ISO 22514-4:2016
(Main)Statistical methods in process management — Capability and performance — Part 4: Process capability estimates and performance measures
Statistical methods in process management — Capability and performance — Part 4: Process capability estimates and performance measures
ISO 22514-4:2016 describes process capability and performance measures that are commonly used.
Méthodes statistiques dans la gestion de processus — Aptitude et performance — Partie 4: Estimations de l'aptitude de processus et mesures de performance
Statistične metode za obvladovanje procesov - Zmogljivost in delovanje - 4. del: Ocene zmogljivosti procesov in merila za delovanje
Ta del standarda ISO 22514 opisuje zmogljivost procesov in merila učinkovitosti, ki se običajno uporabljajo.
General Information
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Standards Content (Sample)
SLOVENSKI STANDARD
01-maj-2017
1DGRPHãþD
SIST-TP ISO/TR 22514-4:2010
6WDWLVWLþQHPHWRGH]DREYODGRYDQMHSURFHVRY=PRJOMLYRVWLQGHORYDQMHGHO
2FHQH]PRJOMLYRVWLSURFHVRYLQPHULOD]DGHORYDQMH
Statistical methods in process management - Capability and performance - Part 4:
Process capability estimates and performance measures
Méthodes statistiques dans la gestion de processus - Aptitude et performance - Partie 4:
Estimations de l'aptitude de processus et mesures de performance
Ta slovenski standard je istoveten z: ISO 22514-4:2016
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 22514-4
First edition
2016-08-01
Statistical methods in process
management — Capability and
performance —
Part 4:
Process capability estimates and
performance measures
Méthodes statistiques dans la gestion de processus — Aptitude et
performance —
Partie 4: Estimations de l’aptitude de processus et mesures de
performance
Reference number
©
ISO 2016
© ISO 2016, Published in Switzerland
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ii © ISO 2016 – All rights reserved
Contents Page
Foreword .v
Introduction .vi
1 Scope . 1
2 Symbols and abbreviated terms . 1
2.1 Symbols . 1
2.2 Abbreviated terms . 3
3 Basic concepts used for process capability and performance . 3
3.1 General . 3
3.2 Location . 3
3.3 Dispersion . 3
3.3.1 Inherent dispersion . 3
3.3.2 Total dispersion . 3
3.3.3 Short-term dispersion . 3
3.4 Mean square error (MSE) . 4
3.5 Reference limits . 4
3.6 Reference interval (also known as process spread) . 4
4 Capability . 4
4.1 General . 4
4.2 Process capability . 6
4.2.1 Normal distribution . 6
4.2.2 Non-normal distribution. 7
4.3 Process location . 7
4.4 Process capability indices for measured data . 8
4.4.1 General. 8
4.4.2 C index (for the normal distribution) . 9
p
4.4.3 C index (for the normal distribution) .10
pk
4.4.4 C index for unilateral tolerances .10
pk
4.5 Process capability indices for measured data (non-normal) .10
4.5.1 General.10
4.5.2 Probability paper method .11
4.5.3 Pearson curves method .11
4.5.4 Distribution identification method .12
4.6 Alternative method for describing and calculating process capability estimates .12
4.7 Other capability measures for continuous data .13
4.7.1 Process capability fraction (PCF) .13
4.7.2 Indices when the specification limit is one-sided or no specification limit
is given .13
4.8 Assessment of proportion out-of-specification (normal distribution) .15
5 Performance .16
5.1 General .16
5.2 Process performance indices for measured data (normal distribution) .16
5.2.1 P index .16
p
5.2.2 P index .17
pk
5.3 Process performance indices for measured data (non-normal distribution) .17
5.3.1 General.17
5.3.2 Probability paper method .17
5.3.3 Pearson curves method .18
5.3.4 Distribution identification method .18
5.4 Other performance indices for measured data .18
5.5 Assessment of proportion out-of-specification for a normal distribution of the
total distribution .18
6 Reporting process capability and performance indices .19
Annex A (informative) Estimating standard deviations .21
Annex B (informative) Estimating capability and performance measures using Pearson
curves — Procedure and example .23
Annex C (informative) Distribution identification .37
Annex D (informative) Confidence intervals .42
Bibliography .44
iv © ISO 2016 – All rights reserved
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
through ISO technical committees. Each member body interested in a subject for which a technical
committee has been established has the right to be represented on that committee. International
organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.
ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of
electrotechnical standardization.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www.iso.org/directives).
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of
any patent rights identified during the development of the document will be in the Introduction and/or
on the ISO list of patent declarations received (see www.iso.org/patents).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation on the meaning of ISO specific terms and expressions related to conformity
assessment, as well as information about ISO’s adherence to the WTO principles in the Technical
Barriers to Trade (TBT) see the following URL: Foreword - Supplementary information.
The committee responsible for this document is Technical Committee ISO/TC 69, Applications of
statistical methods, Subcommittee SC 4, Applications of statistical methods in process management.
This first edition of ISO 22514-4 cancels and replaces ISO/TR 22514-4:2007, which has been technically
revised.
ISO 22514 consists of the following parts, under the general title Statistical methods in process
management — Capability and performance:
— Part 1: General principles and concepts
— Part 2: Process capability and performance of time-dependent process models
— Part 3: Machine performance studies for measured data on discrete parts
— Part 4: Process capability estimates and performance measures
— Part 5: Process capability estimates and performance for attributive characteristics
— Part 6: Process capability statistics for characteristics following a multivariate normal distribution
— Part 7: Capability of measurement processes
— Part 8: Machine performance of a multi-state production process
Introduction
Many organizations have embarked upon a continuous improvement strategy. To comply with such a
strategy, any organization will need to evaluate the capability and performance of its key processes.
The methods described in this part of ISO 22514 are intended to assist any management in this respect.
These evaluations need to be constantly reviewed by the management so that actions compatible with
continuous improvement can be taken when required.
The content of this part of ISO 22514 has been subject to large shifts of opinion during recent times. The
most fundamental shift has been to philosophically separate what is named in this part of ISO 22514 as
capability conditions from performance conditions, the primary difference being whether statistical
stability has been obtained (capability) or not (performance). This naturally leads onto the two sets of
indices that are to be found in their relevant clauses. It has become necessary to draw a firm distinction
between these since it has been observed in the industry that companies have been deceived about
their true capability position due to inappropriate indices being calculated and published.
The progression of this part of ISO 22514 is from the general condition to the specific and this approach
leads to general formulae being presented before their more usual, but specific manifestations.
There exist numerous references that describe the importance of understanding the processes at
work within any organization, be it a manufacturing process or an information handling process. As
organizations compete for sales with each other, it has become increasingly apparent that it is not only
the price paid for a product or service that matters so much, but also what costs will be incurred by the
purchaser from using such a product or service. The objective for any supplier is to continually reduce
variability and not to just satisfy specification.
Continual improvement leads to reductions in the costs of failure and assists in the drive for survival in
an increasingly more competitive world. There will also be savings in appraisal costs for as variation is
reduced, the need to inspect product might disappear or the frequency of sampling might be reduced.
Process capability and performance evaluations are necessary to enable organizations to assess the
capability and performance of their suppliers. Those organizations will find the indices contained
within this part of ISO 22514 useful in this endeavour.
Quantifying the variation present within a process enables judgement of its suitability and ability
to meet some given requirement. The following paragraphs and clauses provide an outline of the
philosophy required to be understood to determine the capability or performance of any process.
All processes will be subject to certain inherent variability. This part of ISO 22514 does not attempt
to explain what is meant by inherent variation, why it exists, where it comes from nor how it affects a
process. This part of ISO 22514 starts from the premise that it exists and is stable.
Process owners should endeavour to understand the sources of variation in their processes. Methods
such as flowcharting the process and identifying the inputs and outputs from a process assist in
identification of these variations together with the appropriate use of cause and effect (fishbone)
diagrams.
It is important for the user of this part of ISO 22514 to appreciate that variations exist that will be of a
short-term nature, as well as those that will be of a long-term nature and that capability determinations
using only the short-term variation might be greatly different to those which have used the long-term
variability.
When considering short-term variation, a study that uses only the shortest-term variation, sometimes
known as a machine study and described in ISO 22514-3, might be carried out. The method required to
carry out such a study will be outside the scope of this part of ISO 22514; however, it should be noted
that such studies are important and useful.
It should be noted that where the capability indices given in this part of ISO 22514 are computed, they
only form point estimates of their true values. It is therefore recommended that, wherever possible, the
vi © ISO 2016 – All rights reserved
indices’ confidence intervals are computed and reported. This part of ISO 22514 describes methods by
which these can be computed.
INTERNATIONAL STANDARD ISO 22514-4:2016(E)
Statistical methods in process management — Capability
and performance —
Part 4:
Process capability estimates and performance measures
1 Scope
This part of ISO 22514 describes process capability and performance measures that are commonly used.
2 Symbols and abbreviated terms
2.1 Symbols
In addition to the symbols listed below, some symbols are defined where they are used within the text.
α fraction or proportion
β shape parameter in a Weibull distribution
β coefficient of kurtosis
c constant based on subgroup size, n (see ISO 7870-2)
C process capability index
p
C minimum process capability index
pk
lower process capability index
C
pk
L
upper process capability index
C
pk
U
C alternative process capability index
pm
C process capability fraction (PCF)
R
d constant based on subgroup size, n (see ISO 7870-2)
e Eulers’s number (approximately 2,718), mathematical constant
Φ distribution function of the standard normal distribution
γ location parameter in a Weibull distribution
γ coefficient of skewness
m number of subgroups
K , K multipliers for estimating the confidence limits for a process capability index
l u
L lower specification limit
P lower 0,135 % percentile
0,135 %
μ location of the process; population mean value
N total sample size
n number of values or subgroup size (for a control chart)
P α percentile
α %
p lower fraction nonconforming
L
P process performance index
p
P minimum process performance index
pk
lower process performance index
P
pk
L
upper process performance index
P
pk
U
p total fraction nonconforming
t
p upper fraction nonconforming
U
P upper 99,865 % percentile
99,865 %
π geometric constant
Q process variation index
k
θ parameter required for the Rayleigh distribution
average subgroup range
R
S standard deviation, sample statistic
S standard deviation, with the subscript ‘t’ indicating total
t
average sample standard deviation
S
th
S observed sample standard deviation of the j subgroup
j
σ standard deviation, population
estimated standard deviation, total
ˆ
σ
t
T target value
U upper specification limit
X α % percentile
α %
th
X i value in a sample
i
arithmetic mean value, sample
X
arithmetic mean, of a number of sample arithmetic means
X
2 © ISO 2016 – All rights reserved
ξ scale parameter in a Weibull distribution
Y , Y values read from a graph
1 2
z quantile of the standardized normal distribution from −∞ to α
α
2.2 Abbreviated terms
MSE mean square error
PCF process capability fraction
PCI process capability index
3 Basic concepts used for process capability and performance
3.1 General
The measures referred to in 4.2 to 4.6 refer only to measured data. They are unsuitable for count or
attributes data and information concerning the expression of measures for such data will be found in
ISO 22514-5.
3.2 Location
The characterization of location is the mean, μ, or the median, X . Although for symmetric
50 %
distributions the mean is the most natural selection, with non-symmetric distributions the median is
the preferred selection.
3.3 Dispersion
3.3.1 Inherent dispersion
The preferred selection to quantify inherent dispersion is the standard deviation σ. This is often
estimated from the mean range value, R , taken from a range (R) chart or S from a standard deviation
(S) chart when the process is stable and in a state of statistical control as indicated in 4.1. Methods used
to estimate the process standard deviation are given in Annex A.
3.3.2 Total dispersion
It is necessary to differentiate between a standard deviation that measures only short-term variation
and that which measures longer-term variation. The total dispersion is the dispersion that is inherent in
the long-term variation. Methods of calculating the standard deviations representing these variations
are given in Annex A. Very often, when data are gathered over a long period of time, the standard
deviation is made larger by the effects of fluctuations in the process, σ .
t
3.3.3 Short-term dispersion
A process may have a short-term dispersion effect that is a part of the total dispersion. Figure 1
illustrates this. The short-term dispersion includes the inherent dispersion and can also include some
short-term instability effect.
Key
1 short-term dispersion
2 overall dispersion
Figure 1 — Short-term dispersion and its relationship to the total dispersion
The total dispersion can be any shape and not necessarily normal as illustrated here.
3.4 Mean square error (MSE)
When minimizing variation, some practitioners use the mean square error as the preferred measure. It
is compatible with the methods used in off-line quality techniques.
3.5 Reference limits
The lower and upper reference limits are respectively defined as the 0,135 % and the 99,865 %
percentiles of the distribution that describe the output of the process characteristic. They are written
as X and X .
0,135 5 % 99,865 %
3.6 Reference interval (also known as process spread)
The reference interval is the interval between the upper and the lower reference limits. The reference
interval includes 99,73 % of the individuals in the population from a process that is in a state of
statistical control.
4 Capability
4.1 General
Process capability is a measure of inherent process variability. The variability that is inherent in a
process when operating in a state of statistical control is known as the inherent process variability. It
represents the variation that remains after all known removable assignable causes have been eliminated.
If the process is monitored using a control chart, the control chart will show an in control state.
Capability is often regarded as being related to the proportion of output that will occur within the
product specification tolerances. Since a process in statistical control should be described by a
predictable distribution, the proportion of out-of-specification outputs can be estimated. As long
as the process remains in statistical control, it will continue to produce the same proportion out-of-
specification.
Management actions to reduce the variation from random causes are required to improve the process’
ability to consistently meet the specification requirements.
4 © ISO 2016 – All rights reserved
In short, the following will be necessary:
— define the process and its operating conditions. If there is a change to those conditions, it will
necessitate a new process study;
— assess the short-term and long-term measurement variabilities as percentages of the total variability
and minimize them;
— preserve the process stability and maintain its statistical control;
— estimate the remaining inherent variation;
— select an appropriate measure of capability.
The following are the conditions that will apply for capability:
— all technical conditions, e.g. temperature and humidity, shall be clearly stated;
— the uncertainty of the measurement system shall be estimated and judged appropriate (see
ISO 22514-7);
— multi-factor, multi-level aspects of the process should be allowed;
— the duration over which the data has been gathered shall be recorded;
— the frequency of sampling and sample size shall be specified and the start and finish dates of data
collection;
— the process shall be controlled with a control chart;
— the process shall be in a state of statistical control.
It is necessary to check the control chart from which the data have been taken for statistical control
and to examine a histogram of the data with any specification limits superimposed upon it. A valid
[15]
test for normality should be used in assessing the data such as the Anderson-Darling test or any
other suitable method. This test is powerful in detecting departures from normality in the tails of
the distribution and is suggested here as this is the region of interest for capability and performance
indices. Additionally, a normal probability plot can be used to look for the following:
a) verification of normality;
b) outliers;
c) data beyond any specification limit;
d) whether the data are well inside the specification limit(s);
e) evidence of asymmetry (i.e. skewness);
f) evidence of “long tails” in the data (i.e. kurtosis);
g) off-centre distribution;
h) any unusual patterns.
Explanations of anomalies should be sought in relation to these mentioned features and appropriate
action taken on the data prior to the calculation of any measure. It would be inappropriate to just discard
data that do not appear to fit any preconceived pattern. Such departures might be very revealing about
the process’ behaviour and should be thoroughly investigated.
4.2 Process capability
4.2.1 Normal distribution
Process capability is defined as a statistical measure of inherent process variability for a given
characteristic. The conventional method is to take the reference interval that describes 99,73 % of the
individual values from a process that is in a state of statistical control with the 0,135 % remaining on
each side. This applies even if the population of individual values is not normally distributed. For a
normal distribution, this process interval is represented by six standard deviations (see Figure 2).
Y₂
Y₁
a
0,135 %
0,135 %
Key
a
Reference interval 99,73 %.
Figure 2 — Normal distribution
On occasions, process capability is taken to account for extra sources of variation such as a multiple
stream process, for example, output from a multi-cavity injection moulding press. Under these
circumstances, the distribution of all values from all cavities could still be approximately normal,
but with extra variability so that the standard deviation shall represent the total variation, σ . It is
t
important to state how the standard deviation has been calculated, as well as the sampling strategy
used, sample size and the quantity and variability of output produced between samples as these will, in
practice, affect the validity of the capability assessment (see ISO 22514-2 for further information).
Data will usually be taken from a control chart. If the control chart had relaxed control lines or modified
control lines, the real process standard deviation will be larger than that estimated from data taken
from a control chart with standard control lines. Issues such as these and those given earlier will
influence the reference interval and it is important that they are stated in any capability assessment.
“Capable” processes will be those whose reference intervals are less than any specified tolerance by a
particular amount. An example of this is shown in Figure 3.
L U
Y₂
Y₁
a
Key
a
Reference interval 99,73 %.
Figure 3 — Normal distribution with specification limits
6 © ISO 2016 – All rights reserved
4.2.2 Non-normal distribution
If the distribution of individual values does not form a normal distribution, but is skewed, then the
reference interval may appear as in Figure 4. The values Y and Y , which will usually be the 0,135 %
1 2
and the 99,865 % percentiles, can be estimated using a suitable probability paper (see Figure 5 for
an example using an extreme value distribution probability paper) or by the use of suitable computer
software. They can also be computed using tabular values (see Annex B) or using the particular
probability function as suggested in Annex C.
Y₁
Y₂
a
0,135 %
0,135 %
Key
a
Reference interval 99,73 %.
Figure 4 — Non-normal distribution
4.3 Process location
Even if a process can be deemed capable by the above definition (in 4.2.1), if the process distribution has
been poorly centred relative to the specification limits, out-of-specification items might be produced.
For this reason, it is necessary to assess the location in addition to the process interval.
Key
best fit line
cumulative percent
Figure 5 — Example using an extreme value distribution probability paper
4.4 Process capability indices for measured data
4.4.1 General
It should be noted that when the capability indices given in this part of ISO 22514 are computed, they
only form point estimates of their true values. It is therefore recommended that wherever possible, the
indices’ confidence intervals are computed and reported. Methods by which these can be computed are
described in Annex D.
It is effective to express the capability of a process with the use of an index number. Several indices are
given. Care shall be taken when handling non-normal distributions.
The process capability indices are only established for a process that is statistically “in control”.
The process capability index often used is the ratio of a specified tolerance to the reference interval and
is known as C . Thus,
p
UL−
C = (1)
p
XX−
99,%865 0,%135
There are other indices that incorporate both the location and the variation. Of these, the most widely
used is the C index. If the observed index is less than a specified value, the process is deemed
pk
unacceptable and might lead to the shipment of a proportion of items outside of the specification or that
function and fit might be compromised.
8 © ISO 2016 – All rights reserved
The C index is the ratio of the difference between a specified tolerance limit and the process location
pk
to the difference between the corresponding natural process limit and the process location.
UX−
50%
C = (2)
pk
U
XX−
99,%865 50%
and
XL−
50%
C =
pk
L
XX−
50%,0135%
The C index is reported as the smaller value of these.
pk
NOTE Some practitioners report both of the above values (that are also known as CPU and CPL, respectively).
This provides information about both sides of the process.
These indices will provide information about whether a process is poorly centred and whether it will
possibly produce out-of-specification items. Even if the C index is high, a low value of the C index will
p pk
reveal a poorly centred process and a high probability of producing out-of-specification items.
4.4.2 C index (for the normal distribution)
p
If the individual values form a normal distribution and come from a statistically stable process, the
length of the reference interval is equal to 6σ, where σ is the inherent process standard deviation.
Therefore, the C index can be expressed as:
p
UL−
C =
p
6σ
ˆ
An estimate (σ ) of the inherent process standard deviation (σ ) is required to obtain an estimate of the
C index. When this has been obtained, usually with data from a control chart once the process is shown
p
to be statistically stable (see 4.1), the index is estimated:
UL−
ˆ
C =
p
ˆ
6σ
4.4.3 C index (for the normal distribution)
pk
When the distribution of individual values forms a normal distribution, the median X is equal to the
50 %
mean (μ). Further, X − X and X − X are each equal to 3σ. Therefore, the C index
99,865 % 50 % 50 % 0,135 % pk
can be expressed as the minimum of:
U −μ
C =
pk
U
3σ
or
μ −L
C =
pk
L
3σ
The estimated C , (using X to estimate μ instead of X ) will be the minimum of:
pk
50%
UX−
ˆ
C =
pk
U ˆ
3σ
or
XL−
ˆ
C =
pk
L ˆ
3σ
In computing a capability index, thought has to be given to the measure of the process variation used in
the denominator. Here, σ is given to represent the variation when the data comes from a process that is
in a state of statistical control.
The data might come from a multiple stream process such as a multi-headed filling machine or a
multi-spindle machine where the total output is treated together, where data from all streams are
simultaneously considered. The lower the index, the greater the proportion of items produced out-of-
specification.
4.4.4 C index for unilateral tolerances
pk
When there is only one specification limit given, it is only possible to calculate a C index. The index
pk
will be calculated using the appropriate limit, either an L or a U.
4.5 Process capability indices for measured data (non-normal)
4.5.1 General
If the distribution of individual values is non-normal, the expressions in Formulae (1) and (2) still
apply, but the estimation of the indices becomes more complicated. Three approaches to estimate the
reference limits are given.
The probability paper method described in 4.5.2 is fairly simple and requires little computation, but is
somewhat crude. The approach given in 4.5.4 is computationally more involved, but is superior to any
other method as far as accuracy is concerned.
10 © ISO 2016 – All rights reserved
4.5.2 Probability paper method
From graphs similar to that shown in Figure 4, estimates of the percentiles X and X can be
0,135 % 99,865 %
obtained. The estimates are denoted by Y and Y , respectively, and Formula (1) becomes:
1 2
UL−
ˆ
C =
p
YY−
In a similar way, the C formulae become:
pk
UX−
50%
ˆ
C =
pk
U
YX−
250%
or
XL−
50%
ˆ
C =
pk
L
XY−
50% 1
whichever gives the lower value.
If the observed index is less than a specified value, the process is deemed unacceptable and might lead
to the shipment of a proportion of items outside of the specification or that function and fit might be
compromised. The proportion nonconforming depends upon the distribution and the value of the index.
The link between the index and the proportion of nonconforming items produced depends on the class
of distributions. Care should be taken not to interpret indices on the basis of cut-off points that have
been derived for the normal distribution and, hence, are only applicable for that distribution.
Note that the probability paper method directly estimates fairly extreme percentiles and this can be
inaccurate.
4.5.3 Pearson curves method
As an alternative to using probability paper, standardized Pearson curves can be used. The method is
described by way of an example (see Annex B). The index is computed using:
UL−
ˆ
C =
p
ˆˆ
XX−
99,%865 0,%135
ˆ ˆ
where X and X are the 0,135 % and 99,865 % percentiles estimated from the standardized
0,%135 99,%865
Pearson curves.
Also, we have the formulae:
ˆ
UX−
50%
ˆ
C =
pk
ˆˆ
U
XX−
99,%865 50%
or
ˆ
XL−
50%
ˆ
C =
pk
ˆˆ
L
XX−
50%,0135%
ˆ
where X is the estimated median.
50%
In order to use this method, it is necessary to establish skewness and kurtosis values in addition to the
mean and standard deviation for the data set upon which the index is to be computed.
This method is not preferred, but is presented here for completeness due to its occasional use.
This approach, and a similar one based on Johnson curves, should be regarded with considerable
caution, especially when it is a procedure within a “black box” computer program used to analyse large
sets of data. Some of the potential difficulties are as follows:
— within a system of distributions, some distributions are more difficult to fit than others. The method
of moments can yield unstable or inefficient curve parameters in some cases;
— unless the estimation technique is applied skilfully, it is possible to obtain fitted curves that are
meaningless over certain ranges of the data. For example, with the method of moments, an easily
made mistake is to fit a Pearson Type III distribution whose estimated threshold is less than the
lower bound for the process output, thereby, invalidating the estimates of X and C ;
0,135 % pk
— the method of moments does not yield estimates of the variability in the estimated indices. Likewise,
these methods do not yield confidence intervals for the indices;
— not every data distribution can be described adequately with a Pearson or Johnson curve;
— goodness-of-fit tests are limited to the chi-squared test since more powerful tests are not generally
available for the Pearson and Johnson systems;
— the “black box” approach tends to displace basic practices, such as plotting the data and applying
simple normalizing transformations, that provide genuine understanding of the process.
4.5.4 Distribution identification method
Annex C describes certain families of distribution functions (such as the log-normal distribution,
the Rayleigh and the Weibull distributions) that are commonly found when investigating process
capability. The method is first to identify the appropriate family of distributions, secondly to estimate
the parameters of the distribution of the family that best explain the data by some efficient estimation
method and, finally, to express the quantiles in terms of the parameters of that distribution.
This is analogous to the procedure adopted in the case of the normal distribution where σ is estimated
and 6σ is represented by (X − X ).
99,865 % 0,135 %
Various types of probability paper might be useful to identify the appropriate family of distributions.
4.6 Alternative method for describing and calculating process capability estimates
The bases for this method are the widely used definitions of C and C for the “ideal process” with a
p pk
normally distributed characteristic, X, where the expectation, μ, and variance, σ , are constant with
time and the corresponding estimates are X and S .
Table 1 — Process capability indices and estimates for the normal distribution
Index Estimate
UL−
UL−
ˆ
C = C =
p
p
6σ 6S
U −μ
UX−
ˆ
C =
C =
pk
pk
U
3σ U
3S
μ −L
XL−
ˆ
C =
C =
pk
pk
L
3σ L
3S
CC= min(,C ) ˆ ˆ ˆ
pk pk pk CC= min( ,C )
LU pk pk pk
LU
This “ideal process” implies that the long-term standard deviation is equal to the short-term standard
deviation.
For the normal distribution, there is an exact relation between the lower fraction nonconforming units
and C and between the upper fraction nonconforming and C . This relation is exploited in 4.8 to
pk pk
L U
12 © ISO 2016 – All rights reserved
calculate the proportion out-of-specification from lower and upper process capability indices. The
relationship is displayed in Table 2 for easy reference.
When these measures of process capability have to be extended to characteristics that are not
normally distributed, the fraction nonconforming item can be transformed to a capability index using
the relationships in Table 2. This method can be applied in particular if the product characteristic is
qualitative.
Table 2 — Process capability indices and estimates for the normal distribution — Equivalent
formulae
Index Estimate
ˆˆ
CC+
CC+
pk pk
pk pk
UL
UL
ˆ
C =
C =
p
p
z
z
1−p ˆ
1−p
U
U
ˆ
C =
C =
pk
pk
U
U
z
z
1−p ˆ
1−p
L
L
ˆ
C =
C =
pk
pk
L
3 L
ˆ ˆ
where p and p are the fractions nonconforming at the upper and lower specification limits and p , p
U L
U L
are the corresponding estimates. The formulae in the above table can be applied to any distribution.
It is assumed that the user has knowledge of the shape of the distribution because of what is known
about the manufacturing process or by some evaluation of a sample by an appropriate probability paper.
For those distributions that are frequently observed (normal, log-normal, Rayleigh and Weibull), the
required relations and formulae are given in Annex C.
4.7 Other capability measures for continuous data
4.7.1 Process capability fraction (PCF)
The PCF is the inverse of the C index:
p
61σ
=
UL− C
p
It can be expressed as a percentage value and occasionally named C (%).
R
4.7.2 Indices when the specification limit is one-si
...
INTERNATIONAL ISO
STANDARD 22514-4
First edition
2016-08-01
Statistical methods in process
management — Capability and
performance —
Part 4:
Process capability estimates and
performance measures
Méthodes statistiques dans la gestion de processus — Aptitude et
performance —
Partie 4: Estimations de l’aptitude de processus et mesures de
performance
Reference number
©
ISO 2016
© ISO 2016, Published in Switzerland
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized otherwise in any form
or by any means, electronic or mechanical, including photocopying, or posting on the internet or an intranet, without prior
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ii © ISO 2016 – All rights reserved
Contents Page
Foreword .v
Introduction .vi
1 Scope . 1
2 Symbols and abbreviated terms . 1
2.1 Symbols . 1
2.2 Abbreviated terms . 3
3 Basic concepts used for process capability and performance . 3
3.1 General . 3
3.2 Location . 3
3.3 Dispersion . 3
3.3.1 Inherent dispersion . 3
3.3.2 Total dispersion . 3
3.3.3 Short-term dispersion . 3
3.4 Mean square error (MSE) . 4
3.5 Reference limits . 4
3.6 Reference interval (also known as process spread) . 4
4 Capability . 4
4.1 General . 4
4.2 Process capability . 6
4.2.1 Normal distribution . 6
4.2.2 Non-normal distribution. 7
4.3 Process location . 7
4.4 Process capability indices for measured data . 8
4.4.1 General. 8
4.4.2 C index (for the normal distribution) . 9
p
4.4.3 C index (for the normal distribution) .10
pk
4.4.4 C index for unilateral tolerances .10
pk
4.5 Process capability indices for measured data (non-normal) .10
4.5.1 General.10
4.5.2 Probability paper method .11
4.5.3 Pearson curves method .11
4.5.4 Distribution identification method .12
4.6 Alternative method for describing and calculating process capability estimates .12
4.7 Other capability measures for continuous data .13
4.7.1 Process capability fraction (PCF) .13
4.7.2 Indices when the specification limit is one-sided or no specification limit
is given .13
4.8 Assessment of proportion out-of-specification (normal distribution) .15
5 Performance .16
5.1 General .16
5.2 Process performance indices for measured data (normal distribution) .16
5.2.1 P index .16
p
5.2.2 P index .17
pk
5.3 Process performance indices for measured data (non-normal distribution) .17
5.3.1 General.17
5.3.2 Probability paper method .17
5.3.3 Pearson curves method .18
5.3.4 Distribution identification method .18
5.4 Other performance indices for measured data .18
5.5 Assessment of proportion out-of-specification for a normal distribution of the
total distribution .18
6 Reporting process capability and performance indices .19
Annex A (informative) Estimating standard deviations .21
Annex B (informative) Estimating capability and performance measures using Pearson
curves — Procedure and example .23
Annex C (informative) Distribution identification .37
Annex D (informative) Confidence intervals .42
Bibliography .44
iv © ISO 2016 – All rights reserved
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
through ISO technical committees. Each member body interested in a subject for which a technical
committee has been established has the right to be represented on that committee. International
organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.
ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of
electrotechnical standardization.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www.iso.org/directives).
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of
any patent rights identified during the development of the document will be in the Introduction and/or
on the ISO list of patent declarations received (see www.iso.org/patents).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation on the meaning of ISO specific terms and expressions related to conformity
assessment, as well as information about ISO’s adherence to the WTO principles in the Technical
Barriers to Trade (TBT) see the following URL: Foreword - Supplementary information.
The committee responsible for this document is Technical Committee ISO/TC 69, Applications of
statistical methods, Subcommittee SC 4, Applications of statistical methods in process management.
This first edition of ISO 22514-4 cancels and replaces ISO/TR 22514-4:2007, which has been technically
revised.
ISO 22514 consists of the following parts, under the general title Statistical methods in process
management — Capability and performance:
— Part 1: General principles and concepts
— Part 2: Process capability and performance of time-dependent process models
— Part 3: Machine performance studies for measured data on discrete parts
— Part 4: Process capability estimates and performance measures
— Part 5: Process capability estimates and performance for attributive characteristics
— Part 6: Process capability statistics for characteristics following a multivariate normal distribution
— Part 7: Capability of measurement processes
— Part 8: Machine performance of a multi-state production process
Introduction
Many organizations have embarked upon a continuous improvement strategy. To comply with such a
strategy, any organization will need to evaluate the capability and performance of its key processes.
The methods described in this part of ISO 22514 are intended to assist any management in this respect.
These evaluations need to be constantly reviewed by the management so that actions compatible with
continuous improvement can be taken when required.
The content of this part of ISO 22514 has been subject to large shifts of opinion during recent times. The
most fundamental shift has been to philosophically separate what is named in this part of ISO 22514 as
capability conditions from performance conditions, the primary difference being whether statistical
stability has been obtained (capability) or not (performance). This naturally leads onto the two sets of
indices that are to be found in their relevant clauses. It has become necessary to draw a firm distinction
between these since it has been observed in the industry that companies have been deceived about
their true capability position due to inappropriate indices being calculated and published.
The progression of this part of ISO 22514 is from the general condition to the specific and this approach
leads to general formulae being presented before their more usual, but specific manifestations.
There exist numerous references that describe the importance of understanding the processes at
work within any organization, be it a manufacturing process or an information handling process. As
organizations compete for sales with each other, it has become increasingly apparent that it is not only
the price paid for a product or service that matters so much, but also what costs will be incurred by the
purchaser from using such a product or service. The objective for any supplier is to continually reduce
variability and not to just satisfy specification.
Continual improvement leads to reductions in the costs of failure and assists in the drive for survival in
an increasingly more competitive world. There will also be savings in appraisal costs for as variation is
reduced, the need to inspect product might disappear or the frequency of sampling might be reduced.
Process capability and performance evaluations are necessary to enable organizations to assess the
capability and performance of their suppliers. Those organizations will find the indices contained
within this part of ISO 22514 useful in this endeavour.
Quantifying the variation present within a process enables judgement of its suitability and ability
to meet some given requirement. The following paragraphs and clauses provide an outline of the
philosophy required to be understood to determine the capability or performance of any process.
All processes will be subject to certain inherent variability. This part of ISO 22514 does not attempt
to explain what is meant by inherent variation, why it exists, where it comes from nor how it affects a
process. This part of ISO 22514 starts from the premise that it exists and is stable.
Process owners should endeavour to understand the sources of variation in their processes. Methods
such as flowcharting the process and identifying the inputs and outputs from a process assist in
identification of these variations together with the appropriate use of cause and effect (fishbone)
diagrams.
It is important for the user of this part of ISO 22514 to appreciate that variations exist that will be of a
short-term nature, as well as those that will be of a long-term nature and that capability determinations
using only the short-term variation might be greatly different to those which have used the long-term
variability.
When considering short-term variation, a study that uses only the shortest-term variation, sometimes
known as a machine study and described in ISO 22514-3, might be carried out. The method required to
carry out such a study will be outside the scope of this part of ISO 22514; however, it should be noted
that such studies are important and useful.
It should be noted that where the capability indices given in this part of ISO 22514 are computed, they
only form point estimates of their true values. It is therefore recommended that, wherever possible, the
vi © ISO 2016 – All rights reserved
indices’ confidence intervals are computed and reported. This part of ISO 22514 describes methods by
which these can be computed.
INTERNATIONAL STANDARD ISO 22514-4:2016(E)
Statistical methods in process management — Capability
and performance —
Part 4:
Process capability estimates and performance measures
1 Scope
This part of ISO 22514 describes process capability and performance measures that are commonly used.
2 Symbols and abbreviated terms
2.1 Symbols
In addition to the symbols listed below, some symbols are defined where they are used within the text.
α fraction or proportion
β shape parameter in a Weibull distribution
β coefficient of kurtosis
c constant based on subgroup size, n (see ISO 7870-2)
C process capability index
p
C minimum process capability index
pk
lower process capability index
C
pk
L
upper process capability index
C
pk
U
C alternative process capability index
pm
C process capability fraction (PCF)
R
d constant based on subgroup size, n (see ISO 7870-2)
e Eulers’s number (approximately 2,718), mathematical constant
Φ distribution function of the standard normal distribution
γ location parameter in a Weibull distribution
γ coefficient of skewness
m number of subgroups
K , K multipliers for estimating the confidence limits for a process capability index
l u
L lower specification limit
P lower 0,135 % percentile
0,135 %
μ location of the process; population mean value
N total sample size
n number of values or subgroup size (for a control chart)
P α percentile
α %
p lower fraction nonconforming
L
P process performance index
p
P minimum process performance index
pk
lower process performance index
P
pk
L
upper process performance index
P
pk
U
p total fraction nonconforming
t
p upper fraction nonconforming
U
P upper 99,865 % percentile
99,865 %
π geometric constant
Q process variation index
k
θ parameter required for the Rayleigh distribution
average subgroup range
R
S standard deviation, sample statistic
S standard deviation, with the subscript ‘t’ indicating total
t
average sample standard deviation
S
th
S observed sample standard deviation of the j subgroup
j
σ standard deviation, population
estimated standard deviation, total
ˆ
σ
t
T target value
U upper specification limit
X α % percentile
α %
th
X i value in a sample
i
arithmetic mean value, sample
X
arithmetic mean, of a number of sample arithmetic means
X
2 © ISO 2016 – All rights reserved
ξ scale parameter in a Weibull distribution
Y , Y values read from a graph
1 2
z quantile of the standardized normal distribution from −∞ to α
α
2.2 Abbreviated terms
MSE mean square error
PCF process capability fraction
PCI process capability index
3 Basic concepts used for process capability and performance
3.1 General
The measures referred to in 4.2 to 4.6 refer only to measured data. They are unsuitable for count or
attributes data and information concerning the expression of measures for such data will be found in
ISO 22514-5.
3.2 Location
The characterization of location is the mean, μ, or the median, X . Although for symmetric
50 %
distributions the mean is the most natural selection, with non-symmetric distributions the median is
the preferred selection.
3.3 Dispersion
3.3.1 Inherent dispersion
The preferred selection to quantify inherent dispersion is the standard deviation σ. This is often
estimated from the mean range value, R , taken from a range (R) chart or S from a standard deviation
(S) chart when the process is stable and in a state of statistical control as indicated in 4.1. Methods used
to estimate the process standard deviation are given in Annex A.
3.3.2 Total dispersion
It is necessary to differentiate between a standard deviation that measures only short-term variation
and that which measures longer-term variation. The total dispersion is the dispersion that is inherent in
the long-term variation. Methods of calculating the standard deviations representing these variations
are given in Annex A. Very often, when data are gathered over a long period of time, the standard
deviation is made larger by the effects of fluctuations in the process, σ .
t
3.3.3 Short-term dispersion
A process may have a short-term dispersion effect that is a part of the total dispersion. Figure 1
illustrates this. The short-term dispersion includes the inherent dispersion and can also include some
short-term instability effect.
Key
1 short-term dispersion
2 overall dispersion
Figure 1 — Short-term dispersion and its relationship to the total dispersion
The total dispersion can be any shape and not necessarily normal as illustrated here.
3.4 Mean square error (MSE)
When minimizing variation, some practitioners use the mean square error as the preferred measure. It
is compatible with the methods used in off-line quality techniques.
3.5 Reference limits
The lower and upper reference limits are respectively defined as the 0,135 % and the 99,865 %
percentiles of the distribution that describe the output of the process characteristic. They are written
as X and X .
0,135 5 % 99,865 %
3.6 Reference interval (also known as process spread)
The reference interval is the interval between the upper and the lower reference limits. The reference
interval includes 99,73 % of the individuals in the population from a process that is in a state of
statistical control.
4 Capability
4.1 General
Process capability is a measure of inherent process variability. The variability that is inherent in a
process when operating in a state of statistical control is known as the inherent process variability. It
represents the variation that remains after all known removable assignable causes have been eliminated.
If the process is monitored using a control chart, the control chart will show an in control state.
Capability is often regarded as being related to the proportion of output that will occur within the
product specification tolerances. Since a process in statistical control should be described by a
predictable distribution, the proportion of out-of-specification outputs can be estimated. As long
as the process remains in statistical control, it will continue to produce the same proportion out-of-
specification.
Management actions to reduce the variation from random causes are required to improve the process’
ability to consistently meet the specification requirements.
4 © ISO 2016 – All rights reserved
In short, the following will be necessary:
— define the process and its operating conditions. If there is a change to those conditions, it will
necessitate a new process study;
— assess the short-term and long-term measurement variabilities as percentages of the total variability
and minimize them;
— preserve the process stability and maintain its statistical control;
— estimate the remaining inherent variation;
— select an appropriate measure of capability.
The following are the conditions that will apply for capability:
— all technical conditions, e.g. temperature and humidity, shall be clearly stated;
— the uncertainty of the measurement system shall be estimated and judged appropriate (see
ISO 22514-7);
— multi-factor, multi-level aspects of the process should be allowed;
— the duration over which the data has been gathered shall be recorded;
— the frequency of sampling and sample size shall be specified and the start and finish dates of data
collection;
— the process shall be controlled with a control chart;
— the process shall be in a state of statistical control.
It is necessary to check the control chart from which the data have been taken for statistical control
and to examine a histogram of the data with any specification limits superimposed upon it. A valid
[15]
test for normality should be used in assessing the data such as the Anderson-Darling test or any
other suitable method. This test is powerful in detecting departures from normality in the tails of
the distribution and is suggested here as this is the region of interest for capability and performance
indices. Additionally, a normal probability plot can be used to look for the following:
a) verification of normality;
b) outliers;
c) data beyond any specification limit;
d) whether the data are well inside the specification limit(s);
e) evidence of asymmetry (i.e. skewness);
f) evidence of “long tails” in the data (i.e. kurtosis);
g) off-centre distribution;
h) any unusual patterns.
Explanations of anomalies should be sought in relation to these mentioned features and appropriate
action taken on the data prior to the calculation of any measure. It would be inappropriate to just discard
data that do not appear to fit any preconceived pattern. Such departures might be very revealing about
the process’ behaviour and should be thoroughly investigated.
4.2 Process capability
4.2.1 Normal distribution
Process capability is defined as a statistical measure of inherent process variability for a given
characteristic. The conventional method is to take the reference interval that describes 99,73 % of the
individual values from a process that is in a state of statistical control with the 0,135 % remaining on
each side. This applies even if the population of individual values is not normally distributed. For a
normal distribution, this process interval is represented by six standard deviations (see Figure 2).
Y₂
Y₁
a
0,135 %
0,135 %
Key
a
Reference interval 99,73 %.
Figure 2 — Normal distribution
On occasions, process capability is taken to account for extra sources of variation such as a multiple
stream process, for example, output from a multi-cavity injection moulding press. Under these
circumstances, the distribution of all values from all cavities could still be approximately normal,
but with extra variability so that the standard deviation shall represent the total variation, σ . It is
t
important to state how the standard deviation has been calculated, as well as the sampling strategy
used, sample size and the quantity and variability of output produced between samples as these will, in
practice, affect the validity of the capability assessment (see ISO 22514-2 for further information).
Data will usually be taken from a control chart. If the control chart had relaxed control lines or modified
control lines, the real process standard deviation will be larger than that estimated from data taken
from a control chart with standard control lines. Issues such as these and those given earlier will
influence the reference interval and it is important that they are stated in any capability assessment.
“Capable” processes will be those whose reference intervals are less than any specified tolerance by a
particular amount. An example of this is shown in Figure 3.
L U
Y₂
Y₁
a
Key
a
Reference interval 99,73 %.
Figure 3 — Normal distribution with specification limits
6 © ISO 2016 – All rights reserved
4.2.2 Non-normal distribution
If the distribution of individual values does not form a normal distribution, but is skewed, then the
reference interval may appear as in Figure 4. The values Y and Y , which will usually be the 0,135 %
1 2
and the 99,865 % percentiles, can be estimated using a suitable probability paper (see Figure 5 for
an example using an extreme value distribution probability paper) or by the use of suitable computer
software. They can also be computed using tabular values (see Annex B) or using the particular
probability function as suggested in Annex C.
Y₁
Y₂
a
0,135 %
0,135 %
Key
a
Reference interval 99,73 %.
Figure 4 — Non-normal distribution
4.3 Process location
Even if a process can be deemed capable by the above definition (in 4.2.1), if the process distribution has
been poorly centred relative to the specification limits, out-of-specification items might be produced.
For this reason, it is necessary to assess the location in addition to the process interval.
Key
best fit line
cumulative percent
Figure 5 — Example using an extreme value distribution probability paper
4.4 Process capability indices for measured data
4.4.1 General
It should be noted that when the capability indices given in this part of ISO 22514 are computed, they
only form point estimates of their true values. It is therefore recommended that wherever possible, the
indices’ confidence intervals are computed and reported. Methods by which these can be computed are
described in Annex D.
It is effective to express the capability of a process with the use of an index number. Several indices are
given. Care shall be taken when handling non-normal distributions.
The process capability indices are only established for a process that is statistically “in control”.
The process capability index often used is the ratio of a specified tolerance to the reference interval and
is known as C . Thus,
p
UL−
C = (1)
p
XX−
99,%865 0,%135
There are other indices that incorporate both the location and the variation. Of these, the most widely
used is the C index. If the observed index is less than a specified value, the process is deemed
pk
unacceptable and might lead to the shipment of a proportion of items outside of the specification or that
function and fit might be compromised.
8 © ISO 2016 – All rights reserved
The C index is the ratio of the difference between a specified tolerance limit and the process location
pk
to the difference between the corresponding natural process limit and the process location.
UX−
50%
C = (2)
pk
U
XX−
99,%865 50%
and
XL−
50%
C =
pk
L
XX−
50%,0135%
The C index is reported as the smaller value of these.
pk
NOTE Some practitioners report both of the above values (that are also known as CPU and CPL, respectively).
This provides information about both sides of the process.
These indices will provide information about whether a process is poorly centred and whether it will
possibly produce out-of-specification items. Even if the C index is high, a low value of the C index will
p pk
reveal a poorly centred process and a high probability of producing out-of-specification items.
4.4.2 C index (for the normal distribution)
p
If the individual values form a normal distribution and come from a statistically stable process, the
length of the reference interval is equal to 6σ, where σ is the inherent process standard deviation.
Therefore, the C index can be expressed as:
p
UL−
C =
p
6σ
ˆ
An estimate (σ ) of the inherent process standard deviation (σ ) is required to obtain an estimate of the
C index. When this has been obtained, usually with data from a control chart once the process is shown
p
to be statistically stable (see 4.1), the index is estimated:
UL−
ˆ
C =
p
ˆ
6σ
4.4.3 C index (for the normal distribution)
pk
When the distribution of individual values forms a normal distribution, the median X is equal to the
50 %
mean (μ). Further, X − X and X − X are each equal to 3σ. Therefore, the C index
99,865 % 50 % 50 % 0,135 % pk
can be expressed as the minimum of:
U −μ
C =
pk
U
3σ
or
μ −L
C =
pk
L
3σ
The estimated C , (using X to estimate μ instead of X ) will be the minimum of:
pk
50%
UX−
ˆ
C =
pk
U ˆ
3σ
or
XL−
ˆ
C =
pk
L ˆ
3σ
In computing a capability index, thought has to be given to the measure of the process variation used in
the denominator. Here, σ is given to represent the variation when the data comes from a process that is
in a state of statistical control.
The data might come from a multiple stream process such as a multi-headed filling machine or a
multi-spindle machine where the total output is treated together, where data from all streams are
simultaneously considered. The lower the index, the greater the proportion of items produced out-of-
specification.
4.4.4 C index for unilateral tolerances
pk
When there is only one specification limit given, it is only possible to calculate a C index. The index
pk
will be calculated using the appropriate limit, either an L or a U.
4.5 Process capability indices for measured data (non-normal)
4.5.1 General
If the distribution of individual values is non-normal, the expressions in Formulae (1) and (2) still
apply, but the estimation of the indices becomes more complicated. Three approaches to estimate the
reference limits are given.
The probability paper method described in 4.5.2 is fairly simple and requires little computation, but is
somewhat crude. The approach given in 4.5.4 is computationally more involved, but is superior to any
other method as far as accuracy is concerned.
10 © ISO 2016 – All rights reserved
4.5.2 Probability paper method
From graphs similar to that shown in Figure 4, estimates of the percentiles X and X can be
0,135 % 99,865 %
obtained. The estimates are denoted by Y and Y , respectively, and Formula (1) becomes:
1 2
UL−
ˆ
C =
p
YY−
In a similar way, the C formulae become:
pk
UX−
50%
ˆ
C =
pk
U
YX−
250%
or
XL−
50%
ˆ
C =
pk
L
XY−
50% 1
whichever gives the lower value.
If the observed index is less than a specified value, the process is deemed unacceptable and might lead
to the shipment of a proportion of items outside of the specification or that function and fit might be
compromised. The proportion nonconforming depends upon the distribution and the value of the index.
The link between the index and the proportion of nonconforming items produced depends on the class
of distributions. Care should be taken not to interpret indices on the basis of cut-off points that have
been derived for the normal distribution and, hence, are only applicable for that distribution.
Note that the probability paper method directly estimates fairly extreme percentiles and this can be
inaccurate.
4.5.3 Pearson curves method
As an alternative to using probability paper, standardized Pearson curves can be used. The method is
described by way of an example (see Annex B). The index is computed using:
UL−
ˆ
C =
p
ˆˆ
XX−
99,%865 0,%135
ˆ ˆ
where X and X are the 0,135 % and 99,865 % percentiles estimated from the standardized
0,%135 99,%865
Pearson curves.
Also, we have the formulae:
ˆ
UX−
50%
ˆ
C =
pk
ˆˆ
U
XX−
99,%865 50%
or
ˆ
XL−
50%
ˆ
C =
pk
ˆˆ
L
XX−
50%,0135%
ˆ
where X is the estimated median.
50%
In order to use this method, it is necessary to establish skewness and kurtosis values in addition to the
mean and standard deviation for the data set upon which the index is to be computed.
This method is not preferred, but is presented here for completeness due to its occasional use.
This approach, and a similar one based on Johnson curves, should be regarded with considerable
caution, especially when it is a procedure within a “black box” computer program used to analyse large
sets of data. Some of the potential difficulties are as follows:
— within a system of distributions, some distributions are more difficult to fit than others. The method
of moments can yield unstable or inefficient curve parameters in some cases;
— unless the estimation technique is applied skilfully, it is possible to obtain fitted curves that are
meaningless over certain ranges of the data. For example, with the method of moments, an easily
made mistake is to fit a Pearson Type III distribution whose estimated threshold is less than the
lower bound for the process output, thereby, invalidating the estimates of X and C ;
0,135 % pk
— the method of moments does not yield estimates of the variability in the estimated indices. Likewise,
these methods do not yield confidence intervals for the indices;
— not every data distribution can be described adequately with a Pearson or Johnson curve;
— goodness-of-fit tests are limited to the chi-squared test since more powerful tests are not generally
available for the Pearson and Johnson systems;
— the “black box” approach tends to displace basic practices, such as plotting the data and applying
simple normalizing transformations, that provide genuine understanding of the process.
4.5.4 Distribution identification method
Annex C describes certain families of distribution functions (such as the log-normal distribution,
the Rayleigh and the Weibull distributions) that are commonly found when investigating process
capability. The method is first to identify the appropriate family of distributions, secondly to estimate
the parameters of the distribution of the family that best explain the data by some efficient estimation
method and, finally, to express the quantiles in terms of the parameters of that distribution.
This is analogous to the procedure adopted in the case of the normal distribution where σ is estimated
and 6σ is represented by (X − X ).
99,865 % 0,135 %
Various types of probability paper might be useful to identify the appropriate family of distributions.
4.6 Alternative method for describing and calculating process capability estimates
The bases for this method are the widely used definitions of C and C for the “ideal process” with a
p pk
normally distributed characteristic, X, where the expectation, μ, and variance, σ , are constant with
time and the corresponding estimates are X and S .
Table 1 — Process capability indices and estimates for the normal distribution
Index Estimate
UL−
UL−
ˆ
C = C =
p
p
6σ 6S
U −μ
UX−
ˆ
C =
C =
pk
pk
U
3σ U
3S
μ −L
XL−
ˆ
C =
C =
pk
pk
L
3σ L
3S
CC= min(,C ) ˆ ˆ ˆ
pk pk pk CC= min( ,C )
LU pk pk pk
LU
This “ideal process” implies that the long-term standard deviation is equal to the short-term standard
deviation.
For the normal distribution, there is an exact relation between the lower fraction nonconforming units
and C and between the upper fraction nonconforming and C . This relation is exploited in 4.8 to
pk pk
L U
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calculate the proportion out-of-specification from lower and upper process capability indices. The
relationship is displayed in Table 2 for easy reference.
When these measures of process capability have to be extended to characteristics that are not
normally distributed, the fraction nonconforming item can be transformed to a capability index using
the relationships in Table 2. This method can be applied in particular if the product characteristic is
qualitative.
Table 2 — Process capability indices and estimates for the normal distribution — Equivalent
formulae
Index Estimate
ˆˆ
CC+
CC+
pk pk
pk pk
UL
UL
ˆ
C =
C =
p
p
z
z
1−p ˆ
1−p
U
U
ˆ
C =
C =
pk
pk
U
U
z
z
1−p ˆ
1−p
L
L
ˆ
C =
C =
pk
pk
L
3 L
ˆ ˆ
where p and p are the fractions nonconforming at the upper and lower specification limits and p , p
U L
U L
are the corresponding estimates. The formulae in the above table can be applied to any distribution.
It is assumed that the user has knowledge of the shape of the distribution because of what is known
about the manufacturing process or by some evaluation of a sample by an appropriate probability paper.
For those distributions that are frequently observed (normal, log-normal, Rayleigh and Weibull), the
required relations and formulae are given in Annex C.
4.7 Other capability measures for continuous data
4.7.1 Process capability fraction (PCF)
The PCF is the inverse of the C index:
p
61σ
=
UL− C
p
It can be expressed as a percentage value and occasionally named C (%).
R
4.7.2 Indices when the specification limit is one-sided or no specification limit is given
4.7.2.1 General
Sometimes, a specification is given that has only one limit, e.g. a maximum value. In these circumstances,
it will only be possible to compute a C or a P index.
pk pk
There will also be situations when specification limits are not given or not known. However, if a target
or nominal value is given for the product characteristic or process parameter, the following measures
might be appropriate. They present a special appeal to those engaged in minimizing process variation
around a target value.
4.7.2.2 Mean square error (MSE)
The mean square error provides a measure that involves both location and variation. It is computed as
follows:
σμ+−()T
In deriving this measure from data, it is necessary to provide estimates of the process standard
deviation and μ using sample data from a control chart.
4.7.2.3 Q index
k
This index uses the mean square error given in 4.7.2.2, but expresses the whole value as a coefficient of
variation and is computed as follows:
100 σμ+−()T
Q = (%)
k
T
for T¹ 0 .
An interesting property of this index is if the process drifts from its target, the index will increase in
value and if the process variation increases, it will also increase the value of the index. The smaller this
index becomes, the better the process is deemed to have performed.
4.7.2.4 C index
pm
T
...
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