ISO/TR 20693:2019

TECHNICAL ISO/TR
REPORT 20693
First edition
2019-04
Statistical methods for
implementation of Six Sigma —
Selected illustrations of distribution
identification studies
Méthodes statistiques pour la mise en œuvre du Six Sigma - Exemples
choisis d'études d'identification de la distribution
Reference number
ISO/TR 20693:2019(E)
©
ISO 2019

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ISO/TR 20693:2019(E)

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ISO/TR 20693:2019(E)

Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols and abbreviated terms . 2
5 Basic principles . 3
5.1 General . 3
5.2 Exploratory data analysis (EDA). 4
5.3 Discrete data case . 4
5.3.1 Graphical methods . 4
5.3.2 Numerical methods . 4
5.4 Continuous data case . 5
5.4.1 Graphical methods . 5
5.4.2 Numerical methods . 5
5.4.3 Distribution family unknown and no prior information available . 5
6 General description of distribution identification . 6
6.1 Overview of the structure of distribution identification . 6
6.2 State overall objectives . 6
6.3 Formulate a model theory . 6
6.4 Collect, prepare and explore data . 7
6.5 Select underlying probability distributions . 8
6.6 Perform goodness of fit test . 8
6.7 Draw conclusions . 8
7 Examples . 9
Annex A (informative) Test uniformity in the Super Lotto .10
Annex B (informative) Distribution of the number of technical issues found after product
release to the field.13
Annex C (informative) Software development effort estimation .18
Annex D (informative) Determining the warranty period of a product .26
Bibliography .33
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Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
through ISO technical committees. Each member body interested in a subject for which a technical
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ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of
electrotechnical standardization.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/directives).
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of
any patent rights identified during the development of the document will be in the Introduction and/or
on the ISO list of patent declarations received (see www .iso .org/patents).
Any trade name used in this document is information given for the convenience of users and does not
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.org/iso/foreword .html.
This document was prepared by Technical Committee ISO/TC 69, Applications of statistical methods,
Subcommittee SC 7, Applications of statistical and related techniques for the implementation of Six Sigma.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www .iso .org/members .html.
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Introduction
Many statistical techniques assume that the data to be analysed come from a given distribution (or
population). Such assumptions are crucial to the effectiveness of subsequent statistical inference
methods. In the Six Sigma community, when using such statistical methods, one needs to consider
whether this assumption is reasonable. More generally, sometimes it is interesting and necessary to
find the distribution which generated the data set (or sample) at hand. Identification of the distribution
may provide some ways to answer this question. It consists of finding a distribution (or a family of
distributions) which provides a good representation of a sample.
The distribution identification within Six Sigma projects should ideally be performed before the end
of the Measure phase and can continue throughout the other phases of the DMAIC. From a Six Sigma
perspective, the distribution identification can have multiple purposes based on the considered phase.
It is used, for example, to characterise a baseline of the process performance, during the Measure
or Analyse phase, to characterise the new process during the Improve phase, and to continuously
monitor the process performance during the Control phase to ensure that the change is sustained.
From a statistical perspective, distribution identification may be helpful to find appropriate statistical
techniques for the related data, since many parametric statistical inference methods need certain
distributional assumptions.
In general, distribution identification methods may be used as a tool to:
a) verify that a distribution used historically is still valid for the current data;
b) choose the appropriate distribution.
The choice of appropriate distribution should be guided by the knowledge of physical phenomena or the
business process. It is recommended to start from a tentative theory to avoid just curve fitting.
In practice, there is always certain context or business background which can be used in determining
the distribution. For example, under some circumstance, one can expect the measurement error is
normally distributed. In reliability fields, the life distributions for certain products are exponential,
lognormal, Weibull, or extreme distributions and so on. However, when such knowledge is not available,
the possible underlying distribution for the data should also be identified if one wants to use parametric
statistical methods. In this case, exploratory data analysis methods should be used to gain a better
understanding. Through graphical visualisation methods, one could form a hypothesis on the possible
distributions, stratification of the data or other aspects. Once the hypothesis is formed, hypothesis
testing, including goodness of fit testing, can be applied to check one’s guess. Finally, a suitable
distribution may be found for the data.
1)
1) 1)
In some commercial software packages including MINITAB , SAS-JMP and Q-DAS , although there
are buttons for distribution identification, one should take knowledge of context and process related
to data into consideration instead of simply relying on the software packages. Otherwise, misleading
results can be given.
1)  MINITAB is the trade name of a product supplied by Minitab Inc. JMP is the trade name of a product supplied
by SAS Institute Inc. Q-DAS is the trade name of a product supplied by Q-DAS GmbH. This information is given for the
convenience of users of this document and does not constitute an endorsement by ISO of these products.
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TECHNICAL REPORT ISO/TR 20693:2019(E)
Statistical methods for implementation of Six Sigma —
Selected illustrations of distribution identification studies
1 Scope
This document provides guidelines for the identification of distributions related to the implementation
of Six Sigma. Examples are given to illustrate the related graphical and numerical procedures.
It only considers one dimensional distribution with one mode. The underlying distribution is either
continuous or discrete.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their content
constitutes requirements of this document. For dated references, only the edition cited applies. For
undated references, the latest edition of the referenced document (including any amendments) applies.
ISO 3534-1:2006, Statistics — Vocabulary and symbols — Part 1: General statistical terms and terms used
in probability
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 3534-1 and the following apply.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https: //www .iso .org/obp
— IEC Electropedia: available at https: //www .electropedia .org/
3.1
population
totality of items under consideration
[SOURCE: ISO 3534-1:2006, 1.1, modified - Notes 1, 2, and 3 deleted.]
3.2
sample
subset of a population (3.1) made up of one or more sampling units
[SOURCE: ISO 3534-1:2006, 1.3, modified - Notes 1and 2 deleted.]
3.3
observed value
obtained value of a property associated with one member of a sample (3.2)
[SOURCE: ISO 3534-1:2006, 1.4, modified - Notes 1 and 2 deleted.]
3.4
family of distributions
distribution family
set of probability distributions
[SOURCE: ISO 3534-1:2006, 2.8, modified - Synonym "distribution family" added; Notes 1 and 2 deleted.]
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3.5
p-value
probability of observing the observed test statistic value or any other value at least as unfavourable to
the null hypothesis
[SOURCE: ISO 3534-1:2006, 1.49, modified - Example and Notes 1 and 2 deleted.]
3.6
descriptive statistics
summary statistics that capture information about the shape, centre or spread of a variable or a
distribution
3.7
frequency distribution
empirical relationship between classes and their number of occurrences or observed values (3.3)
[SOURCE: ISO 3534-1:2006, 1.60]
3.8
histogram
graphical representation of the frequency distribution (3.7) of a data set
3.9
boxplot
horizontal or vertical graphical representation of the five-number summary
[SOURCE: ISO 16269-4:2010, 2.16]
3.10
Q-Q plot
scatter plot for theoretical quantiles and empirical quantiles
3.11
goodness of fit test
hypothesis testing on whether the population (3.1)distribution follows a given distribution or belongs
to a distribution family (3.4)
3.12
normality test
hypothesis testing on whether the population (3.1) distribution belongs to a normal distribution
family (3.4)
4 Symbols and abbreviated terms
X , X , ., X sample or observed values or data
1 2 n
2
χ Chi-square distribution or statistics
ALT accelerated life testing
BB black belt
BTS base transceiver station
CLT central limit theorem
CRM customer relationship management
DMAIC Define, Measure, Analyse, Improve and Control
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EDA exploratory data analysis
EDF empirical distribution function
MIS management information systems
pdf probability density function
TP transaction processing
WEB Web/online
5 Basic principles
5.1 General
The identification of distributions consists in finding a distribution (or a family of distributions) which
best represents a sample (or a group of observed data X , X , ., X ). Based on a priori knowledge or
1 2 n
the state of knowledge on the data-generating process, one may possibly know the distribution family
for the data set. In that case, it is easy to verify (confirm or reject) it. Otherwise, it may be somewhat
complicated to perform distribution identification. At that time, one must narrow down the possible
distribution models to a few likely ones. Here are some general guidelines.
a) Apply basic knowledge about the process.
— If theoretical models exist, they should be applied.
— If the process generates discrete data, limit the test to discrete distributions.
— If the process generates only positive number, limit the test only to positive distributions.
b) Apply the Occam’s Razor — favour a simpler model unless evidence supports a more complex model.
— The exponential distribution family is the simplest positive continuous distribution with one
parameter.
— The normal distribution should be favoured as many natural processes follow a normal
distribution.
— The Poisson distribution is among the simplest discrete distribution with one parameter.
In practice, the general flow chart of the procedures for identification of distributions is given in
Figure 1.
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Figure 1 — General flow chart of the procedures for identification of distributions
5.2 Exploratory data analysis (EDA)
EDA is a collection of techniques for revealing information about the data and methods for visualising
them. Its philosophy is that data should first be explored without assumptions about probabilistic
models, distribution, etc. For one dimensional data, one can consider the following tools.
Descriptive statistics: Mean, standard deviation, skewness, kurtosis, median, max, min, quartile, inter-
quantile range, and range are commonly used. Such statistics give the summary values of the data. Some
information about the distribution can be derived from them. For example, whether the distribution is
symmetric or not. It will be more clearly illustrated by visual tools such as a histogram and a boxplot.
A histogram (or stem-and-leaf plot) is a way to graphically represent the frequency distribution of a
data set. Though the graphical shape of a histogram may be affected by the different width of bins, the
presence of multi-modal behaviour can always be seen from it. The boxplot is another way to display the
distribution of a sample. It may provide insights on skewness, behaviour in the tails, and the presence
of outliers. The Q-Q plot can be used to check normality, or more generally a location-scale distribution
family, or whether two data sets come from the same distribution family.
Since there are some differences between methods of distribution identification for discrete data and
for continuous data, the two cases are treated separately in 5.3 and 5.4.
5.3 Discrete data case
5.3.1 Graphical methods
The barplot and histogram can be used for data generated from discrete distribution.
5.3.2 Numerical methods
2
As a general goodness of fit test statistic, the Pearson χ statistic can be used to test whether the data
set comes from certain discrete distributions.
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5.4 Continuous data case
5.4.1 Graphical methods
Besides the histogram and boxplot, the Q-Q plot can often be used to graphically check whether the
population distribution belongs to a location-scale family. More generally, the Q-Q plot is also used to
check whether two groups of data come from the same family of distributions.
5.4.2 Numerical methods
5.4.2.1 Regression method
If the Q-Q plot is nearly linear, the hypothesis about the population distribution can be accepted. To
estimate the linearity of the Q-Q plot and evaluate the strength of this linear relationship, regression
method may be used. This method is mainly used for testing the location-scale distribution family.
Roughly speaking, by considering regression of order statistic on the corresponding population quantile
or the expectation of standardised order statistic, correlation coefficient between the dependent
variable and the predictor will be used to measure the strength of the linearity in some extent. More
rigidly, generalised least squares estimation can be used. In this way, it can be used for testing the
uniform distribution, normal distribution, exponential distribution, extreme distributions and logistic
distribution. One can refer to [3] for more details.
5.4.2.2 Formal hypothesis testing methods
2
a) χ -type test
2 2 2
χ -type test statistics include the Pearson χ statistic, likelihood ratio statistic, Neyman modified χ
[1],[2]
statistic, Freeman-Tukey statistic, Class of power divergence statistics and so on.
b) EDF-type test
The Kolmogorov-Smirnov (K-S) test is one of test statistics based on empirical distribution function
(EDF). There are still some other complicated test statistics such as supremum EDF type with power
[6] [3]
divergence weight statistics , Cramer-von Mises type statistics , etc.
c) Special test for normality
Because of its special importance in statistics, the test for normality is widely studied in literature.
There are many test statistics for normality testing. Some of them can be found in ISO 5479:1997. The
following list just names a few of them.
a) Testing on skewness or kurtosis (or both at the same time).
b) Shapiro-Wilk test (also known as Shapiro-Francia test).
c) Anderson-Darling test: a modification of Kolmogorov-Smirnov test.
d) Jarque-Bera test or Adjusted Jarque-Bera test.
e) Epps-Pulley test.
f) Cramer-Von Mises test.
g) Kolmogorov-Smirnov test.
5.4.3 Distribution family unknown and no prior information available
In the above, it is supposed that the possible distribution families for the data are known in some
way. Graphical and numerical methods are provided to verify or disprove it. In some cases, there is
no prior knowledge available about the distribution type of the data set. Except for the EDA method,
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data transformation techniques such as the Box-Cox or Johnson transformations can be taken into
consideration. Both graphical and statistical testing methods of identifying distribution may be used
for the transformed data. If it is still hard to identify a good distribution for the data set, the density
estimation methods may be invoked, which belongs to the nonparametric tools. When the kernel density
estimation method, which is a generalisation of histogram estimate method, is chosen, the result is also
affected by the different bandwidths used. The estimated probability density function (pdf) seldom
agrees with a simple known distribution (such as normal, student t and so on). Thus it may not easy to
perform subsequent data analysis in Six Sigma.
6 General description of distribution identification
6.1 Overview of the structure of distribution identification
This document provides general guidelines or principles on distribution identification and illustrates
the steps with distinct applications given in Annexes A through D. Each of these examples follows the
basic structure given in Table 1.
Table 1 — Basic steps for distribution identification
1 State overall objectives
2 Formulate a model theory
3 Collect, prepare and explore data
4 Select underlying probability distributions
5 Perform goodness of fit test
6 Draw conclusions
The steps given in Table 1 provide a general technique and procedures for distribution identification
and how they dovetail with the Six Sigma roadmap (e.g. DMAIC). Each of the six steps of the procedures
in Table 1 is explained in detail in 6.2 to 6.7.
6.2 State overall objectives
Distribution identification is implemented within the Six Sigma project ideally before the end of the
Measure phase and can continue throughout the other phases of the DMAIC.
By the end of Define and Measure phases, the Six Sigma project team has a clear definition of the
problem, the improvement objectives and description of the process structure under study and its
scope. The problem is often related to the process performance which is described qualitatively and
quantitatively by the end of Measure phase. This will lead to a set of measures. These measures may
require identification of the probability distribution for performing further analyses (e.g. capability
analysis).
The Six Sigma project team should link the project objectives and process structure, by which the
data are generated or will be generated, to the motivation for performing the probability distribution
identification. This may be refined or revisited during the following phases of DMAIC as required or
appropriate.
6.3 Formulate a model theory
Starting from the objectives and the process by which the data are generated, will help form a tentative
theory. This is motivated by W. E. Deming saying “Theory comes first”, so avoid simply curve fitting
and instead use the understanding of the data generating process and its structure to identify the most
appropriate distribution.
In a sense, there are some natural or physical phenomena that can be modelled by more appropriate
probability distributions. Similarly, some probability distributions may be more convenient as they
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make less assumptions in terms of parameters (process structure). In this case, use the parsimony
argument: fewer parameters are generally better.
Other information, context or knowledge of data generating process such as data type (e.g. categorical,
discrete or continuous) will have an impact on the selection of the possible potential distribution
families.
One should not solely rely on the data for concluding its type or identifying the distribution, without
referring to the context and the generating process, as this can be misleading. For example, one can
conclude that a given data set is discrete, whilst in reality the values have been rounded due to the
measurement system or by a transformation from one format to another.
Table 2 below lists some of the most common distributions and the motivational theories behind them.
Table 2 — Common distributions and their underlying motivations
Candidate probability
Model theory Justification
distributions
Physical wear out Weibull Flexible distribution
Central limit theorem (CLT), Simple law,
Aggregation Normal appropriate for one dimensional physical
measures (e.g. weight, length)
Multiplication Lognormal Log of product = sum, CLT
Minimum or
Extreme value Asymptotic
maximum
Random
Exponential Memoryless
occurrence times
Random Approximation for binomial and also suitable
Poisson
occurrence counts for (rare) defects per unit
Processes made up
Gamma Poisson, negative binomial Processes made up of sub-processes
of sub-processes
Weibull, lognormal, gamma and
Flexible distribution
Process with
log-logistic
natural lower
For practical reasons due to the ease and/or
boundary as zero, Normal
availability of statistical tools.
e.g. cost, failure
Folded normal distribution, half-normal
times
Truncated distribution
distribution
Weibull, lognormal, gamma and
Process has a natural Distributions with location shift from zero,
log-logistic with a third parameter
lower boundary which and the threshold is only used if physically
representing the threshold or minimum
is not zero relevant.
value are candidates
Process generating
Normal or logistic Central limit theorem (CLT)
symmetric data
NOTE The table is not comprehensive and further justification can be found in other publications.
6.4 Collect, prepare and explore data
This section describes the necessary steps for collecting, characterising, categorising, cleaning and
contextualising the data to enable its analysis.
The data may be generated by the process, as defined during the Define and Measure phases or may be
gathered from a designed experiment.
After collecting the data, it is highly recommended to check it for completeness (non-missing values),
errors or outliers, stability since these types of anomalies may distort the identification of distributions.
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For missing data, one should decide whether to use imputation methods. The erroneous data must be
removed or corrected, whilst for outlier detection and treatment one can refer to ISO 16269-4. Control
charts are suitable for stability check.
One should use EDA methods for visualising data,
...