SIST-TP ISO/TR 8550-3:2010
Guidance on the selection and usage of acceptance sampling systems for inspection of discrete items in lots - Part 3: Sampling by variables
Guidance on the selection and usage of acceptance sampling systems for inspection of discrete items in lots - Part 3: Sampling by variables
The guidance in this part of ISO/TR 8550 is confined to acceptance sampling of products that are supplied in lots and that can be classified as consisting of discrete items (i.e. discrete articles of product). Each item in a lot can be identified and segregated from the other items in the lot and has an equal chance of being included in the sample. Each item of product is countable and has specific characteristics that are measurable on a continuous scale. Each characteristic has, at least to a good approximation, a normal distribution or a distribution that can be transformed so that it closely resembles a normal distribution.
Lignes directrices pour la sélection d'un système, d'un programme ou d'un plan d'échantillonnage pour acceptation pour le contrôle d'unités discrètes en lots - Partie 3: Échantillonnage par variables
Napotek za izbiro in uporabo sistemov prevzemnega vzorčenja za kontrolo diskretnih primerkov v partijah (lotih) - 3. del: Vzorčenje po številskih spremenljivkah
Napotek v tem delu ISO/TR 8550 je omejen na prevzemno vzorčenje produktov, ki se dobavljajo v partijah, in ki se lahko klasificirajo kot sestavljeni iz diskretnih primerkov (tj. diskretni predmeti produkta). Vsak primerek iz partije se lahko prepozna in loči od drugih primerkov iz partije ter ima enako možnost vključitve v vzorec. Vsak primerek produkta je števen in ima določene karakteristike, ki se lahko merijo na nepretrgani skali. Vsaka karakteristika ima, vsaj do dobrega približka, normalno distribucijo oziroma distribucijo, ki se lahko pretvori tako, da je zelo podobna normalni distribuciji.
General Information
Buy Standard
Standards Content (Sample)
SLOVENSKI STANDARD
SIST-TP ISO/TR 8550-3:2010
01-julij-2010
1DSRWHN]DL]ELURLQXSRUDERVLVWHPRYSUHY]HPQHJDY]RUþHQMD]DNRQWUROR
GLVNUHWQLKSULPHUNRYYSDUWLMDK ORWLK GHO 9]RUþHQMHSRãWHYLOVNLK
VSUHPHQOMLYNDK
Guidance on the selection and usage of acceptance sampling systems for inspection of
discrete items in lots - Part 3: Sampling by variables
Lignes directrices pour la sélection d'un système, d'un programme ou d'un plan
d'échantillonnage pour acceptation pour le contrôle d'unités discrètes en lots - Partie 3:
Échantillonnage par variables
Ta slovenski standard je istoveten z: ISO/TR 8550-3:2007
ICS:
03.120.30 8SRUDEDVWDWLVWLþQLKPHWRG Application of statistical
methods
SIST-TP ISO/TR 8550-3:2010 en
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.
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SIST-TP ISO/TR 8550-3:2010
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SIST-TP ISO/TR 8550-3:2010
TECHNICAL ISO/TR
REPORT 8550-3
First edition
2007-06-01
Guidance on the selection and usage of
acceptance sampling systems for
inspection of discrete items in lots —
Part 3:
Sampling by variables
Lignes directrices pour la sélection d'un système, d'un programme ou
d'un plan d'échantillonnage pour acceptation pour le contrôle d'unités
discrètes en lots —
Partie 3: Échantillonnage par variables
Reference number
ISO/TR 8550-3:2007(E)
©
ISO 2007
---------------------- Page: 3 ----------------------
SIST-TP ISO/TR 8550-3:2010
ISO/TR 8550-3:2007(E)
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accepts no liability in this area.
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SIST-TP ISO/TR 8550-3:2010
ISO/TR 8550-3:2007(E)
Contents Page
Foreword. iv
Introduction . v
1 Scope . 1
2 Normative references . 1
3 Normality . 2
4 Types of control . 12
5 Forms of acceptance criteria. 15
6 International Standards for acceptance sampling of lots by variables . 29
7 Effect on the selection process of market and production conditions. 31
Annex A (normative) Normal probability paper . 38
Bibliography . 39
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Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
In exceptional circumstances, when a technical committee has collected data of a different kind from that
which is normally published as an International Standard (“state of the art”, for example), it may decide by a
simple majority vote of its participating members to publish a Technical Report. A Technical Report is entirely
informative in nature and does not have to be reviewed until the data it provides are considered to be no
longer valid or useful.
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/TR 8550-3 was prepared by Technical Committee ISO/TC 69, Applications of statistical methods,
Subcommittee SC 5, Acceptance sampling.
This first edition of ISO/TR 8550-3, together with ISO/TR 8550-1 and ISO/TR 8550-2, cancels and replaces
ISO/TR 8550:1994.
ISO/TR 8550 consists of the following parts, under the general title Guidance on the selection and usage of
acceptance sampling systems for inspection of discrete items in lots:
⎯ Part 1: Acceptance sampling
⎯ Part 3: Sampling by variables
The following part is under preparation:
⎯ Part 2: Sampling by attributes
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Introduction
This part of ISO/TR 8550 gives guidance on the selection of an acceptance sampling system for inspection by
variables. It does this principally by reviewing the available systems specified by various standards and
showing ways in which these can be compared in order to assess their suitability for an intended application. It
is assumed that the choice has already been made to use sampling by variables in preference to sampling by
attributes.
A corresponding guidance document on the selection of a generic acceptance sampling system, scheme or
plan for inspection by attributes is given in ISO/TR 8550-2.
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SIST-TP ISO/TR 8550-3:2010
TECHNICAL REPORT ISO/TR 8550-3:2007(E)
Guidance on the selection and usage of acceptance sampling
systems for inspection of discrete items in lots —
Part 3:
Sampling by variables
1 Scope
The guidance in this part of ISO/TR 8550 is confined to acceptance sampling of products that are supplied in
lots and that can be classified as consisting of discrete items (i.e. discrete articles of product). Each item in a
lot can be identified and segregated from the other items in the lot and has an equal chance of being included
in the sample. Each item of product is countable and has specific characteristics that are measurable on a
continuous scale. Each characteristic has, at least to a good approximation, a normal distribution or a
distribution that can be transformed so that it closely resembles a normal distribution.
Standards on acceptance sampling by variables are applicable to a wide variety of inspection situations.
These include, but are not limited to, the following:
a) end items, such as complete products or sub-assemblies;
b) components and raw materials;
c) services;
d) materials in process;
e) supplies in storage;
f) maintenance operations;
g) data or records;
h) administrative procedures.
Although this part of ISO/TR 8550 is written principally in terms of manufacture and production, it should be
interpreted liberally as it is applicable to the selection of sampling systems, schemes and plans for all types of
product and processes as defined in ISO 9000.
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated
references, only the edition listed applies. For undated references, the latest edition of the referenced
document (including any amendment) applies.
ISO/TR 8550-1:2007, Guidance on the selection and usage of acceptance sampling systems for inspection of
discrete items in lots — Part 1: Acceptance sampling
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3 Normality
3.1 Relationship between form of distribution of quality characteristic and percent
nonconforming
A key aspect of sampling by variables is the form of the distributions of the quality characteristics. Consider a
single quality characteristic. If it is normally distributed and if an upper specification limit is located at the mean
plus two standard deviations, the percent nonconforming is about 2,5 %. If the specification limit is located at
the mean plus three standard deviations, the percent nonconforming is about 0,1 %. However, if the
distribution of the quality characteristic is not normal and has a large positive skewness, i.e. a long tail to the
right, an upper specification limit located at the mean plus three standard deviations could conceivably yield a
percent nonconforming approaching 10 % instead of about 0,1 % (see Figures 1 and 2).
Therefore, whenever a sampling plan for inspection by variables for percent nonconforming is to be employed,
it is highly desirable to check any assumptions about the shape of the distribution, especially in the tails of the
distribution. If the AQL is very small, for example 0,1 %, a study of several thousand items should be made,
including a test of distributional form.
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Key
1 upper specification limit
2 0,1 % above specification
Figure 1 — Normal distribution
Key
1 upper specification limit
Figure 2 — Distribution with large positive skewness
3.2 Identifying departure from normality
3.2.1 Subjective assessment
The degree to which a sample appears to have come from a normal distribution can be subjectively assessed
by means of a normal probability plot. Such a plot is constructed in the following way. Once the random
sample has been selected and the quality characteristic x has been measured for each item, the values x ,
1
x , . . ., x are arranged in ascending order x , x , . . ., x , such that x u x u, . . . u x . The points with
2 n [1] [2] [n] [1] [2] [n]
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3
1
coordinates xi,/−+n are then plotted on a sheet of normal probability paper for i = 1, 2, . . ., n.
()()
{}
[]i
84
To facilitate this process, an A4 sheet of normal probability paper that can be freely photocopied is provided in
Annex A.
Figure 3 shows the normal probability plot of a random sample of size 100 from a normal distribution. The
graph paper is specially designed so that data from a normal distribution tend to lie close to a straight line.
A straight line has been drawn by eye through the data, showing in this case that there are only minor
departures from linearity.
When data originate from a normal distribution, departures of the probability plot from linearity are due solely
to sampling fluctuations. Conversely, data from other types of distribution will tend to show departures from
linearity of a characteristic type, helping in the determination of the family of distributions to which the data
belong. Knowledge of this family can indicate the appropriate transformation to make to the data to bring
these closer to normality.
Figures 4 to 7 show the density functions and examples of normal probability plots based on a random sample
of size 100 for, respectively, a lognormal, Cauchy, Laplace, and exponential distribution, respectively. On the
normal probability plot for Figures 4 to 6, a straight line has been drawn through the data points to aid the eye
in identifying the characteristic differences.
For the lognormal distribution, there is a pronounced downward concavity.
The Cauchy distribution is almost indistinguishable from the normal distribution towards its centre, but the
extra thickness of its tails results in the plot being relatively high for low values of x and relatively low for high
values of x, the extremities of the plot being almost horizontal.
The Laplace distribution is similar, except that there is a shorter region in the normal probability plot where the
distribution is indistinguishable from the normal distribution, and the extremities of the plot are far from
horizontal.
The normal probability plot for the exponential distribution has a very characteristic shape, rising very steeply
at the left and becoming almost horizontal towards the right.
These are a small selection from the many possible distributions from which data might have arisen. In some
cases, e.g. the lognormal distribution, the distribution can be transformed exactly to normality without knowing
its parameters (see 3.3.2 and 3.3.3). In other cases, approximate normality may be achieved, e.g. by using
[20]
the fourth root transformation on exponentially distributed variables, as shown by Kittlitz . In other cases,
acceptance sampling by variables might not be possible without a method tailored to that family of
distributions. If such a method does not exist, acceptance sampling by attributes might have to be used
instead, the loss in efficiency being more than compensated for by the increase in integrity of the acceptance
sampling results.
Figures 4 to 7 show normal probability plots for samples of size 100. Often there is not the luxury of such large
samples. With small samples, it is less clear whether the departures from linearity of the normal probability
plot are due to non-normality or merely to sampling fluctuations. In case of doubt, subjective assessment of
departure from normality should be replaced by objective statistical tests, such as those discussed in 3.2.2.
Further information on tests for departure from normality is given in ISO 5479 and ISO 2854:1976, Clause 2.
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Key
X quality characteristic, x
Y probability density of x
a) Normal distribution
b) Normal probability plot of a random sample of size 100 from a normal distribution
Figure 3 — Normal distribution and normal probability plot
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Key
X quality characteristic, x
Y probability density of x
a) Lognormal distribution
b) Normal probability plot of a random sample of size 100 from a lognormal distribution
Figure 4 — Lognormal distribution and normal probability plot
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Key
X quality characteristic, x
Y probability density of x
a) Cauchy distribution
b) Normal probability plot of a random sample of size 100 from a Cauchy distribution
Figure 5 — Cauchy distribution and normal probability plot
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Key
X quality characteristic, x
Y probability density of x
a) Laplace distribution
b) Normal probability plot of a random sample of size 100 from a Laplace distribution
Figure 6 — Laplace distribution and normal probability plot
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Key
X quality characteristic, x
Y probability density of x
a) Exponential distribution
b) Normal probability plot of a random sample of size 100 from an exponential distribution
Figure 7 — Exponential distribution and normal probability plot
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3.2.2 Statistical tests for departure from normality
3.2.2.1 Directional versus omnibus tests
Statistical tests are used to determine the degree to which the available evidence fails to support a given null
hypothesis, say H . The power of the test is the probability of rejecting the hypothesis H in favour of the
0 0
alternative hypothesis H when the alternative hypothesis is true.
1
When testing for departures from normality, the null hypothesis H is that the distribution is normal while the
0
alternative hypothesis H is that it is not normal. If the alternative hypothesis is more specific, stating the
1
alternative family of distributions to which the distribution belongs, then the test is said to be directional.
Otherwise, it is said to be an omnibus test.
In both cases, a statistic T is calculated from the sample evidence, and H is rejected if the value of T lies in a
0
so-called critical region. The critical region is chosen so that the probability of T falling in the critical region
when H is true is a small quantity, usually 5 %. For an omnibus test, the critical region simply consists of
0
values of T that lie far away from the expected value of T under H . For a directional test, the critical region
0
consists of the values of T for which the power is greatest.
In general, therefore, greater power is achieved by being as specific as can be justified about the alternative
hypothesis, i.e. about the likely nature of the departure from the null hypothesis.
As might be expected, power is generally also increased by increasing the sample size on which T is based.
3.2.2.2 Directional tests
ISO 5479:1997 provides two directional tests. One of these is for skewness, the other for kurtosis (i.e.
peakedness). A simultaneous bi-directional test for skewness and kurtosis is also provided. The skewness
and kurtosis test statistics for n observations x , x , . . ., x are, respectively, the moment coefficients:
1 2 n
3
2
2
bm= /m and bm= /m ,
13 2 24 2
where:
n
1
j
mx=−x for j = 2, 3 and 4.
()
∑
ji
n
i=1
3.2.2.3 Omnibus tests
ISO 5479:1997 also provides two omnibus tests: the Shapiro-Wilk test and the Epps-Pulley test. The test
statistic for the Shapiro-Wilk test is a linear function of the ordered observations. The Epps-Pulley test statistic
is a little more complicated to implement as it involves a sum and a double sum of exponentiated quantities.
A rule of thumb is given for deciding which of these to use in a given situation.
3.3 Transforming to normality
3.3.1 Normalization and variance stabilization
Much analysis of variance is invalidated if the quality characteristic under analysis is heteroscedastic, i.e.
when its variance varies with its mean. A mathematical transformation of the characteristic that roughly
equalizes the variance over all values of the mean is called a variance-stabilizing transformation. It is often the
case that transforming such data to eliminate heteroscedasticity, i.e. to make the data homoscedastic, also
has the effect of making the data more normal. In other words, variance-stabilizing transformations are often
normalizing transformations.
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A rough rule for determining the appropriate transformation is as follows. If the standard deviation σ of a
process characteristic x can be expressed approximately as a function h()µ , where µ is the corresponding
mean of the characteristic, then an approximate variance-stabilizing transformation of x is g(x), where:
x dt
gx()= . (1)
∫
ht
()
Examples of the use of this method are given in 3.3.2 and 3.3.3.
If a test for departure from normality indicates that x is non-normal, the use of y = g(x) should be considered
instead of x.
3.3.2 The square root transformation
Where σ is a constant multiple of µ , i.e. hc(µµ)= , where c is a constant, then, from Expression (1):
xd2t
g()xx== .
∫
c
ct
As the coefficient 2/c has no effect on the stabilization of the variance, it can be ignored. An approximate
variance-stabilizing transformation is therefore the square root transformation:
g(xx)= .
The standard deviation of g(x) is approximately c/2.
3.3.3 The logarithmic transformation
When σ is a constant multiple ofµ , i.e. hc(µµ)= , where c is a constant, then, from Expression (1):
xd1t
g()xx== ln() .
∫
ct c
As the coefficient 1/c has no effect on the stabilization of the variance, it can be ignored. An approximate
variance-stabilizing transformation is therefore the logarithmic transformation:
g(x) = ln(x).
The standard deviation of g(x) is approximately c.
This transformation might be appropriate when the quality characteristic is the sample variance based on a
sample of size n, in which case cn=−21/( ) .
3.3.4 The Box-Cox transformation
[19]
A general transformation, proposed by Box and Cox , is to set
λ
x −1
gx =
()
λ
where λ > 0.
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Note that:
⎯ setting λ equal to 1 simply relocates the existing distribution, leaving its original shape unchanged;
⎯ setting λ equal to 0,5 is equivalent to first using the square root transformation and then relocating and
rescaling the resulting distribution;
⎯ letting λ tend to zero is equivalent to using the logarithmic transformation, i.e. equivalent to setting
g(x) = ln(x).
However, λ is not limited to taking one of these particular values, so the value of λ in excess of zero that best
normalizes the distribution can be found either by trial and error or by some optimization method applied to
past data.
A more general version of the Box-Cox transformation is:
λ
1
()x+−λ 1
2
gx()= for λ > 0 andλ >−x .
1 21[]
λ
1
This transformation effectively relocates the distribution of x by an amount λ before applying the simpler Box-
2
Cox power transformation. A consequence is that the more general transformation does not require all the
data values to be positive. Because the more general transformation has two parameters, it allows a greater
range of distribution families to be transformed to normality.
4 Types of control
4.1 Control of a single quality characteristic
4.1.1 General
Acceptance sampling by variables can become complicated when there are two or more quality
characteristics, so the text first considers the case where only a single quality characteristic is being controlled.
As the acceptance criterion for a single quality characteristic involves either x and s, or x and σ , it can
always be represented diagrammatically as well as algebraically. A diagrammatic representation of an
acceptance criterion is called an acceptance diagram.
Within the case of a single quality characteristic, there are several possible methods of control, which are
described in 4.1.2 and 4.1.3.
4.1.2 Single specification limit
The simplest case of a single quality characteristic is where there is a single specification limit, i.e. where
either an upper limit or a lower limit to values of the characteristic is specified, but not both. Control of such a
characteristic by means of sampling by variables is relatively straightforward, requiring the sample mean to be
within a specification and at least a given multiple, denoted by k, of the sample standard deviation (or process
standard deviation, if known) away from the specification limit. When the acceptance criterion is expressed in
terms of this factor k, the method is described as “Form k” (see 5.2).
4.1.3 Double specification limits
4.1.3.1 General
Rather more complicated is the case of a single quality characteristic with double specification limits, i.e.
where both an upper limit and a lower limit to values of the characteristic are specified. In this case, there are
three modes of control.
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4.1.3.2 Combined control
Double specification limits are said to be under combined control when the fraction nonconforming beyond
both limits belongs to the same class, to which a single AQL applies. By implication, nonconformity beyond
either limit is of roughly the same degree of seriousness.
EXAMPLE 1 A weapon guidance system is to be tested against a moving target. Missing the target either to the left or
to the right is equally unsatisfactory so combined control of both sides of the target might be appropriate in this case.
Form k is inadequate for combined control. Instead, Form p* is used, i.e. the lot is accepted only if an estimate
of the process fraction nonconforming is less than a given value p*. In other words, p* is the maximum
estimate of the process fraction nonconforming that is deemed to be acceptable for the given sample size and
AQL.
4.1.3.3 Separate control
Double specification limits are said to be under separate control when the fraction nonconforming beyond the
two limits belongs to different classes, to which different AQLs apply.
Again, by implication, nonconformity beyond each of the two limits is of a different degree of seriousness. The
AQL for the class of greater seriousness will be smaller than the AQL for the other class.
EXAMPLE 2 In a given bottle-filling plant, overfilling leads to a marginal reduction in profit, whereas underfilling is
much more serious as it could lead to weights and measures violations, financial penalties, bad publicity and loss of
goodwill. The lower specification limit in this case should therefore have a much smaller AQL than the upper specification
limit.
For separate control, a Form k acceptance criterion can be applied separately to each limit. The lot is
accepted if both acceptance criteria are satisfied.
4.1.3.4 Complex control
Double specification limits are said to be under complex control when the fraction nonconforming beyond the
limit of greater seriousness belongs to one class, to which a given AQL applies, and the combined fraction
nonconforming beyond both limits belongs to another class, to which a larger AQL applies. This allows some
trade-off between the fractions nonconforming at both ends of the distribution of values of the quality
characteristic while still maintaining control of the fraction nonconforming at the more important end of the
distribution that is of the greatest concern.
EXAMPLE 3 Wooden strips, supplied in batches and used in the construction of garden furniture, are specified to be
between 86,5 cm and 86,7 cm in length. Strips that are too large can be shortened, but strips that are too short are
unusable and have to be replaced, which is more time-consuming and can interfere with production. An AQL of 2,5 % is
set for both limits combined, with another AQL on the lower limit of 0,65 %.
Complex control is a combination of combined control of both limits with control of just one of those limits.
Form k is therefore again inadequate for this situation, so that Form p* has to be used.
4.2 Control of two or more quality characteristics
4.2.1 General
The number of possible combinations of control soon becomes vast as the number of quality characteristics
increases. The discussion below is therefore confined to providing examples in the case of two quality
characteristics, x and y.
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4.2.2 Exam
...
TECHNICAL ISO/TR
REPORT 8550-3
First edition
2007-06-01
Guidance on the selection and usage of
acceptance sampling systems for
inspection of discrete items in lots —
Part 3:
Sampling by variables
Lignes directrices pour la sélection d'un système, d'un programme ou
d'un plan d'échantillonnage pour acceptation pour le contrôle d'unités
discrètes en lots —
Partie 3: Échantillonnage par variables
Reference number
ISO/TR 8550-3:2007(E)
©
ISO 2007
---------------------- Page: 1 ----------------------
ISO/TR 8550-3:2007(E)
PDF disclaimer
This PDF file may contain embedded typefaces. In accordance with Adobe's licensing policy, this file may be printed or viewed but
shall not be edited unless the typefaces which are embedded are licensed to and installed on the computer performing the editing. In
downloading this file, parties accept therein the responsibility of not infringing Adobe's licensing policy. The ISO Central Secretariat
accepts no liability in this area.
Adobe is a trademark of Adobe Systems Incorporated.
Details of the software products used to create this PDF file can be found in the General Info relative to the file; the PDF-creation
parameters were optimized for printing. Every care has been taken to ensure that the file is suitable for use by ISO member bodies. In
the unlikely event that a problem relating to it is found, please inform the Central Secretariat at the address given below.
COPYRIGHT PROTECTED DOCUMENT
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ISO/TR 8550-3:2007(E)
Contents Page
Foreword. iv
Introduction . v
1 Scope . 1
2 Normative references . 1
3 Normality . 2
4 Types of control . 12
5 Forms of acceptance criteria. 15
6 International Standards for acceptance sampling of lots by variables . 29
7 Effect on the selection process of market and production conditions. 31
Annex A (normative) Normal probability paper . 38
Bibliography . 39
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ISO/TR 8550-3:2007(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
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International 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.
In exceptional circumstances, when a technical committee has collected data of a different kind from that
which is normally published as an International Standard (“state of the art”, for example), it may decide by a
simple majority vote of its participating members to publish a Technical Report. A Technical Report is entirely
informative in nature and does not have to be reviewed until the data it provides are considered to be no
longer valid or useful.
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/TR 8550-3 was prepared by Technical Committee ISO/TC 69, Applications of statistical methods,
Subcommittee SC 5, Acceptance sampling.
This first edition of ISO/TR 8550-3, together with ISO/TR 8550-1 and ISO/TR 8550-2, cancels and replaces
ISO/TR 8550:1994.
ISO/TR 8550 consists of the following parts, under the general title Guidance on the selection and usage of
acceptance sampling systems for inspection of discrete items in lots:
⎯ Part 1: Acceptance sampling
⎯ Part 3: Sampling by variables
The following part is under preparation:
⎯ Part 2: Sampling by attributes
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ISO/TR 8550-3:2007(E)
Introduction
This part of ISO/TR 8550 gives guidance on the selection of an acceptance sampling system for inspection by
variables. It does this principally by reviewing the available systems specified by various standards and
showing ways in which these can be compared in order to assess their suitability for an intended application. It
is assumed that the choice has already been made to use sampling by variables in preference to sampling by
attributes.
A corresponding guidance document on the selection of a generic acceptance sampling system, scheme or
plan for inspection by attributes is given in ISO/TR 8550-2.
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TECHNICAL REPORT ISO/TR 8550-3:2007(E)
Guidance on the selection and usage of acceptance sampling
systems for inspection of discrete items in lots —
Part 3:
Sampling by variables
1 Scope
The guidance in this part of ISO/TR 8550 is confined to acceptance sampling of products that are supplied in
lots and that can be classified as consisting of discrete items (i.e. discrete articles of product). Each item in a
lot can be identified and segregated from the other items in the lot and has an equal chance of being included
in the sample. Each item of product is countable and has specific characteristics that are measurable on a
continuous scale. Each characteristic has, at least to a good approximation, a normal distribution or a
distribution that can be transformed so that it closely resembles a normal distribution.
Standards on acceptance sampling by variables are applicable to a wide variety of inspection situations.
These include, but are not limited to, the following:
a) end items, such as complete products or sub-assemblies;
b) components and raw materials;
c) services;
d) materials in process;
e) supplies in storage;
f) maintenance operations;
g) data or records;
h) administrative procedures.
Although this part of ISO/TR 8550 is written principally in terms of manufacture and production, it should be
interpreted liberally as it is applicable to the selection of sampling systems, schemes and plans for all types of
product and processes as defined in ISO 9000.
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated
references, only the edition listed applies. For undated references, the latest edition of the referenced
document (including any amendment) applies.
ISO/TR 8550-1:2007, Guidance on the selection and usage of acceptance sampling systems for inspection of
discrete items in lots — Part 1: Acceptance sampling
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ISO/TR 8550-3:2007(E)
3 Normality
3.1 Relationship between form of distribution of quality characteristic and percent
nonconforming
A key aspect of sampling by variables is the form of the distributions of the quality characteristics. Consider a
single quality characteristic. If it is normally distributed and if an upper specification limit is located at the mean
plus two standard deviations, the percent nonconforming is about 2,5 %. If the specification limit is located at
the mean plus three standard deviations, the percent nonconforming is about 0,1 %. However, if the
distribution of the quality characteristic is not normal and has a large positive skewness, i.e. a long tail to the
right, an upper specification limit located at the mean plus three standard deviations could conceivably yield a
percent nonconforming approaching 10 % instead of about 0,1 % (see Figures 1 and 2).
Therefore, whenever a sampling plan for inspection by variables for percent nonconforming is to be employed,
it is highly desirable to check any assumptions about the shape of the distribution, especially in the tails of the
distribution. If the AQL is very small, for example 0,1 %, a study of several thousand items should be made,
including a test of distributional form.
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ISO/TR 8550-3:2007(E)
Key
1 upper specification limit
2 0,1 % above specification
Figure 1 — Normal distribution
Key
1 upper specification limit
Figure 2 — Distribution with large positive skewness
3.2 Identifying departure from normality
3.2.1 Subjective assessment
The degree to which a sample appears to have come from a normal distribution can be subjectively assessed
by means of a normal probability plot. Such a plot is constructed in the following way. Once the random
sample has been selected and the quality characteristic x has been measured for each item, the values x ,
1
x , . . ., x are arranged in ascending order x , x , . . ., x , such that x u x u, . . . u x . The points with
2 n [1] [2] [n] [1] [2] [n]
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ISO/TR 8550-3:2007(E)
3
1
coordinates xi,/−+n are then plotted on a sheet of normal probability paper for i = 1, 2, . . ., n.
()()
{}
[]i
84
To facilitate this process, an A4 sheet of normal probability paper that can be freely photocopied is provided in
Annex A.
Figure 3 shows the normal probability plot of a random sample of size 100 from a normal distribution. The
graph paper is specially designed so that data from a normal distribution tend to lie close to a straight line.
A straight line has been drawn by eye through the data, showing in this case that there are only minor
departures from linearity.
When data originate from a normal distribution, departures of the probability plot from linearity are due solely
to sampling fluctuations. Conversely, data from other types of distribution will tend to show departures from
linearity of a characteristic type, helping in the determination of the family of distributions to which the data
belong. Knowledge of this family can indicate the appropriate transformation to make to the data to bring
these closer to normality.
Figures 4 to 7 show the density functions and examples of normal probability plots based on a random sample
of size 100 for, respectively, a lognormal, Cauchy, Laplace, and exponential distribution, respectively. On the
normal probability plot for Figures 4 to 6, a straight line has been drawn through the data points to aid the eye
in identifying the characteristic differences.
For the lognormal distribution, there is a pronounced downward concavity.
The Cauchy distribution is almost indistinguishable from the normal distribution towards its centre, but the
extra thickness of its tails results in the plot being relatively high for low values of x and relatively low for high
values of x, the extremities of the plot being almost horizontal.
The Laplace distribution is similar, except that there is a shorter region in the normal probability plot where the
distribution is indistinguishable from the normal distribution, and the extremities of the plot are far from
horizontal.
The normal probability plot for the exponential distribution has a very characteristic shape, rising very steeply
at the left and becoming almost horizontal towards the right.
These are a small selection from the many possible distributions from which data might have arisen. In some
cases, e.g. the lognormal distribution, the distribution can be transformed exactly to normality without knowing
its parameters (see 3.3.2 and 3.3.3). In other cases, approximate normality may be achieved, e.g. by using
[20]
the fourth root transformation on exponentially distributed variables, as shown by Kittlitz . In other cases,
acceptance sampling by variables might not be possible without a method tailored to that family of
distributions. If such a method does not exist, acceptance sampling by attributes might have to be used
instead, the loss in efficiency being more than compensated for by the increase in integrity of the acceptance
sampling results.
Figures 4 to 7 show normal probability plots for samples of size 100. Often there is not the luxury of such large
samples. With small samples, it is less clear whether the departures from linearity of the normal probability
plot are due to non-normality or merely to sampling fluctuations. In case of doubt, subjective assessment of
departure from normality should be replaced by objective statistical tests, such as those discussed in 3.2.2.
Further information on tests for departure from normality is given in ISO 5479 and ISO 2854:1976, Clause 2.
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ISO/TR 8550-3:2007(E)
Key
X quality characteristic, x
Y probability density of x
a) Normal distribution
b) Normal probability plot of a random sample of size 100 from a normal distribution
Figure 3 — Normal distribution and normal probability plot
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ISO/TR 8550-3:2007(E)
Key
X quality characteristic, x
Y probability density of x
a) Lognormal distribution
b) Normal probability plot of a random sample of size 100 from a lognormal distribution
Figure 4 — Lognormal distribution and normal probability plot
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ISO/TR 8550-3:2007(E)
Key
X quality characteristic, x
Y probability density of x
a) Cauchy distribution
b) Normal probability plot of a random sample of size 100 from a Cauchy distribution
Figure 5 — Cauchy distribution and normal probability plot
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ISO/TR 8550-3:2007(E)
Key
X quality characteristic, x
Y probability density of x
a) Laplace distribution
b) Normal probability plot of a random sample of size 100 from a Laplace distribution
Figure 6 — Laplace distribution and normal probability plot
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ISO/TR 8550-3:2007(E)
Key
X quality characteristic, x
Y probability density of x
a) Exponential distribution
b) Normal probability plot of a random sample of size 100 from an exponential distribution
Figure 7 — Exponential distribution and normal probability plot
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ISO/TR 8550-3:2007(E)
3.2.2 Statistical tests for departure from normality
3.2.2.1 Directional versus omnibus tests
Statistical tests are used to determine the degree to which the available evidence fails to support a given null
hypothesis, say H . The power of the test is the probability of rejecting the hypothesis H in favour of the
0 0
alternative hypothesis H when the alternative hypothesis is true.
1
When testing for departures from normality, the null hypothesis H is that the distribution is normal while the
0
alternative hypothesis H is that it is not normal. If the alternative hypothesis is more specific, stating the
1
alternative family of distributions to which the distribution belongs, then the test is said to be directional.
Otherwise, it is said to be an omnibus test.
In both cases, a statistic T is calculated from the sample evidence, and H is rejected if the value of T lies in a
0
so-called critical region. The critical region is chosen so that the probability of T falling in the critical region
when H is true is a small quantity, usually 5 %. For an omnibus test, the critical region simply consists of
0
values of T that lie far away from the expected value of T under H . For a directional test, the critical region
0
consists of the values of T for which the power is greatest.
In general, therefore, greater power is achieved by being as specific as can be justified about the alternative
hypothesis, i.e. about the likely nature of the departure from the null hypothesis.
As might be expected, power is generally also increased by increasing the sample size on which T is based.
3.2.2.2 Directional tests
ISO 5479:1997 provides two directional tests. One of these is for skewness, the other for kurtosis (i.e.
peakedness). A simultaneous bi-directional test for skewness and kurtosis is also provided. The skewness
and kurtosis test statistics for n observations x , x , . . ., x are, respectively, the moment coefficients:
1 2 n
3
2
2
bm= /m and bm= /m ,
13 2 24 2
where:
n
1
j
mx=−x for j = 2, 3 and 4.
()
∑
ji
n
i=1
3.2.2.3 Omnibus tests
ISO 5479:1997 also provides two omnibus tests: the Shapiro-Wilk test and the Epps-Pulley test. The test
statistic for the Shapiro-Wilk test is a linear function of the ordered observations. The Epps-Pulley test statistic
is a little more complicated to implement as it involves a sum and a double sum of exponentiated quantities.
A rule of thumb is given for deciding which of these to use in a given situation.
3.3 Transforming to normality
3.3.1 Normalization and variance stabilization
Much analysis of variance is invalidated if the quality characteristic under analysis is heteroscedastic, i.e.
when its variance varies with its mean. A mathematical transformation of the characteristic that roughly
equalizes the variance over all values of the mean is called a variance-stabilizing transformation. It is often the
case that transforming such data to eliminate heteroscedasticity, i.e. to make the data homoscedastic, also
has the effect of making the data more normal. In other words, variance-stabilizing transformations are often
normalizing transformations.
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A rough rule for determining the appropriate transformation is as follows. If the standard deviation σ of a
process characteristic x can be expressed approximately as a function h()µ , where µ is the corresponding
mean of the characteristic, then an approximate variance-stabilizing transformation of x is g(x), where:
x dt
gx()= . (1)
∫
ht
()
Examples of the use of this method are given in 3.3.2 and 3.3.3.
If a test for departure from normality indicates that x is non-normal, the use of y = g(x) should be considered
instead of x.
3.3.2 The square root transformation
Where σ is a constant multiple of µ , i.e. hc(µµ)= , where c is a constant, then, from Expression (1):
xd2t
g()xx== .
∫
c
ct
As the coefficient 2/c has no effect on the stabilization of the variance, it can be ignored. An approximate
variance-stabilizing transformation is therefore the square root transformation:
g(xx)= .
The standard deviation of g(x) is approximately c/2.
3.3.3 The logarithmic transformation
When σ is a constant multiple ofµ , i.e. hc(µµ)= , where c is a constant, then, from Expression (1):
xd1t
g()xx== ln() .
∫
ct c
As the coefficient 1/c has no effect on the stabilization of the variance, it can be ignored. An approximate
variance-stabilizing transformation is therefore the logarithmic transformation:
g(x) = ln(x).
The standard deviation of g(x) is approximately c.
This transformation might be appropriate when the quality characteristic is the sample variance based on a
sample of size n, in which case cn=−21/( ) .
3.3.4 The Box-Cox transformation
[19]
A general transformation, proposed by Box and Cox , is to set
λ
x −1
gx =
()
λ
where λ > 0.
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Note that:
⎯ setting λ equal to 1 simply relocates the existing distribution, leaving its original shape unchanged;
⎯ setting λ equal to 0,5 is equivalent to first using the square root transformation and then relocating and
rescaling the resulting distribution;
⎯ letting λ tend to zero is equivalent to using the logarithmic transformation, i.e. equivalent to setting
g(x) = ln(x).
However, λ is not limited to taking one of these particular values, so the value of λ in excess of zero that best
normalizes the distribution can be found either by trial and error or by some optimization method applied to
past data.
A more general version of the Box-Cox transformation is:
λ
1
()x+−λ 1
2
gx()= for λ > 0 andλ >−x .
1 21[]
λ
1
This transformation effectively relocates the distribution of x by an amount λ before applying the simpler Box-
2
Cox power transformation. A consequence is that the more general transformation does not require all the
data values to be positive. Because the more general transformation has two parameters, it allows a greater
range of distribution families to be transformed to normality.
4 Types of control
4.1 Control of a single quality characteristic
4.1.1 General
Acceptance sampling by variables can become complicated when there are two or more quality
characteristics, so the text first considers the case where only a single quality characteristic is being controlled.
As the acceptance criterion for a single quality characteristic involves either x and s, or x and σ , it can
always be represented diagrammatically as well as algebraically. A diagrammatic representation of an
acceptance criterion is called an acceptance diagram.
Within the case of a single quality characteristic, there are several possible methods of control, which are
described in 4.1.2 and 4.1.3.
4.1.2 Single specification limit
The simplest case of a single quality characteristic is where there is a single specification limit, i.e. where
either an upper limit or a lower limit to values of the characteristic is specified, but not both. Control of such a
characteristic by means of sampling by variables is relatively straightforward, requiring the sample mean to be
within a specification and at least a given multiple, denoted by k, of the sample standard deviation (or process
standard deviation, if known) away from the specification limit. When the acceptance criterion is expressed in
terms of this factor k, the method is described as “Form k” (see 5.2).
4.1.3 Double specification limits
4.1.3.1 General
Rather more complicated is the case of a single quality characteristic with double specification limits, i.e.
where both an upper limit and a lower limit to values of the characteristic are specified. In this case, there are
three modes of control.
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4.1.3.2 Combined control
Double specification limits are said to be under combined control when the fraction nonconforming beyond
both limits belongs to the same class, to which a single AQL applies. By implication, nonconformity beyond
either limit is of roughly the same degree of seriousness.
EXAMPLE 1 A weapon guidance system is to be tested against a moving target. Missing the target either to the left or
to the right is equally unsatisfactory so combined control of both sides of the target might be appropriate in this case.
Form k is inadequate for combined control. Instead, Form p* is used, i.e. the lot is accepted only if an estimate
of the process fraction nonconforming is less than a given value p*. In other words, p* is the maximum
estimate of the process fraction nonconforming that is deemed to be acceptable for the given sample size and
AQL.
4.1.3.3 Separate control
Double specification limits are said to be under separate control when the fraction nonconforming beyond the
two limits belongs to different classes, to which different AQLs apply.
Again, by implication, nonconformity beyond each of the two limits is of a different degree of seriousness. The
AQL for the class of greater seriousness will be smaller than the AQL for the other class.
EXAMPLE 2 In a given bottle-filling plant, overfilling leads to a marginal reduction in profit, whereas underfilling is
much more serious as it could lead to weights and measures violations, financial penalties, bad publicity and loss of
goodwill. The lower specification limit in this case should therefore have a much smaller AQL than the upper specification
limit.
For separate control, a Form k acceptance criterion can be applied separately to each limit. The lot is
accepted if both acceptance criteria are satisfied.
4.1.3.4 Complex control
Double specification limits are said to be under complex control when the fraction nonconforming beyond the
limit of greater seriousness belongs to one class, to which a given AQL applies, and the combined fraction
nonconforming beyond both limits belongs to another class, to which a larger AQL applies. This allows some
trade-off between the fractions nonconforming at both ends of the distribution of values of the quality
characteristic while still maintaining control of the fraction nonconforming at the more important end of the
distribution that is of the greatest concern.
EXAMPLE 3 Wooden strips, supplied in batches and used in the construction of garden furniture, are specified to be
between 86,5 cm and 86,7 cm in length. Strips that are too large can be shortened, but strips that are too short are
unusable and have to be replaced, which is more time-consuming and can interfere with production. An AQL of 2,5 % is
set for both limits combined, with another AQL on the lower limit of 0,65 %.
Complex control is a combination of combined control of both limits with control of just one of those limits.
Form k is therefore again inadequate for this situation, so that Form p* has to be used.
4.2 Control of two or more quality characteristics
4.2.1 General
The number of possible combinations of control soon becomes vast as the number of quality characteristics
increases. The discussion below is therefore confined to providing examples in the case of two quality
characteristics, x and y.
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4.2.2 Examples of the control of two independent quality characteristics
4.2.2.1 General
For most cases of two or more quality characteristics, it is necessary to use Form p*. For brevity, all the
examples given for two variables are given in terms of Form p*.
In all cases, it is assumed that a single acceptance criterion is stipulated for each class of nonconformity, and
that a lot is only acceptable if the criterion for each and every class is satisfied.
4.2.2.2 Notation
With two quality characteristics, some new notation is necessary. The two quality characteristics are denoted
by x and y. The lower and upper specification limits on x are denoted by L(x) and U(x) respectively, and on y
by L(y) and U(y). The process fraction nonconforming beyond each of these four limits is denoted by
ˆˆ ˆ ˆ
p ()xp, (xp), (y) and p ()y , respectively, and their estimates by p ()xp, (x),p (y) and p ()y .
LU L U LU L U
Due to the independence of x and y, the total process fraction nonconforming in a class containing
nonconformity at all four of these limits is given by:
p=−11⎡⎤1−px()−p()x⎡−p(y)−p(y)⎤ (2)
LU L U
⎣⎦⎣ ⎦
and its estimate by:
ˆˆ ˆ ˆ ˆ
p=−11⎡⎤−px−p x⎡1−p y−p y⎤. (3)
() () () ()
LU L U
⎣⎦⎣ ⎦
The class of nonconformity is indicated by the appropriate subscript from A, B, etc. to p or pˆ . Expressions (2)
and (3) may be used generally, with the elements not included in the class set to zero. The following examples
demonstrate this.
Note that, if p ()xp, (xp), (y) and p ()y are all very small, then p≅+px() p ()x+p (y)+p (y) ; similarly,
LU L U LU L U
ˆˆ ˆ ˆ ˆˆ ˆ ˆ ˆ
if p ()xp, (x),p (y) and p ()y a
...
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