ISO/TR 16705:2016

Statistical methods for implementation of Six Sigma — Selected illustrations of contingency table analysis

ISO/TR 16705:2016 describes the necessary steps for contingency table analysis and the method to analyse the relation between categorical variables (including nominal variables and ordinal variables). It provides examples of contingency table analysis. Several illustrations from different fields with different emphasis suggest the procedures of contingency table analysis using different software applications. In ISO/TR 16705:2016, only two-dimensional contingency tables are considered.

Méthodes statistiques pour l'implémentation de Six Sigma — Exemples sélectionnés d'application de l'analyse de tableau de contingence

General Information

• Status: Published
• Publication Date: 28-Jul-2016
• ICS: 03.120.30 - Application of statistical methods
• Technical Committee: ISO/TC 69/SC 7 - Applications of statistical and related techniques for the implementation of Six Sigma
• Drafting Committee: ISO/TC 69/SC 7/WG 2 - Process measurement and measurement capability
• Current Stage: 6060 - International Standard published
• Start Date: 29-Jul-2016
• Due Date: 07-Jan-2014
• Completion Date: 07-Jan-2014

Standards Content (Sample)

TECHNICAL ISO/TR
REPORT 16705
First edition
Statistical methods for
implementation of Six Sigma —
Selected illustrations of contingency
table analysis
Méthodes statistiques pour l’implémentation de Six Sigma —
Exemples sélectionnés d’application de l’analyse de tableau de
contingence
PROOF/ÉPREUVE
Reference number
©
ISO 2016
© ISO 2016, Published in Switzerland
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized otherwise in any form
or by any means, electronic or mechanical, including photocopying, or posting on the internet or an intranet, without prior
written permission. Permission can be requested from either ISO at the address below or ISO’s member body in the country of
the requester.
ISO copyright office
Ch. de Blandonnet 8 • CP 401
CH-1214 Vernier, Geneva, Switzerland
Tel. [phone]
Fax [phone]
[e-mail]
www.iso.org
ii © ISO 2016 – All rights reserved

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 General description of contingency table analysis . 2
5.1 Overview of the structure of contingency table analysis . 2
5.2 Overall objectives of contingency table analysis. 3
5.3 List attributes of interest . 3
5.4 State a null hypothesis . 3
5.5 Sampling plan. 3
5.6 Process and analyse data . 4
5.6.1 Chi-squared test. 4
5.6.2 Linear trend test . 6
5.6.3 Correspondence analysis . 6
5.7 Conclusions . 7
6 Description of Annexes A through D . 7
Annex A (informative) Distribution of number of technical issues found after product
release to the field. 8
Annex B (informative) People’s perception about contented life .15
Annex C (informative) Customer satisfaction research on a brand of beer .20
Annex D (informative) Proportions of nonconforming parts of production lines .26
Bibliography .31
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
committee has been established has the right to be represented on that committee. International
organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.
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
constitute an endorsement.
For an explanation on the meaning of ISO specific terms and expressions related to conformity assessment,
as well as information about ISO’s adherence to the World Trade Organization (WTO) principles in the
Technical Barriers to Trade (TBT) see the following URL: www.iso.org/iso/foreword.html.
The committee responsible for this document is ISO/TC 69, Applications of statistical methods,
Subcommittee SC 7, Applications of statistical and related techniques for the implementation of Six Sigma.
iv PROOF/ÉPREUVE © ISO 2016 – All rights reserved

Introduction
The Six Sigma and international statistical standards communities share a philosophy of continuous
improvement and many analytical tools. The Six Sigma community tends to adopt a pragmatic approach
driven by time and resource constraints. The statistical standards community arrives at rigorous
documents through long-term international consensus. The disparities in time pressures, mathematical
rigor, and statistical software usage have inhibited exchanges, synergy, and mutual appreciation
between the two groups.
The present document takes one specific statistical tool (Contingency Table Analysis), develops the
topic somewhat generically (in the spirit of International Standards), then illustrates it through the
use of several detailed and distinct applications. The generic description focuses on the commonalities
across studies designed to assess the association of categorical variables.
The Annexes containing illustrations do not only follow the basic framework, but also identify the
nuances and peculiarities in the specific applications. Each example will offer at least one “winkle” to
the problem, which is generally the case for real Six Sigma and other fields application.
TECHNICAL REPORT ISO/TR 16705:2016(E)
Statistical methods for implementation of Six Sigma —
Selected illustrations of contingency table analysis
1 Scope
This document describes the necessary steps for contingency table analysis and the method to analyse
the relation between categorical variables (including nominal variables and ordinal variables).
This document provides examples of contingency table analysis. Several illustrations from different
fields with different emphasis suggest the procedures of contingency table analysis using different
software applications.
In this document, only two-dimensional contingency tables are considered.
2 Normative references
There are no normative references in this document.
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 3534-1 and ISO 3534-2 and
the following apply.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— IEC Electropedia: available at http://www.electropedia.org/
— ISO Online browsing platform: available at http://www.iso.org/obp
3.1
categorical variable
variable with the measurement scale consisting of a set of categories
3.2
nominal data
variable with a nominal scale of measurement
[SOURCE: ISO 3534-2:2006, 1.1.6]
3.3
ordinal data
variable with an ordinal scale of measurement
[SOURCE: ISO 3534-2:2006, 1.1.7]
3.4
contingency table
tabular representation of categorical data, which shows frequencies for particular combinations of
values of two or more discrete random variables
Note 1 to entry: A table that cross-classifies two variables is called a “two-way contingency table;” the one that
cross-classifies three variables is called a “three-way contingency table.” A two-way table with r rows and c
columns is also named “r × c table.”
EXAMPLE Let n items be classified by categorical variables X and Y with levels X , X and Y , Y , respectively.
1 2 1 2
The number of items with both attribute X and Y is n . Then, a 2 × 2 table is as follows.
i j ij
Table 1 — 2 × 2 contingency table
Variable Y
Variable X
Y Y
1 2
X n n
1 11 12
X n n
2 21 22
3.5
p-value
probability of observing the observed test statistic value or any other value at least as unfavorable to
the null hypothesis
[SOURCE: ISO 3534-1:2006, 1.49]
4 Symbols and abbreviated terms
H null hypothesis
H alternative hypothesis
a
Chi-square statistic
χ
G likelihood-ratio statistic
n total number of cell count
r × c table contingency table with r rows and c columns
DF degree of freedom
5 General description of contingency table analysis
5.1 Overview of the structure of contingency table analysis
This document provides general guidelines on the design, conduct, and analysis of contingency table
analysis and illustrates the steps with distinct applications given in Annexes A through D. Each of these
examples follows the basic structure given in Table 2.
Table 2 — Basic steps for contingency table analysis
1 State the overall objective
2 List attributes of interest
3 State a null hypothesis
4 Sampling plan
5 Process and analyse data
6 Accept or reject the null hypothesis (Conclusions)
Contingency table analysis is used to assess the association of two or more categorical variables. This
document focuses on two-way contingency table analysis, which only considers the relation of two
categorical variables. Particular methods for three or more categorical variables analysis are not
included in this document. The steps given in Table 1 provide general techniques and procedures for
contingency table analysis. Each of the six steps is explained in general in 5.2 to 5.7.
2 PROOF/ÉPREUVE © ISO 2016 – All rights reserved

5.2 Overall objectives of contingency table analysis
1)
Contingency table analysis can be employed in Six Sigma projects in the “Analyse” phase of DMAIC
methodologies, and often used in sampling survey, social science and medical research, etc. Apart
from the usual statistical methods focusing on continuous variables, contingency table analysis mainly
handles the categorical data, including nominal data and ordinal data. In the case that the observed
value is the frequency of certain combinations of several objective conditions, but not the continuous
value from the equipment, the contingency table analysis is needed.
The primary motivation of this method is to test the association of categorical variables, including the
following situations:
a) to assess whether an observed frequency distribution differs from a theoretical distribution;
b) to assess the independence of two categorical variables;
c) to assess the homogeneity of several distributions of same type;
d) to assess the trend association of observations on ordinal variables;
e) to assess extensive association between levels of categorical variables.
5.3 List attributes of interest
This document considers the association of two categorical variables based on the observed frequency
of the characteristic corresponding to combinations of different levels of attributes of interest.
If the association between quantitative variable and categorical variable is of interest (e.g. cup size
versus surface decoration), it is necessary to divide quantitative data into ordinal classes (e.g. small,
medium, large).
5.4 State a null hypothesis
This document is to determine whether row variable and column variable are independent. The null
hypothesis for Chi-square test is
H : the row variable and column variabl
...

TECHNICAL ISO/TR
REPORT 16705
First edition
2016-08-15
Statistical methods for
implementation of Six Sigma —
Selected illustrations of contingency
table analysis
Méthodes statistiques pour l’implémentation de Six Sigma —
Exemples sélectionnés d’application de l’analyse de tableau de
contingence
Reference number
©
ISO 2016
© ISO 2016, Published in Switzerland
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized otherwise in any form
or by any means, electronic or mechanical, including photocopying, or posting on the internet or an intranet, without prior
written permission. Permission can be requested from either ISO at the address below or ISO’s member body in the country of
the requester.
ISO copyright office
Ch. de Blandonnet 8 • CP 401
CH-1214 Vernier, Geneva, Switzerland
Tel. [phone]
Fax [phone]
[e-mail]
www.iso.org
ii © ISO 2016 – All rights reserved

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 General description of contingency table analysis . 2
5.1 Overview of the structure of contingency table analysis . 2
5.2 Overall objectives of contingency table analysis. 3
5.3 List attributes of interest . 3
5.4 State a null hypothesis . 3
5.5 Sampling plan. 3
5.6 Process and analyse data . 4
5.6.1 Chi-squared test. 4
5.6.2 Linear trend test . 6
5.6.3 Correspondence analysis . 6
5.7 Conclusions . 7
6 Description of Annexes A through D . 7
Annex A (informative) Distribution of number of technical issues found after product
release to the field. 8
Annex B (informative) People’s perception about contented life .15
Annex C (informative) Customer satisfaction research on a brand of beer .20
Annex D (informative) Proportions of nonconforming parts of production lines .26
Bibliography .31
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
committee has been established has the right to be represented on that committee. International
organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.
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
constitute an endorsement.
For an explanation on the meaning of ISO specific terms and expressions related to conformity assessment,
as well as information about ISO’s adherence to the World Trade Organization (WTO) principles in the
Technical Barriers to Trade (TBT) see the following URL: www.iso.org/iso/foreword.html.
The committee responsible for this document is ISO/TC 69, Applications of statistical methods,
Subcommittee SC 7, Applications of statistical and related techniques for the implementation of Six Sigma.
iv © ISO 2016 – All rights reserved

Introduction
The Six Sigma and international statistical standards communities share a philosophy of continuous
improvement and many analytical tools. The Six Sigma community tends to adopt a pragmatic approach
driven by time and resource constraints. The statistical standards community arrives at rigorous
documents through long-term international consensus. The disparities in time pressures, mathematical
rigor, and statistical software usage have inhibited exchanges, synergy, and mutual appreciation
between the two groups.
The present document takes one specific statistical tool (Contingency Table Analysis), develops the
topic somewhat generically (in the spirit of International Standards), then illustrates it through the
use of several detailed and distinct applications. The generic description focuses on the commonalities
across studies designed to assess the association of categorical variables.
The Annexes containing illustrations do not only follow the basic framework, but also identify the
nuances and peculiarities in the specific applications. Each example will offer at least one “winkle” to
the problem, which is generally the case for real Six Sigma and other fields application.
TECHNICAL REPORT ISO/TR 16705:2016(E)
Statistical methods for implementation of Six Sigma —
Selected illustrations of contingency table analysis
1 Scope
This document describes the necessary steps for contingency table analysis and the method to analyse
the relation between categorical variables (including nominal variables and ordinal variables).
This document provides examples of contingency table analysis. Several illustrations from different
fields with different emphasis suggest the procedures of contingency table analysis using different
software applications.
In this document, only two-dimensional contingency tables are considered.
2 Normative references
There are no normative references in this document.
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 3534-1 and ISO 3534-2 and
the following apply.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— IEC Electropedia: available at http://www.electropedia.org/
— ISO Online browsing platform: available at http://www.iso.org/obp
3.1
categorical variable
variable with the measurement scale consisting of a set of categories
3.2
nominal data
variable with a nominal scale of measurement
[SOURCE: ISO 3534-2:2006, 1.1.6]
3.3
ordinal data
variable with an ordinal scale of measurement
[SOURCE: ISO 3534-2:2006, 1.1.7]
3.4
contingency table
tabular representation of categorical data, which shows frequencies for particular combinations of
values of two or more discrete random variables
Note 1 to entry: A table that cross-classifies two variables is called a “two-way contingency table;” the one that
cross-classifies three variables is called a “three-way contingency table.” A two-way table with r rows and c
columns is also named “r × c table.”
EXAMPLE Let n items be classified by categorical variables X and Y with levels X , X and Y , Y , respectively.
1 2 1 2
The number of items with both attribute X and Y is n . Then, a 2 × 2 table is as follows.
i j ij
Table 1 — 2 × 2 contingency table
Variable Y
Variable X
Y Y
1 2
X n n
1 11 12
X n n
2 21 22
3.5
p-value
probability of observing the observed test statistic value or any other value at least as unfavorable to
the null hypothesis
[SOURCE: ISO 3534-1:2006, 1.49]
4 Symbols and abbreviated terms
H null hypothesis
H alternative hypothesis
a
Chi-square statistic
χ
G likelihood-ratio statistic
n total number of cell count
r × c table contingency table with r rows and c columns
DF degree of freedom
5 General description of contingency table analysis
5.1 Overview of the structure of contingency table analysis
This document provides general guidelines on the design, conduct, and analysis of contingency table
analysis and illustrates the steps with distinct applications given in Annexes A through D. Each of these
examples follows the basic structure given in Table 2.
Table 2 — Basic steps for contingency table analysis
1 State the overall objective
2 List attributes of interest
3 State a null hypothesis
4 Sampling plan
5 Process and analyse data
6 Accept or reject the null hypothesis (Conclusions)
Contingency table analysis is used to assess the association of two or more categorical variables. This
document focuses on two-way contingency table analysis, which only considers the relation of two
categorical variables. Particular methods for three or more categorical variables analysis are not
included in this document. The steps given in Table 1 provide general techniques and procedures for
contingency table analysis. Each of the six steps is explained in general in 5.2 to 5.7.
2 © ISO 2016 – All rights reserved

5.2 Overall objectives of contingency table analysis
1)
Contingency table analysis can be employed in Six Sigma projects in the “Analyse” phase of DMAIC
methodologies, and often used in sampling survey, social science and medical research, etc. Apart
from the usual statistical methods focusing on continuous variables, contingency table analysis mainly
handles the categorical data, including nominal data and ordinal data. In the case that the observed
value is the frequency of certain combinations of several objective conditions, but not the continuous
value from the equipment, the contingency table analysis is needed.
The primary motivation of this method is to test the association of categorical variables, including the
following situations:
a) to assess whether an observed frequency distribution differs from a theoretical distribution;
b) to assess the independence of two categorical variables;
c) to assess the homogeneity of several distributions of same type;
d) to assess the trend association of observations on ordinal variables;
e) to assess extensive association between levels of categorical variables.
5.3 List attributes of interest
This document considers the association of two categorical variables based on the observed frequency
of the characteristic corresponding to combinations of different levels of attributes of interest.
If the association between quantitative variable and categorical variable is of interest (e.g. cup size
versus surface decoration), it is necessary to divide quantitative data into ordinal classes (e.g. small,
medium, large).
5.4 State a null hypothesis
This document is to determine whether row variable and column variable are independent. The null
hypothesis for Chi-square test is
H : the row variable and column variable are independent;
and the alternative hypothesis is
H : the row v
...