ISO 21747:2006

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

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 2006
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©  ISO 2006
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ii © ISO 2006 – All rights reserved

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

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

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.
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)
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.
2 © ISO 2006 – All rights reserved

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.
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]
4 © ISO 2006 – All rights reserved

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.
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:
SX=−X (4)
()
tti
∑
N −1
where
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
6 © ISO 2006 – All rights reserved

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]
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
SS
R
∑∑ii
S ≈ or or (9)
w
dmc m
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 a
...