreie Methoden in der Biostatistik, 3rd edn. Springer, Heidelberg (2008) 5. Bortz, J., Schuster, C.: Statistik für Human- und Sozialwissenschaftler, 7th edn. Springer, Berlin (2010) 6. Box, G.E.P., Hunter, W.G., Hunter, J.S.: Statistics for experimenters design, innovation and discovery. Wiley, Hoboken, NJ (2005) 7. Büning, H.: Trenkler, Goetz: Nichtparametrische statistische Methoden, 2nd edn. de Gruyter, Berlin (1994) 8. Czado, C., Schmidt, T.: Mathematische Statistik, 1st edn. Springer, Berlin (2011) 9. Dümbgen, L.: Einführung in die Statistik. Springer, Basel (2016) 10. Fahrmeir, L., Künstler, R., Pigeot, I., Tutz, G.: Statistik: Der Weg zur Datenanalyse, 7th edn. Springer, Berlin (2012) 11. Falk, M., Hain, J., Marohn, F., Fischer, H., Michel, R.: Statistik in Theorie und Praxis, 1st edn. Springer, Berlin (2014) 12. Fisz, M.: Wahrscheinlichkeitsrechnung und mathematische Statistik, 11th edn. Dt. Verlag der Wissenschaften, Berlin (1989) 13. Georgii, H.-O.: Stochastik. Einführung in die Wahrscheinlichkeitstheorie und Statistik, 4th edn. Walter de Gruyter, Berlin (2009) 14. Graf, U., Henning, H.-J., Stange, K., Wilrich, P.-T.: Formeln und Tabellen der angewandten mathematischen Statistik, 3rd edn. Springer, Berlin (1998) 15. Hartung, J., Elpelt, B., Klösener, K.-H.: Statistik, 15th edn. Oldenbourg Verlag, Munich (2009) 16. Henze, N.: Stochastik für Einsteiger, 7th edn. Vieweg & Teubner Verlag, Wiesbaden (2008)

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Further Literature

17. Hering, E., Triemel, J., Blank, H.-P. (eds.): Qualitätsmanagement für Ingenieure, 5th edn. Springer, Berlin (2003) 18. Hering, E., Triemel, J., Blank, H.-P. (eds.): Qualitätssicherung für Ingenieure, 5th edn. VDI-Verlag, Düsseldorf (2003) 19. Klein, B.: Versuchsplanung – DoE, 2nd edn. Oldenburg, de Gruyter (2007) 20. Kleppmann, W.: Versuchsplanung. Produkte und Prozesse optimieren, 9th edn. Hanser, Munich (2016) 21. Kohn, W.: Statistik, 1st edn. Springer, Berlin (2005) 22. Liebscher, U.: Anlegen und Auswerten von technischen Versuchen – eine Einführung. FortisVerlag FH (Manz Verlag Schulbuch), Vienna (1999) 23. Mohr, R.: Statistik für Ingenieure und Naturwissenschaftler, 3rd edn. expert Verlag, Renningen (2014) 24. Müller-Funk, U., Wittig, H.: Mathematische Statistik. Teubner Verlag, Stuttgart (1995) 25. Nollau, V.: Statistische Analysen. Mathematische Methoden der Planung und Auswertung von Versuchen, 2nd edn. Basel, Birkhäuser (1979) 26. Papula, L.: Mathematik für Ingenieure und Naturwissenschaftler, vol. Bd. 3, 6th edn. Springer Vieweg, Wiesbaden (2011) 27. Rinne, H., Mittag, H.-J.: Statistische Methoden der Qualitätssicherung, 3rd edn. Hanser, Vienna (1994) 28. Rinne, H.: Taschenbuch der Statistik, 4th edn. Verlag Harri Deutsch, Frankfurt am Main (2008) 29. Rooch, A.: Statistik für Ingenieure. Springer Spektrum, Berlin (2014) 30. Ross, S.M.: Statistik für Ingenieure und Naturwissenschaftler, 3rd edn. Spektrum Akademischer Verlag, Wiesbaden (2006) 31. Sachs, L.: Statistische Methoden 2: Planung und Auswertung. Springer, Berlin (2013) 32. Sachs, L.: Statistische Methoden: Planung und Auswertung, 11th edn. Springer, Berlin (2004) 33. Scheffler, E.: Statistische Versuchsplanung und -auswertung, 3rd edn. Deutscher Verlag für Grundstoffindustrie, Leipzig (1997) 34. Schmeink, L.: Beschreibende Statistik. Verlag Books on Demand, Hamburg (2011) 35. Sieberts, K., van Bebber, D., Hochkirchen, T.: Statistische Versuchsplanung – Design of Experiments (DoE), 1st edn. Springer, Berlin (2010) 36. Storm, R.: Wahrscheinlichkeitsrechnung, mathematische Statistik und statistische Qualitätskontrolle, 12th edn. Hanser, Munich (2007) 37. Vogt, H.: Methoden der Statistischen Qualitätskontrolle. Springer Vieweg, Wiesbaden (1988) 38. Voß, W.: Taschenbuch der Statistik, 1st edn. Fachbuchverlag Leipzig, Leipzig (2000)

Standards 39. DIN 1319: Grundlagen der Messtechnik, Teil 1 bis 4 (Fundamentals of metrology; Part 1–4) 40. DIN 53803: Probenahme, Teil 1–4 (Sampling; Part 1–4) 41. DIN 53804: Statistische Auswertung; Teil 1–4, 13 (Statistical evaluation; Part 1–4, 13) 42. DIN 55303: Statistische Auswertung von Daten, Teil 2, 5, 7 (Statistical interpretation of data; Part 2, 5, 7) 43. DIN 55350: Begriffe der Qualitätssicherung und Statistik, Teil 1, 11–18, 21–24, 31, 33, 34 (Concepts in quality and statistics; Part 1, 11–18, 21–24, 31, 33, 34)

Further Literature

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44. DIN 13303: Stochastik; Teil 1: Wahrscheinlichkeitstheorie, Gemeinsame Grundbegriffe der mathematischen und der beschreibenden Statistik; Begriffe und Zeichen; Teil 2: Mathematische Statistik; Begriffe und Zeichen (Stochastics; Part 1: Probability theory, common fundamental concepts of mathematical and of descriptive statistics; Part 2: Mathematical statistics; concepts, signs and symbols) 45. DIN ISO 2859: Annahmestichprobenprüfung anhand der Anzahl fehlerhafter Einheiten oder Fehler (Attributprüfung) Teil 1 bis 5, 10 (Sampling procedures for inspection by attributes; Part 1–5, 10) 46. DIN ISO 5725: Genauigkeit (Richtigkeit und Präzision) von Meßverfahren und Meßergebnissen, Teil 1–6, 11, 12 (Accuracy (trueness and precision) of measurement methods and results; Part 1–6, 11, 12) 47. DIN ISO 16269: Statistische Auswertung von Daten, Teil 7, 8 (Statistical interpretation of data; Part 7, 8) 48. DIN ISO 18414: Annahmestichprobenverfahren anhand der Anzahl fehlerhafter Einheiten (Acceptance sampling procedures by attributes) 49. ISO 3534: Statistik – Begriffe und Formelzeichen; Teil 1: Wahrscheinlichkeit und allgemeine statistische Begriffe; Teil 2: Angewandte Statistik; Teil 3: Versuchsplanung (Statistics – Vocabulary and symbols; Part 1: General statistical terms and terms used in probability; Part 2: Applied statistics; Part 3: Design of experiments) 50. ISO 3951: Verfahren für die Stichprobenprüfung anhand qualitativer Merkmale (Variablenprüfung) (Sampling procedures for inspection by variables) 51. ISO 5479: Statistische Auswertung von Daten (Statistical interpretation of data) 52. ISO/TR 8550: Leitfaden für die Auswahl und die Anwendung von Annahmestichprobensystemen für die Prüfung diskreter Einheiten in Losen (Guidance on the selection and usage of acceptance sampling systems for inspection of discrete items in lots) 53. VDE/VDI 2620: (nicht mehr gültig), Fortpflanzung von Fehlergrenzen bei Messungen, Blatt 1 und 2 (Propagation of limited errors in measuring – Principles; Sheet 1, 2 (no longer valid))

Index

A

E

Alternative hypotheses, 72–74, 77, 92 Arithmetic means, 22–24, 26, 27, 31, 33–35, 39, 54, 55, 69, 75, 96 Average outgoing quality, 45–50

Empirical covariance, 95–96 Error type I, 72–74 type II, 72, 73 Error calculation, 51–68 Error characteristic fixed-point method, 61 least-squares, 61 tolerance-band method, 62 Error in measured values, 51–54 Error in measurement result due to random errors, 51–54, 59–61 due to systematic errors, 52–53, 55–59 Error limits for measuring chains, 66 Excess, 22, 31–33 Experiment design central composite, 15–20 factorial with center point, 15 fractional factorial, 10, 12, 13 full factorial, 10, 12 orthogonal design, 16, 18 pseudo-orthogonal and rotatable design, 18 rotatable design, 17, 18

B Basic population, 21, 22, 24, 29, 30, 32–35, 38, 45, 72–74, 76, 86 Black box, 1, 2, 4, 95, 99, 100 Blocking, 2–4, 7, 10 Box–Hunter plan, 110, 113

C Cause–effect relationship, 1–3, 14, 99–101, 104 Chi-squared distribution, 81, 84–86, 88, 89, 92, 123 Chi-squared fit/distribution test, 85–90 Chi-squared independence test, 90–93 Chi-squared test, 69, 83–93 Coefficient of determination, 96, 97, 103, 104, 110 Coefficient of variation, 31 Conducting experiments, 1, 4–7 Confidence intervals, 3, 15, 22, 33–38, 54, 59, 79, 80, 115–119 Correlation coefficients, 69, 96–98, 104, 112, 113 Correlations, 4–6, 11, 32, 95–99, 101, 102, 104, 105, 109, 110, 114 Covariance, 2, 95–96

F Fischer, R.A., 80 F-test, 69, 80–83

D Degree of freedom, 21, 34, 38, 77, 84, 85, 89, 90, 92, 118, 119 Design of experiments (DoE) basic principles, 3–4 Dispersion range, 22, 33–38

G Gauss, C.F., 76, 102, 106, 110 Gaussian distribution, 31, 41, 69, 70, 74 Geometric mean, 24–25 Gosset, W.S., 34, 75

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Index

Harmonic mean, 26 Hypotheses, 4, 5, 72–75, 77, 80, 82, 99 Hypotheses on statistical testing, 72–74

Probability of error, 72, 74, 82, 92, 93 Process capability, 42–44 Process controllability, 44 Production variance, 41

I

R

Inaccuracy, 51, 52, 54, 59

Random allocation, 2, 3, 7 Random errors, 51–54, 59–61, 107 Random samples, 3, 21, 29–30, 36, 48 Range coefficient, 28 Ranges, 1–7, 14, 17, 18, 22, 26, 28–29, 33–38, 40, 41, 43, 44, 54, 61–63, 69, 71, 78, 80, 96, 106, 114 Regression linear, 6, 14, 104, 105, 107, 110, 112 multiple linear and nonlinear, 107–109 nonlinear, 6, 105–106 Regression coefficients, 17, 103 Regression function, 6, 14, 16, 103–105 Result Error Limits statistical, 64 Robust arithmetic mean values, 27

L Latin squares, 10–13 Least-squares method, 61, 102 Leibniz, G.W., 99 Level of significance, 72, 73, 86 Linearization, 105–106

M Machine capability, 41–43, 86 Maximum result error limits, 63–64 Means, 1, 3, 6, 8, 21–29, 31, 33, 35, 36, 38, 41–44, 47, 51, 53, 65, 69, 72, 75, 77, 78, 80, 84, 86, 88, 90, 95, 96, 103, 109 α-truncated mean, 27 α-winsorized, 27 Mean values from multiple samples, 27 Measure of dispersion, 21, 28, 34–36 Median, 22, 25, 26, 69 Modal value, 22, 26

S

N

Sample correlation, 96–97 Skewness, 22, 31–33 Standard deviation, 28–30, 40, 43, 53, 54, 70, 75, 76, 86, 96 Standard normal distribution, 70, 71, 76, 86, 115–117, 119 Statistical tests, 1, 69–93 Systematic errors, 51–59

Normal distribution, 21, 22, 31, 33–35, 41, 44, 45, 53, 69, 70, 74, 81, 83, 85, 86, 88, 90 Null hypothesis, 72–74, 77, 78, 85, 86, 91–93

T Transformation, 70, 86, 91, 105, 106 T-test, 69, 72, 74–80, 97

O One-sided test, 34, 72, 74, 77, 78, 82, 92, 93 Operation characteristic curve, 45–50

U Uncertainty, 51, 53, 54, 59–61, 74, 82

P Parameter-bound statistical test, 69–71 Partial correlation, 97–98 Partial correlation coefficient, 97–98

V Variability, 31 Variation range, 29