# ISO 21747:2006

(Main)## Statistical methods — Process performance and capability statistics for measured quality characteristics

## Statistical methods — Process performance and capability statistics for measured quality characteristics

ISO 21747:2006 describes a procedure for the determination of statistics in order to estimate the quality capability of product and process characteristics. The process results of these quality characteristics are tabularized into eight possible distribution types. Calculation formulae for the statistical values are placed with every distribution.These statistics relate to continuous quality characteristics exclusively. ISO 21747:2006 is applicable to processes in any industrial or economical sector.

## Méthodes statistiques — Performances de processus et statistiques d'aptitude pour les caractéristiques de qualité mesurées

## Statistične metode – Statistike delovanja in sposobnosti procesa za merjene karakteristike kakovosti

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INTERNATIONAL ISO

STANDARD 21747

First edition

2006-07-01

Statistical methods — Process

performance and capability statistics

for measured quality characteristics

Méthodes statistiques — Performances de processus et statistiques

d'aptitude pour les caractéristiques de qualité mesurées

Reference number

ISO 21747:2006(E)

©

ISO 2006

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ISO 21747:2006(E)

PDF disclaimer

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© ISO 2006

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ii © ISO 2006 – All rights reserved

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ISO 21747:2006(E)

Contents Page

Foreword. iv

Introduction . v

1 Scope . 1

2 Normative references . 1

3 Terms and definitions. 1

3.1.1 Variation-related concepts. 1

3.1.2 Fundamental process performance and process capability related terms. 3

3.1.3 Process performance — measured data. 6

3.1.4 Process capability — measured data . 8

3.2 Specifications, values and test results. 10

3.2.1 Specification-related concepts. 10

4 Symbols and abbreviated terms . 12

5 Process analysis. 13

6 Time-dependent distribution models. 13

7 Process capability and performance indices . 22

7.1 Methods for the determination of performance and capability indices — Overview . 22

7.2 General geometric method (M1 ). 23

l,d

7.3 Explicit inclusion of additional variation (M2 ). 26

l,d,a

7.4 Alternative method of explicit inclusion of additional variation (M3 ). 27

l,d,a

7.5 Calculation of fractions nonconforming (M4). 28

7.6 One-sided specification limits. 29

8 Reporting process performance/capability indices . 31

Bibliography . 32

© ISO 2006 – All rights reserved iii

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ISO 21747:2006(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 21747 was prepared by Technical Committee ISO/TC 69, Application of Statistical Methods,

Subcommittee SC 4, Application of Statistical Methods and Process Management.

iv © ISO 2006 – All rights reserved

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ISO 21747:2006(E)

Introduction

Many standards have been created concerning the quality capability/performance of processes by

international, regional and national standardization bodies and also by industry. However, all of them assume

that the process is in a state of statistical control, with stationary, normal processes behaviour. However, a

comprehensive analysis of production processes shows that it is very rare for processes to remain in a

normally distributed, stationary state. In recognition of this fact, this International Standard provides a

framework for estimating the quality capability/performance of industrial processes for an array of standard

processes. These standard processes are categorized by the stability of the first and second distributional

moments, as to whether they are constant, change systematically, or randomly. As such, the quality

capability/performance can be assessed for very differently shaped distributions with respect to time.

© ISO 2006 – All rights reserved v

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INTERNATIONAL STANDARD ISO 21747:2006(E)

Statistical methods — Process performance and capability

statistics for measured quality characteristics

1 Scope

This International Standard describes a procedure for the determination of statistics in order to estimate the

quality capability of product and process characteristics. The process results of these quality characteristics

are tabularized into eight possible distribution types. Calculation formulae for the statistical values are placed

with every distribution.

These statistics relate to continuous quality characteristics exclusively. This International Standard is

applicable to processes in any industrial or economical sector.

NOTE This method is usually applied in case of a great number of serial process results, but it can also be used for

small series (a small number of process results).

2 Normative references

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 9000:2005, Quality management systems — Fundamentals and vocabulary

3 Terms and definitions

For the purpose of this document, the terms and definitions given in ISO 9000 and the following apply.

3.1

quality characteristic

inherent characteristic of a product, process or system related to a requirement

NOTE 1 Inherent means existing in something, especially as a permanent characteristic.

NOTE 2 A characteristic assigned to a product, process or system (e.g. the price of a product, the owner of a product)

is not a quality characteristic of that product, process or system.

[ISO 9000:2005, 3.5.2]

3.1.1 Variation-related concepts

3.1.1.1

variation

difference between values of a characteristic

NOTE Variation is often expressed as a variance or standard deviation.

1)

[ISO 3534-2:— , 2.2.1]

1) To be published. (Revision of ISO 3534-2:1993)

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ISO 21747:2006(E)

3.1.1.2

inherent process variation

variation (3.1.1.1) in a process when the process is operating in a state of statistical control

NOTE 1 When it is expressed in terms of standard deviation, the subscript “w” is applied, (e.g. σ , S , or s ), indicating

w w w

inherent. See also 3.1.4.1, NOTE 2.

NOTE 2 This variation corresponds to “within subgroup variation”.

[ISO 3534-2:—, 2.2.2]

3.1.1.3

total process variation

variation (3.1.1.1) in a process due to both special causes (3.1.1.4) and random causes (3.1.1.5)

NOTE 1 When it is expressed in terms of standard deviation, the subscript “t” is applied (e.g. σ , S or s ), indicating total.

t t t

See also 3.1.3.1, Note 3.

NOTE 2 This variation corresponds with the combination of the “within-subgroup variation” and the “between-subgroup

variation”.

[ISO 3534-2:—, 2.2.3]

3.1.1.4

special cause

〈process variation〉 source of process variation other than inherent process variation (3.1.1.2)

NOTE 1 Sometimes “special cause” is taken to be synonymous with “assignable cause”. However, a distinction is

recognized. A special cause is assignable only when it is specifically identified.

NOTE 2 A special cause arises because of specific circumstances that are not always present. As such, in a process

subject to special causes, the magnitude of the variation from time to time is unpredictable.

[ISO 3534-2:—, 2.2.4]

3.1.1.5

random cause

common cause

chance cause

〈process variation〉 source of process variation that is inherent in a process over time

NOTE 1 In a process subject only to random cause variation, the variation is predictable within statistically established

limits.

NOTE 2 The reduction of these causes gives rise to process improvement. However, the extent of their identification,

reduction and removal is the subject of cost/benefit analysis in terms of technical tractability and economics.

[ISO 3534-2:—, 2.2.5]

3.1.1.6

stable process

process in a state of statistical control

〈constant mean〉 process subject to only random causes (3.1.1.5)

NOTE 1 A stable process will generally behave as though the samples from the process at any time are simple random

samples from the same population.

NOTE 2 This state does not imply that the random variation is large or small, within or outside of specification, but

rather that the variation (3.1.1.1) is predictable using statistical techniques.

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ISO 21747:2006(E)

NOTE 3 The process capability (3.1.4.1) of a stable process is usually improved by fundamental changes that reduce

or remove some of the random causes present and/or adjusting the mean towards the preferred value.

NOTE 4 In some processes, the mean of a characteristic can have a drift or the standard deviation can increase due,

for example, to wear out of tools or depletion of concentration in a solution. A progressive change in the mean or standard

deviation of such a process is considered due to systematic and not random causes. The results, then, are not simple

random samples from the same population.

[ISO 3534-2:—, 2.2.7]

3.1.1.7

out-of-control criteria

set of decision rules for identifying the presence of special causes (3.1.1.4)

NOTE Decision rules may include those relating to points outside of control limits, runs, trends, cycles, periodicity,

concentration of points near the centre line or control limits, unusual spread of points within control limits (large or small

dispersion) and relationships among values within subgroups.

[ISO 3534-2:—, 2.2.8]

3.1.2 Fundamental process performance and process capability related terms

3.1.2.1

distribution

〈of a characteristic〉 information on the probabilistic behaviour of a characteristic

NOTE 1 The distribution of a characteristic can be represented, for example, by ranking of the values of the

characteristic and showing the resulting pattern of measures or scores in the form of a tally chart or histogram. Such a

pattern provides all of the numerical value information on the characteristic except for the serial order in which the data

arises.

NOTE 2 The distribution of a characteristic is dependent on prevailing conditions. Thus, if meaningful information about

the distribution of a characteristic is desired, the conditions under which the data is collected should be specified.

NOTE 3 It is important to know the class of distribution, for instance, normal or log-normal, before predicting or

estimating process capability and performance measures and indices or fraction nonconforming.

[ISO 3534-2:—, 2.5.1]

3.1.2.2

class of distributions

particular family of distributions (3.1.2.1) each member of which has the same common attributes by which

the family is fully specified

EXAMPLE 1 The two-parameter, symmetrical bell-shaped, normal distribution with parameters mean and standard

deviation.

EXAMPLE 2 The three-parameter Weibull distribution with parameters location, shape and scale.

EXAMPLE 3 The unimodal continuous distributions.

NOTE The class of distributions can often be fully specified through the values of appropriate parameters.

[ISO 3534-2:—, 2.5.2]

3.1.2.3

distribution model

specified distribution (3.1.2.1) or class of distributions (3.1.2.2)

EXAMPLE 1 A model for the distribution of a product characteristic, the diameter of a bolt, might be the normal

distribution with mean 15 mm and standard deviation 0,05 mm. Here the model is a fully specified one.

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ISO 21747:2006(E)

EXAMPLE 2 A model for the diameter of bolts as in Example 1 could be the class of normal distributions without

attempting to specify a particular distribution. Here the model is the class of normal distributions.

[ISO 3534-2:—, 2.5.3]

3.1.2.4

upper fraction nonconforming

p

U

fraction of the distribution (3.1.2.1) of a characteristic that is greater than the upper specification limit

(3.2.1.3), U

EXAMPLE In a normal distribution, with mean, µ, and standard deviation, σ :

⎛⎞UU−−µµ⎛⎞

p =−1ΦΦ= (1)

U ⎜⎟ ⎜⎟

σσ

⎝⎠ ⎝⎠

where

p is the upper fraction nonconforming;

U

Φ is the distribution function of the standard normal distribution;

U is the upper specification limit.

NOTE 1 Tables (or functions in statistical computer packages) of the standard normal distribution are readily available

which give the proportion of process output expected beyond a particular value of interest, such as a specification limit

(3.2.1.2), in terms of standard deviations away from the process mean. This obviates the need to work out the statistical

distribution function given in the example.

NOTE 2 The function relates to a theoretical distribution. In practice, with empirical distributions, the parameters are

replaced by their estimates.

[ISO 3534-2:—, 2.5.4]

3.1.2.5

lower fraction nonconforming

p

L

fraction of the distribution (3.1.2.1) of a characteristic that is less than the lower specification limit (3.2.1.4),

L

EXAMPLE In a normal distribution (3.1.2.1), with mean, µ, and standard deviation, σ :

L − µ

⎛⎞

p =Φ (2)

L ⎜⎟

σ

⎝⎠

where

p is the lower fraction nonconforming;

L

Φ is the distribution function of the standard normal distribution;

L is the lower specification limit.

NOTE 1 Tables (or functions in statistical computer packages) of the standard normal distribution are readily available

which give the proportion of process output expected beyond a particular value of interest, such as a specification limit

(3.2.1.2), in terms of standard deviations away from the process mean. This obviates the need to work out the statistical

distribution function given in the example.

NOTE 2 The function relates to a theoretical distribution. In practice, with empirical distributions, the parameters are

replaced by their estimates.

[ISO 3534-2:—, 2.5.5]

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ISO 21747:2006(E)

3.1.2.6

total fraction nonconforming

p

t

sum of upper fraction nonconforming (3.1.2.4) and lower fraction nonconforming (3.1.2.5)

EXAMPLE In a normal distribution, with mean, µ, and standard deviation, σ :

⎛⎞µµ−−UL⎛ ⎞

p=+ΦΦ (3)

t ⎜⎟ ⎜ ⎟

σσ

⎝⎠ ⎝ ⎠

where

p is the total fraction nonconforming;

t

Φ is the distribution function of the standard normal distribution;

L is the lower specification limit;

U is the upper specification limit.

NOTE 1 Tables (or functions in statistical computer packages) of the standard normal distribution are readily available

which give the proportion of process output expected beyond a particular value of interest, such as a specification limit

(3.2.1.2), in terms of standard deviations away from the process mean. This obviates the need to work out the statistical

distribution function given in the example.

NOTE 2 The function relates to a theoretical distribution. In practice, with empirical distributions, the parameters are

replaced by their estimates.

[ISO 3534-2:—, 2.5.6]

3.1.2.7

reference interval

interval bounded by the 99,865 % distribution quantile, X , and the 0,135 % distribution quantile,

99,865 %

X

0,135 %

NOTE 1 The interval can be expressed by (X , X ) and the length of the interval is X – X .

99,865 % 0,135 % 99,865 % 0,135 %

NOTE 2 This term is used only as an arbitrary, but standardized, basis for defining the process performance index

(3.1.3.2) and process capability index (3.1.4.2).

NOTE 3 For a normal distribution (3.1.2.1), the length of the reference interval can be expressed in terms of six

standard deviations, 6σ, or 6S, when estimated from a sample.

NOTE 4 For a non-normal distribution, the length of the reference interval can be estimated by means of appropriate

probability papers (e.g. log-normal) or from the sample kurtosis and sample skewness using the methods described in

2)

ISO/TR 12783 .

NOTE 5 A quantile or fractile indicates division of a distribution into equal units or fractions, e.g. percentiles. Quantile is

defined in ISO 3534-1.

[ISO 3534-2:—, 2.5.7]

3.1.2.8

lower reference interval

interval bounded by the 50 % distribution quantile, X and the 0,135 % distribution quantile, X

50 % 0,135 %

NOTE 1 The interval can be expressed by (X , X ) and the length of the interval is X – X .

50 % 0,135 % 50 % 0,135 %

2) Under preparation.

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ISO 21747:2006(E)

NOTE 2 This term is used only as an arbitrary, but standardized, basis for defining the lower process performance

index (3.1.3.3) and lower process capability index (3.1.4.3).

NOTE 3 For a normal distribution (3.1.2.1), the length of the lower reference interval can be expressed in terms of

standard deviations as 3σ, or an estimated 3S, and X represents both the mean and the median.

50 %

NOTE 4 For a non-normal distribution, the 50 % distribution quantile, X , namely the median, and the 0,135 %

50 %

distribution quantile, X , can be estimated by means of appropriate probability papers (e.g. log-normal) or from the

0,135 %

2)

sample kurtosis and sample skewness using the methods described in ISO/TR 12783 .

[ISO 3534-2:—, 2.5.8]

3.1.2.9

upper reference interval

interval bounded by the 99,865 % distribution quantile, X , and the 50 % distribution quantile, X

99,865 % 50 %

NOTE 1 The interval can be expressed by (X , X ) and the length of the interval is X – X .

99,865 % 50 % 99,865 % 50 %

NOTE 2 This term is used only as an arbitrary, but standardized, basis for defining the upper process performance

index (3.1.3.4) and upper process capability index (3.1.4.4).

NOTE 3 For a normal distribution (3.1.2.1), the length of the upper reference interval can be expressed in terms of

standard deviations as 3σ, or an estimated 3S, and X represents both the mean and the median.

50 %

NOTE 4 For a non-normal distribution, the 50 % distribution quantile, X , namely the median, and the 99,865 %

50 %

distribution quantile, X , can be estimated by means of appropriate probability papers (e.g. log-normal) or from the

99,865 %

2)

sample kurtosis and sample skewness using the methods described in ISO/TR 12783 .

[ISO 3534-2:—, 2.5.9]

3.1.3 Process performance — Measured data

3.1.3.1

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

NOTE 1 The outcome is a distribution (3.1.2.1), the class of which needs determination and its parameters assessed.

NOTE 2 Care should be exercised in using this measure as it may contain a component of variability due to special

causes (3.1.1.4), the value of which is not predictable.

NOTE 3 For a normal distribution described in terms of the standard deviation, S , assessed from only one sample of

t

size N, the standard deviation is expressed thus:

2

1

SX=−X (4)

()

tti

∑

N −1

where

1

X = X (5)

t ∑ i

N

This descriptor, S , takes into account the variation due to random (common) causes (3.1.1.5) together with any special

t

causes that may be present. S is used here instead of σ as the standard deviation is a statistical descriptive measure. The

t t

sample size N can be made up of m subgroups, each of size n.

NOTE 4 For a normal distribution, process performance can be assessed from the expression:

process performance=±X (zS )

tt

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ISO 21747:2006(E)

and, “z” is dependent on the particular parts per million performance requirement. Typically “z” takes the value of 3, 4 or 5.

If the process performance coincides with the specified requirements, a z value of 3 indicates an expected 2 700 parts per

million outside of specification. Similarly, a z of 4 indicates an expected 64 parts per million and a z of 5 an expected

0,6 parts per million outside of specification.

NOTE 5 For a non-normal distribution, process performance can be assessed using, for example, an appropriate

probability paper or from the parameters of the distribution fitted to the data. The expression for process performance

takes the form:

+a

process performance = X

t −b

+a

The notation, , is in the same style as standard drawing office practice for expressing specified tolerances about a

−b

nominal, or preferred, value for a characteristic, when the preferred value is not equidistant from each limit. The equivalent

notation for limits symmetrical about the preferred value is ±. This enables a direct comparison to be made between the

dimensional performance of a characteristic and its specified requirements in terms of both location and dispersion.

[ISO 3534-2:—, 2.6.1]

3.1.3.2

process performance index

P

p

index describing process performance (3.1.3.1) in relation to specified tolerance

NOTE 1 Frequently, the process performance index is expressed as the value of the specified tolerance divided by a

measure of the length of the reference interval (3.1.2.7), namely as:

UL−

P = (6)

p

XX−

99,865 % 0,135 %

NOTE 2 For a normal distribution (3.1.2.1), the length of the reference interval is equal to 6S (see 3.1.3.1, Note 3).

t

NOTE 3 For a non-normal distribution, the length of the reference interval can be estimated using, for example, the

2)

method described in ISO/TR 12783 .

[ISO 3534-2:—, 2.6.2]

3.1.3.3

lower process performance index

P

pkL

index describing process performance (3.1.3.1) in relation to the lower specification limit (3.2.1.4), L

NOTE 1 Frequently, the lower process performance index is expressed by the difference between the 50 % distribution

quantile, X , and lower specification limit (3.2.1.4) divided by a measure of the length of the lower reference interval

50 %

(3.1.2.8), namely as:

XL−

50 %

P =

pkL

XX−

50 % 0,135 %

(7)

NOTE 2 For the symmetrical normal distribution (3.1.2.1), the length of the lower reference interval is equal to 3S

t

(see 3.1.3.1, Note 3) and X represents both the mean and the median.

50 %

NOTE 3 For a non-normal distribution, the length of the lower reference range can be estimated using the method

2)

described in ISO/TR 12783 and X represents the median.

50 %

[ISO 3534-2:—, 2.6.3]

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ISO 21747:2006(E)

3.1.3.4

upper process performance index

P

pkU

index describing process performance (3.1.3.1) in relation to the upper specification limit (3.2.1.3), U

NOTE 1 Frequently, the upper process performance index is expressed as the difference between the upper

specification limit and the 50 % distribution quantile, X , divided by a measure of the length of the upper reference

50 %

interval (3.1.2.9), namely as:

UX−

50 %

(8)

P =

pkU

XX−

99,865 % 50 %

NOTE 2 For a normal distribution (3.1.2.1), the length of the upper reference interval is equal to 3S (see 3.1.3.1,

t

Note 3) and X represents both the mean and the median.

50 %

NOTE 3 For a non-normal distribution, the length of the upper reference interval can be estimated using the method

2)

described in ISO/TR 12783 and X represents the median.

50 %

[ISO 3534-2:—, 2.6.4]

3.1.3.5

minimum process performance index

P

pk

smaller of upper process performance index (3.1.3.4) and lower process performance index (3.1.3.3)

[ISO 3534-2:—, 2.6.5]

3.1.4 Process capability — Measured data

3.1.4.1

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 fulfil the

requirements for that characteristic

NOTE 1 The outcome is a distribution (3.1.2.1), the class of which needs determination and its parameters estimated.

NOTE 2 For a normal distribution, the process overall standard deviation, σ , can be estimated using the formula for S

t

t

(see 3.1.3.1, Note 3).

Alternatively, in certain circumstances, the standard deviation, S , which represents only within-subgroup variation, can

w

replace S as an estimator.

t

2

SS

R

∑∑ii

S ≈ or or (9)

w

dmc m

24

where

R is the average range calculated from a set of m subgroup ranges;

S is the observed sample standard deviation of the ith subgroup;

i

m is the number of subgroups of the same size, n;

d , c are constants based on subgroup size, n (see ISO 8258).

2 4

The value of the estimators S and S converge for a process in a state of statistical control. So, a comparison of the two

t w

gives an indication of the degree of stability of the process. For an out-of-control process about a constant mean, or, for a

process that is subject to systematic change in the mean (see 3.1.1.6, Note 4), the value of S is likely to significantly

w

underestimate the process standard deviation.

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ISO 21747:2006(E)

Hence S should be used with extreme caution. Sometimes, too, the estimator S is preferred to S because it has more

w t w

tractable statistical properties (e.g. facilitating the calculation of confidence limits).

NOTE 3 For a normal distribution, process capability can be assessed from the expression:

process capability = X ±(zS ) (10)

t

where

1

X = X (11)

∑ i

m

X is the observed mean of the ith subgroup. Note that X gives identical results to X (see 3.1.3.1, Note 3).

i t

The choice of the value of “z” depends on the particular parts per million capability standard used. Typically “z” takes the

value of 3, 4 or 5. If the process capability meets the specified requirements, a z value of 3 indicates an expected

2 700 parts per million outside of spec

**...**

SLOVENSKI STANDARD

SIST ISO 21747:2006

01-september-2006

6WDWLVWLþQHPHWRGH±6WDWLVWLNHGHORYDQMDLQVSRVREQRVWLSURFHVD]DPHUMHQH

NDUDNWHULVWLNHNDNRYRVWL

Statistical methods -- Process performance and capability statistics for measured quality

characteristics

Méthodes statistiques -- Performances de processus et statistiques d'aptitude pour les

caractéristiques de qualité mesurées

Ta slovenski standard je istoveten z: ISO 21747:2006

ICS:

03.120.30 8SRUDEDVWDWLVWLþQLKPHWRG Application of statistical

methods

SIST ISO 21747:2006 en

2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

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SIST ISO 21747:2006

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SIST ISO 21747:2006

INTERNATIONAL ISO

STANDARD 21747

First edition

2006-07-01

Statistical methods — Process

performance and capability statistics

for measured quality characteristics

Méthodes statistiques — Performances de processus et statistiques

d'aptitude pour les caractéristiques de qualité mesurées

Reference number

ISO 21747:2006(E)

©

ISO 2006

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SIST ISO 21747:2006

ISO 21747:2006(E)

PDF disclaimer

This PDF file may contain embedded typefaces. In accordance with Adobe's licensing policy, this file may be printed or viewed but

shall not be edited unless the typefaces which are embedded are licensed to and installed on the computer performing the editing. In

downloading this file, parties accept therein the responsibility of not infringing Adobe's licensing policy. The ISO Central Secretariat

accepts no liability in this area.

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ii © ISO 2006 – All rights reserved

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SIST ISO 21747:2006

ISO 21747:2006(E)

Contents Page

Foreword. iv

Introduction . v

1 Scope . 1

2 Normative references . 1

3 Terms and definitions. 1

3.1.1 Variation-related concepts. 1

3.1.2 Fundamental process performance and process capability related terms. 3

3.1.3 Process performance — measured data. 6

3.1.4 Process capability — measured data . 8

3.2 Specifications, values and test results. 10

3.2.1 Specification-related concepts. 10

4 Symbols and abbreviated terms . 12

5 Process analysis. 13

6 Time-dependent distribution models. 13

7 Process capability and performance indices . 22

7.1 Methods for the determination of performance and capability indices — Overview . 22

7.2 General geometric method (M1 ). 23

l,d

7.3 Explicit inclusion of additional variation (M2 ). 26

l,d,a

7.4 Alternative method of explicit inclusion of additional variation (M3 ). 27

l,d,a

7.5 Calculation of fractions nonconforming (M4). 28

7.6 One-sided specification limits. 29

8 Reporting process performance/capability indices . 31

Bibliography . 32

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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 21747 was prepared by Technical Committee ISO/TC 69, Application of Statistical Methods,

Subcommittee SC 4, Application of Statistical Methods and Process Management.

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Introduction

Many standards have been created concerning the quality capability/performance of processes by

international, regional and national standardization bodies and also by industry. However, all of them assume

that the process is in a state of statistical control, with stationary, normal processes behaviour. However, a

comprehensive analysis of production processes shows that it is very rare for processes to remain in a

normally distributed, stationary state. In recognition of this fact, this International Standard provides a

framework for estimating the quality capability/performance of industrial processes for an array of standard

processes. These standard processes are categorized by the stability of the first and second distributional

moments, as to whether they are constant, change systematically, or randomly. As such, the quality

capability/performance can be assessed for very differently shaped distributions with respect to time.

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SIST ISO 21747:2006

INTERNATIONAL STANDARD ISO 21747:2006(E)

Statistical methods — Process performance and capability

statistics for measured quality characteristics

1 Scope

This International Standard describes a procedure for the determination of statistics in order to estimate the

quality capability of product and process characteristics. The process results of these quality characteristics

are tabularized into eight possible distribution types. Calculation formulae for the statistical values are placed

with every distribution.

These statistics relate to continuous quality characteristics exclusively. This International Standard is

applicable to processes in any industrial or economical sector.

NOTE This method is usually applied in case of a great number of serial process results, but it can also be used for

small series (a small number of process results).

2 Normative references

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 9000:2005, Quality management systems — Fundamentals and vocabulary

3 Terms and definitions

For the purpose of this document, the terms and definitions given in ISO 9000 and the following apply.

3.1

quality characteristic

inherent characteristic of a product, process or system related to a requirement

NOTE 1 Inherent means existing in something, especially as a permanent characteristic.

NOTE 2 A characteristic assigned to a product, process or system (e.g. the price of a product, the owner of a product)

is not a quality characteristic of that product, process or system.

[ISO 9000:2005, 3.5.2]

3.1.1 Variation-related concepts

3.1.1.1

variation

difference between values of a characteristic

NOTE Variation is often expressed as a variance or standard deviation.

1)

[ISO 3534-2:— , 2.2.1]

1) To be published. (Revision of ISO 3534-2:1993)

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3.1.1.2

inherent process variation

variation (3.1.1.1) in a process when the process is operating in a state of statistical control

NOTE 1 When it is expressed in terms of standard deviation, the subscript “w” is applied, (e.g. σ , S , or s ), indicating

w w w

inherent. See also 3.1.4.1, NOTE 2.

NOTE 2 This variation corresponds to “within subgroup variation”.

[ISO 3534-2:—, 2.2.2]

3.1.1.3

total process variation

variation (3.1.1.1) in a process due to both special causes (3.1.1.4) and random causes (3.1.1.5)

NOTE 1 When it is expressed in terms of standard deviation, the subscript “t” is applied (e.g. σ , S or s ), indicating total.

t t t

See also 3.1.3.1, Note 3.

NOTE 2 This variation corresponds with the combination of the “within-subgroup variation” and the “between-subgroup

variation”.

[ISO 3534-2:—, 2.2.3]

3.1.1.4

special cause

〈process variation〉 source of process variation other than inherent process variation (3.1.1.2)

NOTE 1 Sometimes “special cause” is taken to be synonymous with “assignable cause”. However, a distinction is

recognized. A special cause is assignable only when it is specifically identified.

NOTE 2 A special cause arises because of specific circumstances that are not always present. As such, in a process

subject to special causes, the magnitude of the variation from time to time is unpredictable.

[ISO 3534-2:—, 2.2.4]

3.1.1.5

random cause

common cause

chance cause

〈process variation〉 source of process variation that is inherent in a process over time

NOTE 1 In a process subject only to random cause variation, the variation is predictable within statistically established

limits.

NOTE 2 The reduction of these causes gives rise to process improvement. However, the extent of their identification,

reduction and removal is the subject of cost/benefit analysis in terms of technical tractability and economics.

[ISO 3534-2:—, 2.2.5]

3.1.1.6

stable process

process in a state of statistical control

〈constant mean〉 process subject to only random causes (3.1.1.5)

NOTE 1 A stable process will generally behave as though the samples from the process at any time are simple random

samples from the same population.

NOTE 2 This state does not imply that the random variation is large or small, within or outside of specification, but

rather that the variation (3.1.1.1) is predictable using statistical techniques.

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NOTE 3 The process capability (3.1.4.1) of a stable process is usually improved by fundamental changes that reduce

or remove some of the random causes present and/or adjusting the mean towards the preferred value.

NOTE 4 In some processes, the mean of a characteristic can have a drift or the standard deviation can increase due,

for example, to wear out of tools or depletion of concentration in a solution. A progressive change in the mean or standard

deviation of such a process is considered due to systematic and not random causes. The results, then, are not simple

random samples from the same population.

[ISO 3534-2:—, 2.2.7]

3.1.1.7

out-of-control criteria

set of decision rules for identifying the presence of special causes (3.1.1.4)

NOTE Decision rules may include those relating to points outside of control limits, runs, trends, cycles, periodicity,

concentration of points near the centre line or control limits, unusual spread of points within control limits (large or small

dispersion) and relationships among values within subgroups.

[ISO 3534-2:—, 2.2.8]

3.1.2 Fundamental process performance and process capability related terms

3.1.2.1

distribution

〈of a characteristic〉 information on the probabilistic behaviour of a characteristic

NOTE 1 The distribution of a characteristic can be represented, for example, by ranking of the values of the

characteristic and showing the resulting pattern of measures or scores in the form of a tally chart or histogram. Such a

pattern provides all of the numerical value information on the characteristic except for the serial order in which the data

arises.

NOTE 2 The distribution of a characteristic is dependent on prevailing conditions. Thus, if meaningful information about

the distribution of a characteristic is desired, the conditions under which the data is collected should be specified.

NOTE 3 It is important to know the class of distribution, for instance, normal or log-normal, before predicting or

estimating process capability and performance measures and indices or fraction nonconforming.

[ISO 3534-2:—, 2.5.1]

3.1.2.2

class of distributions

particular family of distributions (3.1.2.1) each member of which has the same common attributes by which

the family is fully specified

EXAMPLE 1 The two-parameter, symmetrical bell-shaped, normal distribution with parameters mean and standard

deviation.

EXAMPLE 2 The three-parameter Weibull distribution with parameters location, shape and scale.

EXAMPLE 3 The unimodal continuous distributions.

NOTE The class of distributions can often be fully specified through the values of appropriate parameters.

[ISO 3534-2:—, 2.5.2]

3.1.2.3

distribution model

specified distribution (3.1.2.1) or class of distributions (3.1.2.2)

EXAMPLE 1 A model for the distribution of a product characteristic, the diameter of a bolt, might be the normal

distribution with mean 15 mm and standard deviation 0,05 mm. Here the model is a fully specified one.

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EXAMPLE 2 A model for the diameter of bolts as in Example 1 could be the class of normal distributions without

attempting to specify a particular distribution. Here the model is the class of normal distributions.

[ISO 3534-2:—, 2.5.3]

3.1.2.4

upper fraction nonconforming

p

U

fraction of the distribution (3.1.2.1) of a characteristic that is greater than the upper specification limit

(3.2.1.3), U

EXAMPLE In a normal distribution, with mean, µ, and standard deviation, σ :

⎛⎞UU−−µµ⎛⎞

p =−1ΦΦ= (1)

U ⎜⎟ ⎜⎟

σσ

⎝⎠ ⎝⎠

where

p is the upper fraction nonconforming;

U

Φ is the distribution function of the standard normal distribution;

U is the upper specification limit.

NOTE 1 Tables (or functions in statistical computer packages) of the standard normal distribution are readily available

which give the proportion of process output expected beyond a particular value of interest, such as a specification limit

(3.2.1.2), in terms of standard deviations away from the process mean. This obviates the need to work out the statistical

distribution function given in the example.

NOTE 2 The function relates to a theoretical distribution. In practice, with empirical distributions, the parameters are

replaced by their estimates.

[ISO 3534-2:—, 2.5.4]

3.1.2.5

lower fraction nonconforming

p

L

fraction of the distribution (3.1.2.1) of a characteristic that is less than the lower specification limit (3.2.1.4),

L

EXAMPLE In a normal distribution (3.1.2.1), with mean, µ, and standard deviation, σ :

L − µ

⎛⎞

p =Φ (2)

L ⎜⎟

σ

⎝⎠

where

p is the lower fraction nonconforming;

L

Φ is the distribution function of the standard normal distribution;

L is the lower specification limit.

NOTE 1 Tables (or functions in statistical computer packages) of the standard normal distribution are readily available

which give the proportion of process output expected beyond a particular value of interest, such as a specification limit

(3.2.1.2), in terms of standard deviations away from the process mean. This obviates the need to work out the statistical

distribution function given in the example.

NOTE 2 The function relates to a theoretical distribution. In practice, with empirical distributions, the parameters are

replaced by their estimates.

[ISO 3534-2:—, 2.5.5]

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3.1.2.6

total fraction nonconforming

p

t

sum of upper fraction nonconforming (3.1.2.4) and lower fraction nonconforming (3.1.2.5)

EXAMPLE In a normal distribution, with mean, µ, and standard deviation, σ :

⎛⎞µµ−−UL⎛ ⎞

p=+ΦΦ (3)

t ⎜⎟ ⎜ ⎟

σσ

⎝⎠ ⎝ ⎠

where

p is the total fraction nonconforming;

t

Φ is the distribution function of the standard normal distribution;

L is the lower specification limit;

U is the upper specification limit.

NOTE 1 Tables (or functions in statistical computer packages) of the standard normal distribution are readily available

which give the proportion of process output expected beyond a particular value of interest, such as a specification limit

(3.2.1.2), in terms of standard deviations away from the process mean. This obviates the need to work out the statistical

distribution function given in the example.

NOTE 2 The function relates to a theoretical distribution. In practice, with empirical distributions, the parameters are

replaced by their estimates.

[ISO 3534-2:—, 2.5.6]

3.1.2.7

reference interval

interval bounded by the 99,865 % distribution quantile, X , and the 0,135 % distribution quantile,

99,865 %

X

0,135 %

NOTE 1 The interval can be expressed by (X , X ) and the length of the interval is X – X .

99,865 % 0,135 % 99,865 % 0,135 %

NOTE 2 This term is used only as an arbitrary, but standardized, basis for defining the process performance index

(3.1.3.2) and process capability index (3.1.4.2).

NOTE 3 For a normal distribution (3.1.2.1), the length of the reference interval can be expressed in terms of six

standard deviations, 6σ, or 6S, when estimated from a sample.

NOTE 4 For a non-normal distribution, the length of the reference interval can be estimated by means of appropriate

probability papers (e.g. log-normal) or from the sample kurtosis and sample skewness using the methods described in

2)

ISO/TR 12783 .

NOTE 5 A quantile or fractile indicates division of a distribution into equal units or fractions, e.g. percentiles. Quantile is

defined in ISO 3534-1.

[ISO 3534-2:—, 2.5.7]

3.1.2.8

lower reference interval

interval bounded by the 50 % distribution quantile, X and the 0,135 % distribution quantile, X

50 % 0,135 %

NOTE 1 The interval can be expressed by (X , X ) and the length of the interval is X – X .

50 % 0,135 % 50 % 0,135 %

2) Under preparation.

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NOTE 2 This term is used only as an arbitrary, but standardized, basis for defining the lower process performance

index (3.1.3.3) and lower process capability index (3.1.4.3).

NOTE 3 For a normal distribution (3.1.2.1), the length of the lower reference interval can be expressed in terms of

standard deviations as 3σ, or an estimated 3S, and X represents both the mean and the median.

50 %

NOTE 4 For a non-normal distribution, the 50 % distribution quantile, X , namely the median, and the 0,135 %

50 %

distribution quantile, X , can be estimated by means of appropriate probability papers (e.g. log-normal) or from the

0,135 %

2)

sample kurtosis and sample skewness using the methods described in ISO/TR 12783 .

[ISO 3534-2:—, 2.5.8]

3.1.2.9

upper reference interval

interval bounded by the 99,865 % distribution quantile, X , and the 50 % distribution quantile, X

99,865 % 50 %

NOTE 1 The interval can be expressed by (X , X ) and the length of the interval is X – X .

99,865 % 50 % 99,865 % 50 %

NOTE 2 This term is used only as an arbitrary, but standardized, basis for defining the upper process performance

index (3.1.3.4) and upper process capability index (3.1.4.4).

NOTE 3 For a normal distribution (3.1.2.1), the length of the upper reference interval can be expressed in terms of

standard deviations as 3σ, or an estimated 3S, and X represents both the mean and the median.

50 %

NOTE 4 For a non-normal distribution, the 50 % distribution quantile, X , namely the median, and the 99,865 %

50 %

distribution quantile, X , can be estimated by means of appropriate probability papers (e.g. log-normal) or from the

99,865 %

2)

sample kurtosis and sample skewness using the methods described in ISO/TR 12783 .

[ISO 3534-2:—, 2.5.9]

3.1.3 Process performance — Measured data

3.1.3.1

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

NOTE 1 The outcome is a distribution (3.1.2.1), the class of which needs determination and its parameters assessed.

NOTE 2 Care should be exercised in using this measure as it may contain a component of variability due to special

causes (3.1.1.4), the value of which is not predictable.

NOTE 3 For a normal distribution described in terms of the standard deviation, S , assessed from only one sample of

t

size N, the standard deviation is expressed thus:

2

1

SX=−X (4)

()

tti

∑

N −1

where

1

X = X (5)

t ∑ i

N

This descriptor, S , takes into account the variation due to random (common) causes (3.1.1.5) together with any special

t

causes that may be present. S is used here instead of σ as the standard deviation is a statistical descriptive measure. The

t t

sample size N can be made up of m subgroups, each of size n.

NOTE 4 For a normal distribution, process performance can be assessed from the expression:

process performance=±X (zS )

tt

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and, “z” is dependent on the particular parts per million performance requirement. Typically “z” takes the value of 3, 4 or 5.

If the process performance coincides with the specified requirements, a z value of 3 indicates an expected 2 700 parts per

million outside of specification. Similarly, a z of 4 indicates an expected 64 parts per million and a z of 5 an expected

0,6 parts per million outside of specification.

NOTE 5 For a non-normal distribution, process performance can be assessed using, for example, an appropriate

probability paper or from the parameters of the distribution fitted to the data. The expression for process performance

takes the form:

+a

process performance = X

t −b

+a

The notation, , is in the same style as standard drawing office practice for expressing specified tolerances about a

−b

nominal, or preferred, value for a characteristic, when the preferred value is not equidistant from each limit. The equivalent

notation for limits symmetrical about the preferred value is ±. This enables a direct comparison to be made between the

dimensional performance of a characteristic and its specified requirements in terms of both location and dispersion.

[ISO 3534-2:—, 2.6.1]

3.1.3.2

process performance index

P

p

index describing process performance (3.1.3.1) in relation to specified tolerance

NOTE 1 Frequently, the process performance index is expressed as the value of the specified tolerance divided by a

measure of the length of the reference interval (3.1.2.7), namely as:

UL−

P = (6)

p

XX−

99,865 % 0,135 %

NOTE 2 For a normal distribution (3.1.2.1), the length of the reference interval is equal to 6S (see 3.1.3.1, Note 3).

t

NOTE 3 For a non-normal distribution, the length of the reference interval can be estimated using, for example, the

2)

method described in ISO/TR 12783 .

[ISO 3534-2:—, 2.6.2]

3.1.3.3

lower process performance index

P

pkL

index describing process performance (3.1.3.1) in relation to the lower specification limit (3.2.1.4), L

NOTE 1 Frequently, the lower process performance index is expressed by the difference between the 50 % distribution

quantile, X , and lower specification limit (3.2.1.4) divided by a measure of the length of the lower reference interval

50 %

(3.1.2.8), namely as:

XL−

50 %

P =

pkL

XX−

50 % 0,135 %

(7)

NOTE 2 For the symmetrical normal distribution (3.1.2.1), the length of the lower reference interval is equal to 3S

t

(see 3.1.3.1, Note 3) and X represents both the mean and the median.

50 %

NOTE 3 For a non-normal distribution, the length of the lower reference range can be estimated using the method

2)

described in ISO/TR 12783 and X represents the median.

50 %

[ISO 3534-2:—, 2.6.3]

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3.1.3.4

upper process performance index

P

pkU

index describing process performance (3.1.3.1) in relation to the upper specification limit (3.2.1.3), U

NOTE 1 Frequently, the upper process performance index is expressed as the difference between the upper

specification limit and the 50 % distribution quantile, X , divided by a measure of the length of the upper reference

50 %

interval (3.1.2.9), namely as:

UX−

50 %

(8)

P =

pkU

XX−

99,865 % 50 %

NOTE 2 For a normal distribution (3.1.2.1), the length of the upper reference interval is equal to 3S (see 3.1.3.1,

t

Note 3) and X represents both the mean and the median.

50 %

NOTE 3 For a non-normal distribution, the length of the upper reference interval can be estimated using the method

2)

described in ISO/TR 12783 and X represents the median.

50 %

[ISO 3534-2:—, 2.6.4]

3.1.3.5

minimum process performance index

P

pk

smaller of upper process performance index (3.1.3.4) and lower process performance index (3.1.3.3)

[ISO 3534-2:—, 2.6.5]

3.1.4 Process capability — Measured data

3.1.4.1

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 fulfil the

requirements for that characteristic

NOTE 1 The outcome is a distribution (3.1.2.1), the class of which needs determination and its parameters estimated.

NOTE 2 For a normal distribution, the process overall standard deviation, σ , can be estimated using the formula for S

t

t

(see 3.1.3.1, Note 3).

Alternatively, in certain circumstances, the standard deviation, S , which represents only within-subgroup variation, can

w

replace S as an estimator.

t

2

SS

R

∑∑ii

S ≈ or or (9)

w

dmc m

24

where

R is the average range calculated from a set of m subgroup ranges;

S is the observed sample standard deviation of the ith subgroup;

i

m is the number of subgroups of the same size, n;

d , c are constants based on subgroup size, n (see ISO 8258).

2 4

The value of the estimators S and S converge for a process in a state of statistical control. So, a comparison of the two

t w

gives an indicati

**...**

FINAL

INTERNATIONAL ISO/FDIS

DRAFT

STANDARD 21747

ISO/TC 69/SC 4

Statistical methods — Process

Secretariat: AFNOR

performance and capability statistics

Voting begins on:

for measured quality characteristics

2006-01-09

Voting terminates on:

Méthodes statistiques — Performances de processus et statistiques

2006-03-09

d'aptitude pour les caractéristiques de qualité mesurées

Please see the administrative notes on page iii

RECIPIENTS OF THIS DRAFT ARE INVITED TO

SUBMIT, WITH THEIR COMMENTS, NOTIFICATION

OF ANY RELEVANT PATENT RIGHTS OF WHICH

THEY ARE AWARE AND TO PROVIDE SUPPORT-

ING DOCUMENTATION.

IN ADDITION TO THEIR EVALUATION AS

Reference number

BEING ACCEPTABLE FOR INDUSTRIAL, TECHNO-

ISO/FDIS 21747:2006(E)

LOGICAL, COMMERCIAL AND USER PURPOSES,

DRAFT INTERNATIONAL STANDARDS MAY ON

OCCASION HAVE TO BE CONSIDERED IN THE

LIGHT OF THEIR POTENTIAL TO BECOME STAN-

DARDS TO WHICH REFERENCE MAY BE MADE IN

©

NATIONAL REGULATIONS. ISO 2006

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ISO/FDIS 21747:2006(E)

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ISO/FDIS 21747:2006(E)

In accordance with the provisions of Council Resolution 15/1993, this document is circulated in the

English language only.

© ISO 2006 – All rights reserved iii

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ISO/FDIS 21747:2006(E)

Contents Page

Foreword. v

Introduction . vi

1 Scope. 1

2 Normative references . 1

3 Terms and definitions. 1

3.1.1 Variation-related concepts. 1

3.1.2 Fundamental process performance and process capability related terms . 3

3.1.3 Process performance — measured data . 6

3.1.4 Process capability — measured data . 8

3.2 Specifications, values and test results. 10

3.2.1 Specification-related concepts. 10

4 Symbols and abbreviated terms. 12

5 Process analysis . 13

6 Time-dependent distribution models. 13

7 Process capability and performance indices. 22

7.1 Methods for the determination of performance and capability indices — Overview. 22

7.2 General geometric method (M1 ) . 23

l,d

7.3 Explicit inclusion of additional variation (M2 ). 26

l,d,a

7.4 Alternative method of explicit inclusion of additional variation (M3 ) . 27

l,d,a

7.5 Calculation of fractions nonconforming (M4) . 28

7.6 One-sided specification limits . 29

8 Reporting process performance/capability indices . 31

Bibliography . 32

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ISO/FDIS 21747:2006(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

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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 21747 was prepared by Technical Committee ISO/TC 69, Application of Statistical Methods,

Subcommittee SC 4, Application of Statistical Methods and Process Management.

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ISO/FDIS 21747:2006(E)

Introduction

Many standards have been created concerning the quality capability/performance of processes by

international, regional and national standardization bodies and also by industry. However, all of them assume

that the process is in a state of statistical control, with stationary, normal processes behaviour. However, a

comprehensive analysis of production processes shows that it is very rare for processes to remain in a

normally distributed, stationary state. In recognition of this fact, this International Standard provides a

framework for estimating the quality capability/performance of industrial processes for an array of standard

processes. These standard processes are categorized by the stability of the first and second distributional

moments, as to whether they are constant, change systematically, or randomly. As such, the quality

capability/performance can be assessed for very differently shaped distributions with respect to time.

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FINAL DRAFT INTERNATIONAL STANDARD ISO/FDIS 21747:2006(E)

Statistical methods — Process performance and capability

statistics for measured quality characteristics

1 Scope

This International Standard describes a procedure for the determination of statistics in order to estimate the

quality capability of product and process characteristics. The process results of these quality characteristics

are tabularized into eight possible distribution types. Calculation formulae for the statistical values are placed

with every distribution.

These statistics relate to continuous quality characteristics exclusively. This International Standard is

applicable to processes in any industrial or economical sector.

NOTE This method is usually applied in case of a great number of serial process results, but it can also be used for

small series (a small number of process results).

2 Normative references

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 8258, Shewhart control charts

ISO 9000, Quality management systems — Fundamentals and vocabulary

3 Terms and definitions

For the purpose of this document, the terms and definitions given in ISO 9000 and the following apply.

3.1

quality characteristic

inherent characteristic of a product, process or system related to a requirement

NOTE 1 Inherent means existing in something, especially as a permanent characteristic.

NOTE 2 A characteristic assigned to a product, process or system (e.g. the price of a product, the owner of a product)

is not a quality characteristic of that product, process or system.

[ISO 9000]

3.1.1 Variation-related concepts

3.1.1.1

variation

difference between values of a characteristic

NOTE Variation is often expressed as a variance or standard deviation.

[ISO 3534-2]

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ISO/FDIS 21747:2006(E)

3.1.1.2

inherent process variation

variation (3.1.1.1) in a process when a process is operating in a state of statistical control

NOTE 1 When it is expressed in terms of standard deviation, the subscript “w” is applied, (e.g. σ , S , or s ), indicating

w w w

inherent. See also 3.1.4.1, NOTE 2.

NOTE 2 This variation corresponds to “within subgroup variation”.

[ISO 3534-2]

3.1.1.3

total process variation

variation (3.1.1.1) in a process due to both special causes (3.1.1.4) and random causes (3.1.1.5)

NOTE 1 When it is expressed in terms of standard deviation, the subscript “t” is applied (e.g. σ , S or s ), indicating total.

t t t

NOTE 2 This variation is correspondent to the sum of the “within subgroup variation” and the “between subgroup

variation”.

[ISO 3534-2]

3.1.1.4

special cause

〈process variation〉 source of process variation other than inherent process variation (3.1.1.2)

NOTE 1 Sometimes “special cause” is taken to be synonymous with “assignable cause”. However, a distinction is

recognized. A special cause is assignable only when it is specifically identified.

NOTE 2 A special cause arises because of specific circumstances which are not always present. As such, in a process

subject to special causes, the magnitude of the variation from time to time is unpredictable.

[ISO 3534-2]

3.1.1.5

random cause

common cause

chance cause

〈process variation〉 source of process variation that is inherent in a process over time

NOTE 1 In a process subject only to random cause variation, the variation is predictable within statistically-established

limits.

NOTE 2 The reduction of these causes gives rise to process improvement. However, the extent of their identification,

reduction and removal will be the subject of cost/benefit analysis in terms of technical tractability and economics.

[ISO 3534-2]

3.1.1.6

stable process

process in a state of statistical control

〈constant mean〉 process subject to only random causes (3.1.1.5)

NOTE 1 A stable process will generally behave as though the samples from the process at any time are simple random

samples from the same population.

NOTE 2 This state does not imply that the random variation is large or small, within or outside of specification, but

rather that the variation (3.1.1.1) is predictable using statistical techniques.

NOTE 3 The process capability (3.1.4.1) of a stable process is usually improved by fundamental changes that reduce

or remove some of the random causes present and/or adjusting the mean towards the preferred value.

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ISO/FDIS 21747:2006(E)

NOTE 4 In some processes, the mean of a characteristic can have a drift or the standard deviation can increase due,

for example, to wear out of tools or depletion of concentration in a solution. A progressive change in the mean or standard

deviation of such a process is considered due to systematic and not random causes. The results, then, are not simple

random samples from the same population.

[ISO 3534-2]

3.1.1.7

out-of-control criteria

set of decision rules for identifying the presence of special causes (3.1.1.4)

NOTE Decision rules may include those relating to points outside of control limits, runs, trends, cycles, periodicity,

concentration of points near the centre line or control limits, unusual spread of points within control limits (dispersion large

or small) and relationships among values within sub-groups.

[ISO 3534-2]

3.1.2 Fundamental process performance and process capability related terms

3.1.2.1

distribution

〈of a characteristic〉 information on the probabilistic behaviour of a characteristic

NOTE 1 The distribution of a characteristic can be represented, for example, by ranking of the values of the

characteristic and showing the resulting pattern of measures or scores in the form of a tally chart or histogram. Such a

pattern provides all of the numerical value information on the characteristic except for the serial order in which the data

arises.

NOTE 2 The distribution of a characteristic is dependent on prevailing conditions. Thus, if meaningful information about

the distribution of a characteristic is desired the conditions under which the data is collected should be specified.

NOTE 3 It is important to know the class of distribution, for instance, normal or log normal, before predicting or

estimating process capability and performance measures and indices or fraction non-conforming.

[ISO 3534-2]

3.1.2.2

class of distributions

particular family of distributions (3.1.2.1) each member of which have the same common attributes by which

the family is fully specified

EXAMPLE 1 The two-parameter, symmetrical bell-shaped, normal distribution with parameters (mean and standard

deviation).

EXAMPLE 2 The three-parameter Weibull distribution with parameters (location, shape and scale).

EXAMPLE 3 The unimodal continuous distributions.

NOTE The class of distributions can often be fully specified through the values of appropriate parameters.

[ISO 3534-2]

3.1.2.3

distribution model

specified distribution (3.1.2.1) or class of distributions (3.1.2.2)

EXAMPLE 1 A model for the distribution of a product characteristic, the diameter of a bolt, might be the normal

distribution with mean 15 mm and standard deviation 0,05 mm. Here the model is a fully specified one.

EXAMPLE 2 A model for the diameter of bolts as in EXAMPLE 1 could be the class of normal distributions without

attempting to specify a particular distribution. Here the model is the class of normal distributions.

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ISO/FDIS 21747:2006(E)

[ISO 3534-2]

3.1.2.4

upper fraction nonconforming

fraction of the distribution (3.1.2.1) of a characteristic that is greater than the upper specification limit

(3.2.1.3)

EXAMPLE In a normal distribution, with mean, µ, and standard deviation, σ:

⎛⎞UU−−µµ⎛⎞

p =−1ΦΦ= (1)

U ⎜⎟ ⎜⎟

σσ

⎝⎠ ⎝⎠

where

p is the upper fraction nonconforming;

U

Φ is the distribution function of the standard normal distribution;

U is the upper specification limit.

NOTE 1 Tables (or functions in statistical computer packages) of the standard normal distribution are readily available

which give the proportion of process output expected beyond a particular value of interest, such as a specification limit, in

terms of standard deviations away from the process mean. This obviates the need to work out the statistical distribution

function given in the example.

NOTE 2 The function relates to a theoretical distribution. In practice, with empirical distributions, the parameters are

replaced by estimates.

[ISO 3534-2]

3.1.2.5

lower fraction nonconforming

fraction of the distribution (3.1.2.1) of a characteristic that is less than the lower specification limit (3.2.1.4)

EXAMPLE In a normal distribution (3.1.2.1), with mean, µ, and standard deviation, σ:

⎛⎞L − µ

p =Φ (2)

L ⎜⎟

σ

⎝⎠

where

p is the lower fraction nonconforming;

L

Φ is the distribution function of the standard normal distribution;

L is the lower specification limit.

NOTE 1 Tables (or functions in statistical computer packages) of the standard normal distribution are readily available

which give the proportion of process output expected beyond a particular value of interest, such as a specification limit

(3.2.1.2), in terms of standard deviations away from the process mean. This obviates the need to work out the statistical

distribution function given in the example.

NOTE 2 The function relates to a theoretical distribution. In practice, with empirical distributions, the parameters are

replaced by estimates.

[ISO 3534-2]

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ISO/FDIS 21747:2006(E)

3.1.2.6

total fraction nonconforming

sum of upper fraction nonconforming (3.1.2.4) and lower fraction nonconforming (3.1.2.5)

EXAMPLE In a normal distribution, with mean, µ, and standard deviation, σ:

⎛⎞µµ−−UL⎛ ⎞

p=+ΦΦ (3)

t ⎜⎟ ⎜ ⎟

σσ

⎝⎠ ⎝ ⎠

where

p is the total fraction nonconforming;

t

Φ is the distribution function of the standard normal distribution;

U is the upper specification limit;

L is the lower specification limit.

NOTE 1 Tables (or functions in statistical computer packages) of the standard normal distribution are readily available

which give the proportion of process output expected beyond a particular value of interest, such as a specification limit, in

terms of standard deviations away from the process mean. This obviates the need to work out the statistical distribution

function given in the example.

NOTE 2 The function relates to a theoretical distribution. In practice, with empirical distributions, the parameters are

replaced by estimates.

[ISO 3534-2]

3.1.2.7

reference interval

interval bounded by the 99,865 % distribution fractile, X , and the 0,135 % distribution fractile, X

99,865 % 0,135 %

NOTE 1 The interval can be expressed by (X , X ) and the length of the interval is X – X .

99,865 % 0,135 % 99,865 % 0,135 %

NOTE 2 This term is used only as an arbitrary, but standardized, basis for defining the process performance index

(3.1.3.2) and process capability index (3.1.4.2).

NOTE 3 For a normal distribution (3.1.2.1), the reference interval may be expressed in terms of six standard

deviations, 6σ, or 6S, when estimated from a sample.

NOTE 4 For a non-normal distribution, the reference interval may be estimated by means of appropriate probability

papers (e.g. log-normal) or from the sample kurtosis and sample skewness using the methods described in

1)

ISO/TR 12783 .

NOTE 5 A quantile or fractile indicates division of a distribution into equal units or fractions, e.g. percentiles. Quantile is

defined in ISO 3534-1.

[ISO 3534-2]

3.1.2.8

lower reference interval

interval bounded by the 50 % distribution fractile, X and the 0,135 % distribution fractile, X

50 % 0,135 %

NOTE 1 The interval can be expressed by (X , X ) and the length of the interval is X – X .

50 % 0,135 % 50 % 0,135 %

NOTE 2 This term is used only as an arbitrary, but standardized, basis for defining the lower process performance

index (3.1.3.3) and lower process capability index (3.1.4.3).

1) Under preparation.

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ISO/FDIS 21747:2006(E)

NOTE 3 For a normal distribution (3.1.2.1), the length of the lower reference interval can be expressed in terms of

standard deviations as, 3σ or an estimated 3S, and X represents both the mean and the median.

50 %

NOTE 4 For a non-normal distribution, the 50 % distribution fractile, X , namely the median, and the 0,135 %

50 %

distribution fractile, X , may be estimated by means of appropriate probability papers (e.g. log-normal) or from the

0,135 %

1)

sample kurtosis and sample skewness using the methods described in ISO/TR 12783 .

[ISO 3534-2]

3.1.2.9

upper reference interval

interval bounded by the 99,865 % distribution fractile, X , and the 50 % distribution fractile, X

99,865 % 50 %

NOTE 1 The interval can be expressed by (X , X ) and the length of the interval is X – X .

99,865 % 50 % 99,865 % 50 %

NOTE 2 This term is used only as an arbitrary, but standardized, basis for defining the upper process performance

index (3.1.3.4) and upper process capability index (3.1.4.4).

NOTE 3 For a normal distribution (3.1.2.1), the upper reference interval may be expressed in terms of standard

deviations as, 3σ or 3S,and X represents both the mean and the median.

50 %

NOTE 4 For a non-normal distribution, the 50 % distribution fractile, X , namely the median, and the 99,865 %

50 %

distribution fractile, X , may be estimated by means of appropriate probability papers (e.g. log-normal) or from the

99,865 %

1)

sample kurtosis and sample skewness using the methods described in ISO/TR 12783 .

[ISO 3534-2]

3.1.3 Process performance — Measured data

3.1.3.1

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

NOTE 1 The outcome is a distribution (3.1.2.1) the class of which needs determination and its parameters assessed.

NOTE 2 Care should be exercised in using this measure as it may contain a component of variability due to special

causes (3.1.1.4) the value of which is not predictable.

NOTE 3 For a normal distribution described in terms of the standard deviation, S , estimated from only one sample of

t

size N, the standard deviation is expressed thus:

2

1

SX=−X (4)

t

t ()i

∑

N −1

where

1

X = X (5)

t

∑ i

N

This descriptor, S , takes into account the variation both due to random (common) causes (3.1.1.5) together with any

t

special causes that may be present. S is used here instead of σ as the standard deviation is a statistical descriptive

t t

measure. The sample size N can be made up of k subgroups each of size n.

NOTE 4 For a normal distribution, process performance can be assessed from the expression:

X ± zS

tt

and, “z” is dependent on the particular parts per million performance requirement. Typically “z” takes the value of 3, 4 or 5.

If the process performance coincides with the specified requirements, a z value of 3 indicates an expected 2 700 parts per

million outside of specification. Similarly a z of 4 indicates an expected 64 parts per million and a z of 5 an expected

0,6 parts per million outside of specification.

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ISO/FDIS 21747:2006(E)

NOTE 5 For a non-normal distribution, process performance can be assessed using, for example, an appropriate

probability paper or from the parameters of the distribution fitted to the data. The expression for process performance

takes the form:

+a

X

t −b

+a

The notation, , is in the same style as standard drawing office practice for expressing specified tolerances about a

−b

nominal, or preferred, value for a characteristic, when the preferred value is not equidistant from each limit. The equivalent

notation for limits symmetrical about the preferred value is, ±. This enables a direct comparison to be made between the

dimensional performance of a characteristic and its specified requirements in terms of both location and dispersion.

[ISO 3534-2]

3.1.3.2

process performance index

index describing process performance (3.1.3.1) in relation to specified tolerance

NOTE 1 Frequently, the process performance index is expressed as the value of the specified tolerance divided by a

measure of the length of the reference interval (3.1.2.7), namely as:

UL−

P = (6)

p

XX−

99,865 % 0,135 %

NOTE 2 For a normal distribution (3.1.2.1), the length of the reference interval is equal to 6S (see 3.1.3.1, NOTE 3).

t

NOTE 3 For a non-normal distribution, the reference interval can be estimated using, for example, the method

1)

described in ISO/TR 12783 .

[ISO 3534-2]

3.1.3.3

lower process performance index

index describing process performance (3.1.3.1) in relation to the lower specification limit (3.2.1.4)

NOTE 1 Frequently, the lower process performance index is expressed by the difference between the 50 % distribution

fractile, X , and lower specification limit (3.2.1.4) divided by a measure of the length of the lower reference interval

50 %

(3.1.2.8), namely as:

XL−

50 %

P = (7)

pkL

XX−

50 % 0,135 %

NOTE 2 For the symmetrical normal distribution (3.1.2.1), the length of the lower reference interval is equal to 3S

t

(see 3.1.3.1, NOTE 3) and X represents both the mean and the median.

50 %

NOTE 3 For a non-normal distribution, the lower reference range can be estimated using the method described in

1)

ISO/TR 12783 and X represents the median.

50 %

[ISO 3534-2]

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ISO/FDIS 21747:2006(E)

3.1.3.4

upper process performance index

index describing process performance (3.1.3.1) in relation to the upper specification limit (3.2.1.3)

NOTE 1 Frequently, the upper process performance index is expressed as the difference between the upper

specification limit and the 50 % distribution fractile, X , divided by a measure of the length of the upper reference

50 %

interval (3.1.2.9), namely as:

UX−

50 %

P = (8)

pkU

XX−

99,865 % 50 %

NOTE 2 For a normal distribution (3.1.2.1), the length of the upper reference interval is equal to 3S (see 3.1.3.1,

t

NOTE 3) and X represents both the mean and the median.

50 %

NOTE 3 For a non-normal distribution, the length of the upper reference interval can be estimated using the method

1)

described in ISO/TR 12783 and X represents the median.

50 %

[ISO 3534-2]

3.1.3.5

minimum process performance index

smaller of upper process performance index (3.1.3.4) and lower process performance index (3.1.3.3)

NOTE The symbol often used for this index is P .

pk

[ISO 3534-2]

3.1.4 Process capability — Measured data

3.1.4.1

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 ability to realize a characteristic that will fulfil the

requirements for that characteristic

NOTE 1 The outcome is a distribution (3.1.2.1) the class of which needs determination and its parameters estimated.

NOTE 2 For a normal distribution, the process overall standard deviation, σ , can be estimated using the formula for S

t

t

(see 3.1.3.1, NOTE 3).

Alternatively, in certain circumstances, the standard deviation, S , which represents only within subgroup variation, can

w

replace S as an estimator.

t

2

SS

R

∑∑ii

S ≈ or or (9)

w

dkc k

24

where

R is the average range calculated from a set of k subgroup ranges;

S is the sample standard deviation of the ith subgroup;

i

k is the number of subgroups of the same size, n;

d , c are constants based on subgroup size, n (see ISO 8258).

2 4

The value of the estimators S and S converge for a process in a state of statistical control. So a comparison of the two

t w

gives an indication of the degree of stability of the process. For an out of control process about a constant mean, or, for a

process that is subject to systematic change in the mean (see 3.1.1.6, NOTE 4), the value of S is likely to significantly

w

underestimate the process standard deviation.

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