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
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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
vii
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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.3
Dispersion Range and Confidence Interval . . . . . . . . . . . . . . . . . . .
2.3.1 Dispersion Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Confidence Interval (Confidence Range) . . . . . . . . . . . . . . .
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6
Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1 Covariance, Empirical Covariance . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Correlation (Sample Correlation), Empirical Correlation Coefficient .
6.3 Partial Correlation Coefficient, Partial Correlation . . . . . . . . . . . . . .
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1 Cause–Effect Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Nonlinear Regression (Linearization) . . . . . . . . . . . . . . . . . . . . . . .
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7.4 Multiple Linear and Nonlinear Regression . . . . . . . . . . . . . . . . . . . . .
7.5 Examples of Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
107
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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
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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
xiii
xiv
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
1
2
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
4
1
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.
8
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
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
1.33
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 ¼ ∂y
∂y
∂y
∂y
∙ G1 þ
∙ G2 þ . . . þ
∙ Gi þ . . . þ
∙ Gk
∂x1
∂x2
∂xi
∂xk
The relative maximum error limit of the result
Gym
y
is:
Gym
∂y G1
∂y G2
∂y Gi
∂y Gk
∙
∙
∙
∙
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:
Gym
∂y x1 G1
∂y x2 G2
∂y xi Gi
∂y xk Gk
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
Gym
¼
y
G1
G
þ 2
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)
1
2 ∙ R3 þ 2 ∙ R1
∂G
∂G
G¼
¼
; GRm ¼ ∙ G1 þ
∙ G3
RG
R1 ∙ R3
∂R1
∂R3
GRm ¼ 2
2
2 ∙ G1 þ 2 ∙ G3
R1
R3
¼ 0:0015 Ω1
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:
Gym
¼
y
GK 1
G
GK i
þ K2 þ . . . þ
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
5
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
95
96
6
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
99
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)
# 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
125
126
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
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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
127
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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130
H
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