SIST ISO 22514-6:2014

INTERNATIONAL ISO
STANDARD 22514-6
First edition
2013-02-15
Statistical methods in process
management — Capability and
performance —
Part 6:
Process capability statistics
for characteristics following a
multivariate normal distribution
Méthodes statistiques dans la gestion des processus — Capabilité et
performance —
Partie 6: Statistiques de capabilité pour un processus caractérisé par
une distribution normale multivariée
Reference number
ISO 22514-6:2013(E)
©
ISO 2013

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ISO 22514-6:2013(E)

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ISO 22514-6:2013(E)

Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Abbreviated terms . 3
5 Process analysis . 4
6 Use of multivariate process capability and performance assessment .4
7 Calculation of process capability and process performance . 4
7.1 Description of Types I and II . 4
7.2 Designation and symbols of the indices . 5
7.3 Types Ιc and ΙΙc process capability index . 8
7.4 Types ΙΙa and Type ΙΙb process capability index .10
8 Examples .11
8.1 Two-dimensional position tolerances .11
8.2 Position and dimension of a slot .16
Annex A (informative) Derivation of formulae .20
Annex B (informative) Shaft imbalance example .25
Annex C (informative) Hole position example .29
Annex D (informative) Construction of the quality function .33
Bibliography .34
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ISO 22514-6:2013(E)

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.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International
Standards adopted by the technical committees are circulated to the member bodies for voting.
Publication as an International Standard requires approval by at least 75 % of the member bodies
casting a vote.
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.
ISO 22514-6 was prepared by Technical Committee ISO/TC 69, Applications of statistical methods,
Subcommittee SC 4, Applications of statistical methods in process management.
ISO 22514 consists of the following parts, under the general title Statistical methods in process
management — Capability and performance:
— Part 1: General principles and concepts
— Part 2: Process capability and performance of time-dependent process models
— Part 3: Machine performance studies for measured data on discrete parts
— Part 4: Process capability estimates and performance measures [Technical Report]
— Part 5: Process capability statistics for attribute characteristics
— Part 6: Process capability statistics for characteristics following a multivariate normal distribution
— Part 7: Capability of measurement processes
— Part 8: Machine performance of a multi-state production process
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ISO 22514-6:2013(E)

Introduction
Due to the increased complexity of the production methods and the increasing quality requirements for
products and processes, a process analysis based on univariate quantities is in many cases not sufficient.
Instead, it may be necessary to analyse the process on the basis of multivariate product quantities. This
can, for instance, be in such cases where geometric tolerances, dynamic magnitudes such as imbalance,
correlated quantities of materials or other procedural products are observed.
By analogy with ISO 22514-2, ISO 22514-6 provides calculation formulae for process performance and
process capability indices, which take into consideration process dispersion as well as process location
as an extension to the corresponding indices for univariate quantities. The indices proposed are indeed
based on the classical C and C indices for the one-dimensional case. The motivation for the extension
p pk
to the multivariate case is explained in Annex A.
Examples of possible applications are two-dimensional or three-dimensional positions, imbalance or
several correlated quantities of chemical products.
The dispersion of the measuring results comprises the dispersion of the product realization process and
the precision of the measuring process. It is assumed that the capability of the used measuring system
was demonstrated prior to the determination of the capability of the product realization process.
The calculation method described here should be used to support an unambiguous decision, especially if
— limiting values for process capability indices for multivariate, continuous product quantities are
specified as part of a contract between customers and suppliers, or
— the capabilities of different constructions, production methods or suppliers are to be compared, or
— production processes are to be approved, or
— problems are to be analysed and decisions made in complaint cases or damage events.
NOTE Product realization processes include e.g. manufacturing processes, service processes, product
assembly processes.
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INTERNATIONAL STANDARD ISO 22514-6:2013(E)
Statistical methods in process management — Capability
and performance —
Part 6:
Process capability statistics for characteristics following a
multivariate normal distribution
1 Scope
This part of ISO 22514 provides methods for calculating performance and capability statistics for process
or product quantities where it is necessary or beneficial to consider a family of singular quantities in
relation to each other. The methods provided here mostly are designed to describe quantities that follow
a bivariate normal distribution.
NOTE In principle, this part of ISO 22514 can be used for multivariate cases.
This part of ISO 22514 does not offer an evaluation of the different provided methods with respect to
different situations of possible application of each method. For the current state, the selection of one
preferable method might be done following the users preferences.
The purpose is to give definitions for different approaches of index calculation for performance and
capability in the case of a multiple process or product quantity description.
2 Normative references
The following documents, in whole or in part, are normatively referenced in this document and are
indispensable for its application. For dated references, only the edition cited applies. For undated
references, the latest edition of the referenced document (including any amendments) applies.
ISO 22514-1, Statistical methods in process management — Capability and performance — Part 1: General
principles and concepts
ISO 22514-2, Statistical methods in process management — Capability and performance — Part 2: Process
capability and performance of time-dependent process models
3 Terms and definitions
For the purpose of this document, the terms and definitions given in ISO 22514-1 and ISO 22514-2 and
the following apply.
3.1
quantity
property of a phenomenon, body, or substance, where the property has a magnitude that can be expressed
as a number and a reference
[ISO/IEC Guide 99:2007, 1.1]
3.2
multivariate quantity
set of distinguishing features
Note 1 to entry: The set can be expressed by a d-tuple, i.e. an ordered set consisting of d elements.
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ISO 22514-6:2013(E)

Note 2 to entry: If the single quantities in the set are denoted by x where i = 1, 2…d, the multivariate quantity is
i
T
expressed as the vector x = (x , x , … x ) . Thus, a multivariate quantity can be considered as a feature vector of
1 2 d
a product. The value of the multivariate quantity is represented by a point in the d-dimensional feature space.
Note 3 to entry: The selection of the quantities in a vector is made for specific technical reason.
Note 4 to entry: All single quantities combined in the vector of a multivariate must be measurable in the same
product or object.
Note 5 to entry: If the multivariate quantity is to be described by means of statistics, the vector is to be considered
as a random vector following a d-dimensional multivariate distribution.
EXAMPLE 1 A number of d = 3 quantities like x = colour, x = mass and x = number of defects are combined in
1 2 3
order to use only one statistic for process assessment. The dimension of vector x is d = 3.
EXAMPLE 2 In order to evaluate a boring process, the position of the borehole axis is measured in an
x-coordinate and y-coordinate. The coordinates are combined to the two-dimensional multivariate quantity x
where the component x is the x-coordinate and x is the y-coordinate.
1 2
EXAMPLE 3 Imbalance of a wheel.
3.3
tolerance region
region in the feature space that contains all permitted values of the multivariate quantity (3.2)
Note 1 to entry: The region is limited by lines, surfaces or hyper-surfaces in the d-dimensional space and not
necessarily closed. The form and extension of the region are specified by one or more parameters.
Note 2 to entry: Typical shapes of tolerance regions are rectangles, ellipses (or circles) in the two-dimensional
case, cuboids or hyper-cuboids, ellipsoids or hyper-ellipsoids or composite prismatic shapes. Figure 1 shows
examples of tolerance regions in the two-dimensional space.
Note 3 to entry: The tolerance region is specified based on the required function of the product. Products showing
values outside the region are assumed to not fulfil functional requirements. Those products are considered to be
nonconforming parts.
Note 4 to entry: In order to assess a product with respect to the limits of the tolerance region, the order of the
single quantity in the multivariate quantity and the number d of dimension must be equal to that of the tolerance
region description.
EXAMPLE A tolerance zone as it is defined in ISO 1101 for geometrical product features can be considered as
a tolerance region. In that case, limiting geometrically perfect lines or surfaces correspond to the boundary and
the tolerance correspond to the parameter of the tolerance region.
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ISO 22514-6:2013(E)

A) B) C)
a a
d
1 1
x x
x
X
2
X
1
Key
A rectangular tolerance region with parameters a , a , x and y
1 2
B circular tolerance region with parameters d, x and y
C triangularly extended rectangular region with parameters a , a , b, x and y
1 2
Figure 1 — Examples of tolerance regions in the two-dimensional space of the bivariate
T
quantity (x , x )
1 2
3.4
process capability
distribution of measured quantity (3.1) values from a process that has been demonstrated to be in
statistical control and which describes the ability of a process to produce quantity values that will fulfil
the requirements for that quantity
Note 1 to entry: The process capability index provides the ability to meet requirements of the measured quantity.
Note 2 to entry: The abbreviation for process capability index is PCI.
3.5
estimated process capability
statistical description of a process capability (3.4)
3.6
process performance
distribution of measured quantity (3.1) values from a process
Note 1 to entry: The process may not have been demonstrated to be in statistical control.
3.7
estimated process performance
statistical description of a process performance (3.6)
4 Abbreviated terms
MMC maximum material condition
PCI process capability index
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a
2
y
y
y
a
2
b

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ISO 22514-6:2013(E)

5 Process analysis
The purpose of process analysis is to obtain sound knowledge of a process. This knowledge is necessary for
controlling the process efficiently, so that the products realized by the process fulfil the quality requirement.
A process analysis is always an analysis of one or more quantities that are considered to be important
to the process.
Product quantities can often be analysed instead of process quantities because product quantities not
only characterize the products, but due to their correlation with process quantities they also characterize
the process creating these products.
The values of the quantities under consideration are typically determined on the basis of samples
taken from the process flow. The sample size and frequency should be chosen depending on the type of
process and the type of product so that all important changes are detected in time. The samples should
be representative for the multivariate quantities under consideration. (Univariate quantity values are
considered in ISO 22514-2.) This part of ISO 22514 describes multivariate capability statistics.
To estimate the PCI, the sample size should preferably be at least 125.
6 Use of multivariate process capability and performance assessment
The purpose of a process capability index is to reflect how well or how badly a process generates qualified
products. The use of PCI for multivariate quantities should reflect this process behaviour better than
PCI for single quantities would. Since a variety of multivariate PCI definitions exists, the selection of a
specific definition to be used will remain in the user’s accountability. However, the following guidance is
given as to when a multivariate PCI should be preferred at all.
A multivariate assessment of process capability and performance is suitable if at least one of the
following circumstances is applicable.
— It is found to be advantageous to describe process capability and performance with only one
comprehensive statistic instead of a high number of single statistics for each product quality quantity.
— The boundary of the tolerance region cannot be expressed independently for all quantities, i.e. at
least one tolerance limit for one quantity is a function of another quantity. This is the case if the
tolerance region is not of rectangular or cuboid shape.
— The single quantities that could be combined to a multivariate one appear to be correlated
among each other.
EXAMPLE In the case of a two-dimensional position tolerance for a borehole axis, the tolerance region is a
circle with defined distances in an x- and y-coordinate direction from the references, see 8.1. The result of the hole
axis measurement will be a value for the x- and y-coordinate. The tolerance limit for the x-coordinate cannot be
expressed independently from the y-coordinate. Thus, a bivariate assessment is to be applied.
7 Calculation of process capability and process performance
7.1 Description of Types I and II
In the multivariate domain, different approaches exist for measuring process capability and process
performance. This part of ISO 22514 describes examples of two different types of indices: Type Ι and
Type ΙΙ. The distinction between the types is based on whether the index is defined based on probability
or defined geometrically by relating the area or volume of a tolerance or process region.
The following description of the types applies:
— Type Ι Based on the probability of conforming or non-conforming products P, the index is calculated
using the relationship between the index and the said probability in a univariate normal case.
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ISO 22514-6:2013(E)

— Type ΙΙ The index is calculated as the ratio of the area or volume of the tolerance region to the
area or volume of the region covered by the process variation.
For practical reasons, the multivariate normal distribution mode has been chosen for the calculation
of the statistics which are described in this clause. However, the choice of normal distribution does not
exclude that in special cases other model distributions will describe the reality better. Also, for practical
reasons, in this part of ISO 22514, the process variation region has been chosen to be of ellipsoid shape.
The most important properties of the multivariate normal distribution are explained in Annex A.
Because of that choice, additional transformations should be applied to make the shape of the process
variation intervals comparable to the shape of the tolerance region. Thus, three further principles are to
be distinguished. These are the principles of transforming the shape of the
a) tolerance region into the shape of the process variation interval,
b) process variation interval into the shape of the tolerance region, and
c) tolerance region and/or the process variation into a new function-oriented dimension.
Both the above-mentioned types and the principles can be combined to define a multivariate PCI. Each
combination, however, may not be useful. There is, for instance, no known definition of a type Ιb PCI.
The term “capability” can only be used for processes that have been demonstrated to be in statistical
control using control charts. In the multivariate case, the distinction between special and common
causes is usually more difficult than in the univariate case. If the process has not been demonstrated to
be in statistical control, the term “performance” is used in this part of ISO 22514.
7.2 Designation and symbols of the indices
7.2.1 General
Different symbols are currently used for multivariate index definitions in industry and science. Currently
used symbols try to distinguish between the types of calculation or to specify their use. This part of
ISO 22514 uses the designation C and/or C for basic definitions of calculation. Furthermore, it will
p pk
be distinguished between process capability and process performance in applying the indices by using
capitals “C” for capability and “P” for performance.
7.2.2 Process capability index
Consider a d-dimensional normal distribution N (μ, Σ) with the mean vector μ and covariance matrix Σ.
d
If the tolerance region is not of elliptic shape (circle, ellipse if d = 2 or sphere, ellipsoid if d = 3 or hyper-
sphere, hyper-ellipsoid if d > 3), it is to be transformed into a modified tolerance region that is of elliptic
shape. This is to be done by determining the largest ellipse (or ellipsoid, hyper-ellipsoid) that is centred
at the target and completely fits into the original tolerance region.
In order to calculate the multivariate C index, the normal distribution shall be centred to have the mean
p
at the centre of the elliptic tolerance region. For that normal distribution, determine the largest contour
ellipsoid that is completely contained in the elliptic tolerance region and calculate the probability of the
volume bounded by that contour ellipse under the d-dimensional normal distribution with covariance
matrix Σ and mean at the centre of the elliptic tolerance region. Denote that probability by P. Then, the
multivariate C index is
p
1 P+1
 
−1
C = Φ
p  
3 2
 
The calculation of P, the probability for observations of x within the determined contour ellipse (ellipsoid/
hyper-ellipsoid) for any d can be done by using the relation to the F-distribution. The explanation is
given in Clause A.1.
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ISO 22514-6:2013(E)

In order to estimate a C index from d-dimensional data, start by estimating the covariance matrix of the
p
ˆ
multivariate normal distribution from the data. Denote the estimate by Σ and use that covariance matrix
ˆ
to determine the contour ellipsoid and its probability P . Finally, the estimated multivariate C index is
p
ˆ
 
1 P +1
−1
ˆ
C = Φ
 
p
3 2
 
L U
10 U
1
5
0
2
−5
3
L
−10
−10 −5 0 5 10
X-coordinate
Key
1 elliptic tolerance zone
2 contour ellipse used for the calculation of the capability index
3 contour ellipse corresponding to the probability zone 99,73 %
Figure 2 — Contour ellipse and tolerance zone used to calculate the capability index for d = 2
In Figure 1, the contour ellipse with probability 99,73 % is completely contained in the contour ellipse
used for the calculation of the index. When this is the case, the index will be larger than 1.
We use the symbol C for this index as for the classical capability index for the univariate normal
p
distribution. The reason is that this calculation method in the one-dimensional case gives the classical
C index. This is explained in Clause A.1.
p
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Y-coordinate

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ISO 22514-6:2013(E)

7.2.3 Minimum process capability index
–
L X U
10 U
1
5




 










 –



 Y 2



 

 


 






0 






3
−5
−10 L
−10 −5 0 5 10
X-coordinate
Key
1 elliptic tolerance zone
2 contour ellipse used for the calculation of the C index
pk
3 contour ellipse corresponding to the probability 99,73 %
Figure 3 — Tolerance zone and contour ellipse used to calculate the capability index for d = 2
Calculation of the C index involves both the mean and the variance of the distribution, so consider
pk
again a d-dimensional normal distribution with mean μ and covariance matrix Σ. For the N (μ,Σ)
d
distribution, calculate
— the largest contour ellipse (ellipsoid, hyper-ellipsoid) that is completely contained in the elliptic
tolerance region, if μ is contained in the tolerance region, or
— the largest contour ellipse (ellipsoid, hyper-ellipsoid) that is not contained in the tolerance region, if
μ is not contained in the tolerance zone.
Now, the probability, P, of the area (volume) contained in the contour ellipse (ellipsoid, hyper-ellipsoid)
under the N (μ,Σ) distribution is calculated. Finally, the C index is calculated as
d pk
1 P+1
−1
C = Φ
pk  
3 2
 
if μ is in the tolerance region and as
1 1−P
 
−1
C = Φ
pk
 
3 2
 
if μ is not in the tolerance region.
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Y-coordinate

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ISO 22514-6:2013(E)

We use the same symbol as for the classical C index for the one-dimensional normal distribution. The
pk
reason is that this method of calculation gives the classical index in the one-dimensional case. This is
explained in Clause A.1.
NOTE The described Type Ia finds applications in geometrical dimensioning and tolerancing of position
deviations. Here, the tolerance region usually describes a circular tolerance zone. The symbols often used in that
case are C and C for C and C respectively.
Po Pok p pk
7.3 Types Ιc and ΙΙc process capability index
Type Ιc as well as Type ΙΙc capability indices are characterized by a function-oriented transformation
of the multiple feature characterization into a single feature characterization. By that type, the
multivariate aspect is expressed in the definition of the transforming function q(x), where x describes
the multivariate quantity. This transformation shall represent the functional importance of the single
quantities in x and their interplay. For example, it describes a model for the tolerance region and can be
interpreted as a weighing function, e.g. a loss function or a quantification function that quantifies the
technical functionality.
The calculation of Type Ιc and ΙΙc indices follows four steps; see Figure 4.
Modeling Sampling parts Estimating the Calculating the
tolerance and trans- univariate PCI
region forming data distribution (Type I or II)
Figure 4 — Steps for calculating the type Ιc or ΙΙc process capability indices
The first step concerns the definition of the technical qualification function q(x) over the d-dimensional
tolerance region. This function has a maximum with the value q at the target in the tolerance region.
max
At the boundary of the tolerance region, the q(x) has the value q . In some cases, q and q
bound max bound
can be derived from the technical context of all single quantities in x. In other cases, appropriate values
are q = 1 and q = 0,5. The function q(x) may be expressed in a closed equation or piecewise
max bound
composed. An example for a piecewise composed linear function is given in Figure 5.
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ISO 22514-6:2013(E)

Dimenstions in millimetres
20 ±0,2
⊕ 0.1 M A
(d)(d)
Y
20,3 0,33
1
0,5
20,2
0,67


20,1 
0,83 


2 


















20,0 

1
0,95
50
A 0,83
19,9
0,67
0,5
19,8
0,33
0,17
19,7 3
19,6
0,00,1 0,20,3 0,40,5
X
Key
X position
Y width
1 q - contour; limits of the tolerance region
bound
2 q = 1, target
max
3 q(x) contour lines
Figure 5 — Example of a qualification function for a width/symmetry tolerance region under MMC
In Figure 5, the multivariate quantity consists of two geometrical features: width and position. The
tolerance region by utilizing the maximum-material-condition is of composite rectangular and triangular
shape. Further explanations about the example are given in Clause 8. The target where q = 1 is situated
max
at the nominal values. From that point, the qualification function is decreasing towards the tolerance
limits. Thereby different trends may be defined: linear, exponential and others. In Figure 5, the function
q(x) is composed of three linear functions. Completely different
...

SLOVENSKI STANDARD
oSIST ISO/DIS 22514-6:2010
01-julij-2010
6WDWLVWLþQHPHWRGH]DREYODGRYDQMHSURFHVRY6SRVREQRVWLQGHORYDQMHGHO
6WDWLVWLNHSURFHVQHVSRVREQRVWLNDUDNWHULVWLNSRUD]GHOMHQLKSRPXOWLYDULDWQL
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Statistical methods in process management - Capability and performance - Part 6:
Process capability statistics for characteristics following a multivariate normal distribution
Méthodes statistiques dans la gestion de processus - Aptitude et performance - Partie 6:
Statistiques de capacité opérationnelle d'un processus pour les caractéristiques qui
suivent une distribution normale à plusieurs variables
Ta slovenski standard je istoveten z: ISO/DIS 22514-6
ICS:
03.120.30 8SRUDEDVWDWLVWLþQLKPHWRG Application of statistical
methods
oSIST ISO/DIS 22514-6:2010 en
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

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oSIST ISO/DIS 22514-6:2010

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oSIST ISO/DIS 22514-6:2010
DRAFT INTERNATIONAL STANDARD ISO/DIS 22514-6
ISO/TC 69/SC 4 Secretariat: ANSI
Voting begins on: Voting terminates on:
2010-05-14 2010-10-14
INTERNATIONAL ORGANIZATION FOR STANDARDIZATION • МЕЖДУНАРОДНАЯ ОРГАНИЗАЦИЯ ПО СТАНДАРТИЗАЦИИ • ORGANISATION INTERNATIONALE DE NORMALISATION
Statistical methods in process management — Capability and
performance —
Part 6:
Process capability statistics for characteristics following a
multivariate normal distribution
Méthodes statistiques dans la gestion de processus — Aptitude et performance —
Partie 6: Statistiques de capacité opérationnelle d'un processus pour les caractéristiques qui suivent une
distribution normale à plusieurs variables
ICS 03.120.30

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©
International Organization for Standardization, 2010

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oSIST ISO/DIS 22514-6:2010
ISO/DIS 22514-6
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ii ISO 2010 – All rights reserved

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oSIST ISO/DIS 22514-6:2010
ISO/DIS 22514-6
Contents Page
Contents .iii
Foreword .iv
Introduction.v
1 Scope.1
2 Normative Reference.1
3 Terms and Definitions.1
4 Process analysis .2
5 Calculation of process capability and performance.3
5.1 Process capability index .3
5.2 Minimum process capability index.5
6 Example.6
6.1 Two dimensional position tolerances .6
6.2 Calculating the capability index using the distance from target.8
Annex A (informative).10
A.1 Useful properties of the multivariate normal distribution in calculating capability
indices .10
A.2 Motivation for the definitions of multivariate capability.11
Annex B .14
B.1 Example unbalance.14
Annex C .18
C.1 Numerical example distance from target.18
C.2 Practical example .19
Bibliography.21
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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 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.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
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.
ISO 22514-6 was prepared by Technical Committee ISO/TC 69, Applications of statistical methods, SC 4,
Applications of statistical methods in process management.
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Introduction
Due to the increased complexity of the production methods and the increasing quality requirements for
products and processes, a process analysis based on univariate characteristics is in many cases not
sufficient.
Instead it may be necessary to analyse the process on the basis of multivariate product characteristics. This
can for instance be in cases where geometric tolerances, dynamic magnitudes as unbalance, correlated
characteristics of materials or other procedural products are observed.
By analogy with ISO 22514-2, this standard provides calculation formulae for process performance and
process capability indices, which take into consideration process dispersion as well as process location as
extension to the corresponding indices for univariate characteristics. The indices proposed are indeed based
on the classical C and C indices for the one-dimensional case. The motivation for the extension to the
p pk
multivariate case is explained in Annex A.
Examples of possible applications are two-dimensional or three-dimensional positions, unbalance or several
correlated characteristics of chemical products.
The dispersion of the measuring results is composed by the dispersion of the product realization process and
the precision of the measuring process. It is assumed that the capability of the used measuring system was
demonstrated prior to the determination of the capability of the product realization process.
This standard is applicable if the distribution of the values for the observed product characteristics by means
of the parameters of their distribution model are analysed and evaluated in relation to specified values.
The calculation method described here should be used to support an unambiguous decision, especially if
• limiting values for process capability indices for multivariate, continuous product characteristics are
specified as part of a contract between customers and suppliers, or
• the capabilities of different constructions, production methods or suppliers are to be compared, or
• production processes are to be approved, or
• problems are to be analysed and decisions are made in cases of complaints or damage events.
NOTE: Product realization processes include e.g. manufacturing processes, service processes, product assembly
processes.

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DRAFT INTERNATIONAL STANDARD ISO/DIS 22514-6

Statistical methods in process management — Capability and
performance —
Part 6:
Process capability statistics for characteristics following a
multivariate normal distribution
1 Scope
This International Standard provides methods for calculating performance and capability statistics for process
or product characteristics that follow a multivariate normal distribution.
NOTE In principle this International Standard can be used for bivariate up to d variate cases.
2 Normative Reference
The following referenced documents are indispensable for the application 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 1101, Geometrical Product Specifications (GPS) — Geometrical tolerancing — Tolerances of form,
orientation, location and run-out
3 Terms and Definitions
For the purpose of this document, the terms and definitions given in ISO 9000 and ISO 22514-1 and the
following apply.
3.1
process capability estimate
statistical estimate of the outcome of a characteristic from a process which has been demonstrated to be in a
state of statistical control and which describes that ability of the process to realise a characteristic that will fulfil
the requirements for that characteristic
[ISO 22514-1 3.3.3]
3.2
characteristic
distinguishing feature
[ISO 9000:2005, definition 3.5.1]
NOTE 1 A characteristic can be inherent or assigned.
NOTE 2 A characteristic can be nominal, ordinal, differential or rational.
NOTE 3 There are various classes of characteristic, such as the following:
— physical (e. g. mechanical, electrical, chemical or biological characteristics);
— sensory (e. g. related to smell, touch, taste, sight, hearing);
— behavioral (e. g. courtesy, honesty, veracity);
— temporal (e. g. punctuality, reliability, availability);
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— ergonomic (e.g. physiological characteristic, or related to human safety);
— functional (e. g. maximum speed of an aircraft).
EXAMPLE In the multivariate case of dimension k characteristics are given by a k-tuple, i.e. an ordered set
consisting of k elements. For k = 3 an example is (colour, mass, number of defects).
3.3
tolerance zone
space limited by one or several geometrically perfect lines or surfaces, and characterized by a linear dimension, called a
tolerance
[ISO 1101 2004]
NOTE 1 In the univariate case the tolerance zone is an interval.
NOTE 2 In the bi-variate case, the tolerance zone is a tolerance ellipse or alternatively a tolerance circle (when the
tolerance zone for the both univariate characteristics has the same size.).
NOTE 3 In the tri-variate case, the tolerance zone corresponds to a tolerance ellipsoid or alternatively a tolerance sphere
(when the tolerance zone for the three components have the same size).
3.4
process capability
statistical estimate of the outcome of a characteristic from a process which has been demonstrated to be in a
state of statistical control and which describes that process's ability to realize a characteristic that will fulfill the
requirements for that characteristic
[ISO 3534-2: 2006 2.7.1]
3.5
process performance
statistical measure of the outcome of a characteristic from a process, which may not have been demonstrated
to be in a state of statistical control.
[ISO 3534-2: 2006 2.6.1]
4 Process analysis
The purpose of process analysis is to obtain sound knowledge of a process. This knowledge is necessary for
controlling the process efficiently so that the products realized by the process fulfil the quality requirement.
A process analysis is always an analysis of one or more characteristics that are considered to be important to
the process.
Product characteristics can often be analysed instead of process characteristics because product
characteristics not only characterise the products, but due to their correlation with process characteristics they
also characterise the process creating these products.
The values of the characteristics under consideration are typically determined on the basis of samples taken
from the process flow. The sample size and frequency should be chosen depending on the type of process
and the type of product so that all important changes are detected in time. The samples should be
representative for the multivariate characteristic under consideration, whereas univariate characteristic values
are considered in ISO 22514-2. This standard describes multivariate capability statistics.
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5 Calculation of process capability and performance
The term “multivariate” may refer to dimensions in space, for example the coordinate measuring technique, as
well as measuring technique for multivariate variables (e.g. unbalance of a wheel).
For practical reasons, the model for multivariate normal distribution has been chosen for the calculation of the
statistics which are described in this clause. However, the choice of normal distribution does not exclude that
in special cases other model distributions will describe the reality better. Also, for practical reasons, in this
standard the process variation intervals have been chosen as ellipsoids.
The most important properties of the multivariate normal distribution are explained in annex A.
The term capability can only be used for processes that have been demonstrated to be in control using control
charts. In the multivariate case the distinction between special and common causes is usually more difficult
than in the univariate case. If the process has not been demonstrated to be in control we use the term
performance in this standard.
5.1 Process capability index
Consider a d-dimensional normal distribution with covariance matrix Σ . In order to calculate the multivariate
C index the normal distribution must be centred to have mean at the centre of the tolerance zone. For that
p
normal distribution determine the largest contour ellipsoid that is completely contained in the tolerance zone
and calculate the probability of the volume bounded by that contour ellipse under the d-dimensional normal
distribution with covariance matrix Σ and mean at the centre of the tolerance zone. Denote that probability by
P. Then the multivariate C index is
p
11P +
⎛⎞
−1
C =Φ
p ⎜⎟
32
⎝⎠

In order to estimate a C index from d-dimensional data start by estimating the covariance matrix of the
p
ˆ
multivariate normal distribution from the data. Denote the estimate by Σand use that covariance matrix to
ˆ
determine the contour ellipsoid and its probability P . Finally the estimated multivariate C index is
p
ˆ
⎛⎞
11P +
−1
ˆ
C =Φ
⎜⎟
p
32
⎝⎠

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U
U
Tolerance
Y
Hyper ellipsoid
Contour ellipse
used for the
calculation of
capability index
Contour
X-coordinate
ellipsecorrespon
ding to the
probability
99,73%
X-coordinate

Figure 1 — Contour ellipse and tolerance zone used to calculate the capability index
In Figure 1 the contour ellipse with probability 99,73 % is completely contained in the contour ellipse used for
the calculation of the index. When this is the case the index will be larger that 1.
We use the symbol C for this index as for the classical capability index for the univariate normal distribution.
p
The reason is that this calculation method in the one-dimensional case gives the classical C index. This is
p
explained in clause A1 of Annex 1.


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5.2 Minimum process capability index

U
Ellipsoid tolerance
U
zone
Contour ellipse
used for the
calculation of the
Y
C index
pk


Contour ellipse
corresponding to
the probability
99,73%.
X-coordinate

Figure 2 — Tolerance zone and countour ellipse used to calculate the capability index
Calculation of the C index involves both the mean and the variance of the distribution, so consider a d-
pk
dimensional normal distribution with mean and covariance matrix . For the N()µ,Σ distribution
µ Σ
d
calculate the largest contour ellipsoid that is complete contained in the tolerance zone, if µ is contained in the
tolerance zone, or the largest contour ellipsoid that is not contained in the tolerance zone. If µ is not contained
in the tolerance zone. Now the probability, P, of the volume contained in the contour ellipsoid under the
N()µ,Σ distribution is calculated. Finally the C index is calculated as
pk
d
11P +
⎛⎞
−1
C =Φ
pk ⎜⎟
32
⎝⎠

if µ is in the tolerance zone and as
11− P
⎛⎞
−1
C =Φ
pk ⎜⎟
32
⎝⎠

if µ is not in the tolerance zone.
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We use the same symbol as for the classical C index for the one-dimensional normal distribution. The
pk
reason is that this method of calculation gives the classical index in the one-dimensional case. This is
explained in clause A1 of Annex A.
6 Example
6.1 Two dimensional position tolerances
On a produced part, the midpoint of a drilled hole is measured. The nominal value is 80 mm in the X direction
and –116.5 mm in the Y direction.

Figure 3 — Measurement task is position of a hole
100 set of values from produced parts are measured (see table 1).
The X- and Y- values are plotted in a diagram (figure 4.) and the variation interval calculated.
The formulas used can be found in annex A.

Number of measured parts n = 100
Specification limits:    Lx = 79.750 Ux = 80.250
      Ly = −116.750 Uy = −116.250

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Table 1. Measured values:
Table of the measured values and a calculated deviation
Nr. dev. D X-coord. Y-coord.  Nr. dev. D X-coord. Y-coord. Nr. Dev D X-coord. Y-coord.
1 0.038 79.976 −116.470  36 0.090 79.995 −116.410 71 0.107 79.986 −116.394
2 0.094 79.993 −116.406  37 0.097 80.002 −116.403 72 0.073 80.016 −116.429
3 0.086 80.031 −116.420  38 0.113 80.027 −116.390 73 0.069 79.995 −116.431
4 0.041 79.968 −116.475  39 0.021 79.995 −116.520 74 0.108 79.975 −116.395
5 0.105 79.973 −116.399  40 0.085 80.010 −116.416 75 0.118 79.965 −116.387
6 0.092 79.983 −116.410  41 0.110 80.005 −116.390 76 0.122 79.971 −116.382
7 0.099 80.008 −116.401  42 0.081 80.004 −116.419 77 0.119 79.978 −116.383
8 0.086 80.014 −116.415  43 0.055 79.966 −116.457 78 0.118 79.999 −116.382
9 0.075 80.020 −116.428  44 0.097 80.013 −116.404 79 0.024 80.008 −116.477
10 0.076 79.979 −116.427  45 0.078 80.021 −116.425 80 0.094 80.005 −116.406
11 0.064 79.978 −116.440  46 0.118 79.989 −116.383 81 0.056 80.007 −116.444
12 0.086 80.016 −116.416  47 0.111 79.988 −116.390 82 0.093 80.032 −116.413
13 0.067 79.990 −116.434  48 0.057 79.987 −116.445 83 0.139 79.958 −116.368
14 0.120 79.992 −116.380  49 0.101 80.012 −116.400 84 0.122 79.990 −116.378
15 0.103 79.999 −116.397  50 0.067 80.017 −116.435 85 0.126 79.994 −116.374
16 0.119 80.016 −116.382  51 0.099 80.000 −116.401 86 0.089 80.029 −116.416
17 0.086 80.038 −116.423  52 0.101 79.995 −116.399 87 0.110 80.000 −116.390
18 0.118 80.018 −116.383  53 0.139 79.999 −116.361 88 0.084 80.010 −116.417
19 0.116 80.005 −116.384  54 0.086 80.002 −116.414 89 0.121 80.000 −116.379
20 0.118 80.071 −116.406  55 0.095 80.068 −116.433 90 0.131 79.992 −116.369
21 0.072 79.941 −116.458  56 0.103 79.990 −116.397 91 0.122 79.992 −116.378
22 0.097 79.984 −116.404  57 0.178 80.035 −116.325 92 0.062 79.990 −116.439
23 0.029 79.986 −116.475  58 0.107 79.980 −116.395 93 0.098 79.999 −116.402
24 0.093 80.043 −116.418  59 0.182 79.978 −116.319 94 0.086 79.986 −116.415
25 0.047 80.027 −116.538  60 0.099 80.000 −116.401 95 0.097 79.986 −116.404
26 0.090 80.031 −116.415  61 0.080 79.995 −116.420 96 0.092 80.020 −116.410
27 0.097 80.005 −116.403  62 0.133 79.996 −116.367 97 0.095 79.984 −116.406
28 0.122 80.024 −116.380  63 0.088 80.000 −116.412 98 0.133 79.980 −116.369
29 0.081 80.040 −116.430  64 0.107 79.948 −116.406 99 0.132 79.981 −116.369
30 0.094 80.006 −116.406  65 0.101 80.015 −116.400 100 0.058 80.033 −116.452
31 0.099 79.986 −116.402  66 0.081 79.990 −116.420
32 0.094 79.982 −116.408  67 0.087 80.009 −116.413
33 0.111 79.942 −116.405  68 0.067 80.004 −116.433
34 0.135 79.975 −116.367  69 0.130 79.960 −116.376
35 0.103 80.014 −116.398  70 0.121 80.007 −116.379

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U
U
X-coordinate (mm)

Figure 4 — Graphical presentation of the position tolerances with reference zone and specified
tolerance
Results:
Process capability index: C  = 2.43
P
Minimum capability index: C =1.48
PK
6.2 Calculating the capability index using the distance from target
The target location (x , y ) = (80, -116,5) is specified for the centre of the hole in Figure 3. The location of
0 0
each (x, y) is measured. The coordinates (x, y) denote the centre of the hole drilled. The deviation from
the target location is
2 2
D = (x − x ) + (y − y )
0 0

The actual calculated values for the distances D can be found in Table 1.
All deviations are plotted in a histogram shown in Figure 5. The maximal acceptable deviation is 0.25 mm,
because the tolerance zone is a circle centred on the target and with a diameter of 0.5 mm. The maximal
acceptable deviation is the radius of that circle.
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The distribution model for the actual dataset is a Rayleigh distribution.
Calculation of the capability indices:
Capability index can not be calculated because of no lower specification limit exist.
U − X 0,25 − 0,096
50%
Minimum capability index:
C = = = 1,81
PK
X − X 0,181 − 0,096
99,865% 50%

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Annex A
(informative)
A.1 Useful properties of the multivariate normal distribution in calculating capability
indices
The density of the d-dimensional normal distribution with mean µ and covariance matrix Σ is
11⎛ T ⎞
−1 d
fxxx=−exp −µ Σ (xµ− ) ,   ∈ R ,
() ()
⎜⎟
d 1
2
⎝⎠
2 2
2π Σ
()

−1
if the covariance matrix Σ is positive definite such that its inverse Σ exists. Here x and µ are d -
dimensional vectors and Σ is a dd× matrix. Vectors are column vectors and T denotes the transpose of a
T
matrix or a vector, i.e. x is x written as a row vector. The d-dimensional normal distribution with mean
µ
and covariance matrix Σ is denoted by N()µ,Σ .
d
The contours of constant density are
T
−12
x|)xµ−−Σ (xµ= c
()
{ }

and they are intervals for d =1, ellipses for d = 2 and ellipsoids for d ≥ 3.
2
The probability of the area bounded by the contour ellipsoids can be calculated from the χ distribution on d
degrees of freedom. If X follows a d-dimensional normal distribution with mean and covariance matrix ,
µ Σ
then
T
−12 2
PFXµ−−Σ()Xµ≤cc= ()
() 2
()
χ ()d

2
where F denotes the distribution function of the χ distribution on d degrees of freedom.
2
χ ()d
It follows that the contour ellipsoid
2
T
⎧⎫
−−11
x|(xµ− Σ()xµ−= Fp)
()
⎨⎬2
( )
χ ()d
⎩ ⎭

−1 2
is the boundary of an area with a probability of p. Here Fp() is the p-fractile of the χ distribution on d
2
χ ()d
2
degrees of freedom. This quantity is sometimes denoted by χ ()d .
p
If xx,,… is a sample from a d -dimensional normal distribution with mean µ and covariance matrix Σ then
1 n
µ and Σ are estimated as
n
1
µµ←=ˆ =xx
· ∑ i
n
i=1

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and
n
1
T
ˆ
Σ←=S Σ= ()xx−()x−x
∑ii··
n −1
i=1

where “ ← ” is read as “is estimated by”.
A.2 Motivation for the definitions of multivariate capability
Consider first the C index in the one-dimensional case.The tolerance interval is the interval [L,U]. Let X ~
p
2
N(µ,σ ) with µ=(U+L)/2, i.e. the distribution is centred on the midpoint of the tolerance interval. The probability
of the tolerance interval is
L++ULU L+U
⎛⎞
PP= L 22()
⎜⎟
NU((++L)/2,σσ) N ((U L)/2, )
22 2
⎝⎠
⎛⎞LU+
X −
⎜⎟
L−−U UL UL− U−L
⎛⎞⎛ ⎞
2
=

2
⎜⎟
⎜⎟⎜ ⎟
NU(( +L)/2,σ )
22σσ σ 2σ 2σ
⎝⎠⎝ ⎠
⎜⎟
⎝⎠
UL− ⎡ UL− ⎤ U −L
⎛ ⎞ ⎛⎞ ⎛⎞
=Φ −12−Φ = Φ −1=2Φ3C −1
()
p
⎜ ⎟⎢⎥⎜⎟ ⎜⎟
22σσ 2σ
⎝ ⎠ ⎝⎠ ⎝⎠
⎣⎦

It follows that
11P +
⎛⎞
−1
C =Φ
p ⎜⎟
32
⎝⎠

where P is the probability of the tolerance interval for a normal distribution cen
...