ISO 12491:1997

Statistical methods for quality control of building materials and components

Méthodes statistiques de contrôle de la qualité des matériaux et éléments de construction

General Information

• Status: Published
• Publication Date: 30-Apr-1997
• ICS: 91.200 - Construction technology
• Technical Committee: ISO/TC 98/SC 2 - Reliability of structures
• Drafting Committee: ISO/TC 98/SC 2 - Reliability of structures
• Current Stage: 9093 - International Standard confirmed
• Start Date: 02-Jun-2025
• Completion Date: 13-Dec-2025

Overview

ISO 12491:1997 - Statistical methods for quality control of building materials and components - provides general principles and operational techniques for applying statistical quality control in the construction sector. Intended to support the safety and serviceability principles of ISO 2394, the standard is applicable to all buildings and civil engineering works and to common materials such as concrete, steel, wood and bricks. It focuses on classical statistical methods (with a Bayesian option) to make quality control more efficient and cost-effective, especially where tests are expensive or destructive.

Key topics

• Probability distributions: treatment of normal and log‑normal distributions and procedures for normality tests.
• Estimation and hypothesis testing: principles for estimating population mean, variance, fractiles, and constructing confidence intervals (one- and two‑sided).
• Bayesian approach: use of Bayesian inference for parameter estimation and prediction of fractiles.
• Sampling inspection: methods by variables and by attributes, including plans for isolated lots and ongoing series of lots.
• Sampling plans and acceptance criteria: sample sizes, acceptance constants (k1, k2), acceptance numbers (AC), and rules for lot acceptance.
• Operating Characteristic (OC) curves: characterization of sampling-plan performance and identification of producer’s risk and consumer’s risk points.
• Comparison tests: comparison of means and variances, treatment of outliers, and use of common distributions (t, χ2, F, noncentral t) in inference.
• Definitions and vocabulary: clear definitions for terms such as lot, population, sample, conforming/nonconforming unit, and fractile (aligned with ISO 3534 parts 1 & 2).

Applications and users

ISO 12491:1997 is practical for:

• Manufacturers and suppliers of building materials establishing internal QA/QC procedures.
• Testing laboratories designing statistically valid test programs, especially when tests are destructive or costly.
• Construction contractors and procurement teams setting acceptance criteria and sampling plans for incoming materials or subcontracted components.
• Structural and materials engineers using statistical evidence to support reliability and safety assessments.
• Regulators and certification bodies developing inspection regimes and conformity assessment procedures.

Benefits include optimized sampling (reducing test costs), objective lot-acceptance decisions, and defensible statistical documentation for quality claims.

Related standards

• ISO 2394 - General principles on reliability for structures
• ISO 3534‑1 and ISO 3534‑2 - Statistics vocabulary and symbols (probability and statistical quality control)

Keywords: ISO 12491:1997, statistical methods, quality control, building materials, sampling inspection, acceptance sampling, OC curve, Bayesian estimation, normal distribution, log-normal distribution.

Standards Content (Sample)

IS0
INTERNATIONAL
STANDARD
First edition
1997-05-01
Statistical methods for quality control of
building materials and components
M&hodes sta tistiques de con tr6le de la qua/it6 des ma tkiaux et &ments
de construction
Reference number
IS0 12491:1997(E)
IS0 12491:1997(E)
Page
Contents
........................................................................................
Scope
Normative references .
.................................................................................. 1
Definitions
a
Population and sample .
............................................................................. a
4.1 General
a
4.2 Normal distribution .
...................................................... 9
4.3 Log-normal distribution
.................................................................. 9
4.4 Normality tests
Methods of statistical quality control .
......................................................... 9
5.1 Quality requirements
..................................................
5.2 Basic statistical methods
5.3 Bayesian approach .
...........................................................
5.4 Additional methods
........................................... 12
Estimation and tests of parameters
..................................... 12
6.1 Principles of estimation and tests
6.2 Estimation of the mean .
.................................................. 13
6.3 Estimation of the variance
.......................................................
6.4 Comparison of means
.................................................. 15
6.5 Comparison of variances
......................................................... 15
6.6 Estimation of fractiles
......... 16
6.7 Prediction of fractiles using the Bayesian approach
ia
Sampling inspection .
ia
....................................................
7.1 Variables and attributes
.............................................. ia
7.2 Inspection of an isolated lot
..................... 19
7.3 Sampling inspection by variables: CJ known.
Sampling inspection by variables: o unknown . 20
7.4
..................................... 20
7.5 Sampling inspection by attributes
Annex
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Bibliography
. . . . . . .~.~.
Alphabetical index
0 IS0 1997
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced
or utilized in any form or by any means, electronic or mechanical, including photocopying and
microfilm, without permission in writing from the publisher.
International Organization for Standardization
Case postale 56 l CH-1211 Geneve 20 l Switzerland
Internet central @ iso.ch
c=ch; a=40Onet; p=iso; o=isocs; s=central
x.400
Printed in Switzerland
ii
IS0 12491 :1997(E)
@ IS0
Foreword
IS0 (the International Organization for Standardization) is a worldwide
federation of national standards bodies (IS0 member bodies). The work of
preparing International Standards is normally carried out through IS0
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. IS0
collaborates closely with the International Electrotechnical Commission
(IEC) on all matters of electrotechnical standardization.
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.
International Standard IS0 12491 was prepared by Technical Committee
ISOmC 98, Bases for design of structures, Subcommittee SC 2, Reliability
of structures.
Annex A of this International Standard is for information only.

IS0 12491:1997(E) @ IS0
Introduction
Quality control of building materials and components is, according to
IS0 2394, an indispensable part of an overall concept of structural
reliability. As quality control is generally a time-consuming and expensive
task, various operational techniques and activities have been developed to
fulfil quality requirements in building. It appears that properly employed
statistical methods can provide efficient, economic and effective means of
quality control, particularly when expensive and destructive tests are to be
performed. The purpose of this International Standard is to provide general
techniques for quality control of building materials and components used in
building or other civil engineering works.
statistical methods .
Described techniques consist predominantly of classical
of common interest for all the participants in the building process. For other
statistical
more sophisticated techniques and specific problems, existing
standards listed in annex A should be applied.
iv
INTERNATIONAL STANDARD 0 IS0 IS0 12491:1997(E)
Statistical methods for quality control of building
materials and components
1 Scope
This International Standard gives general principles for the application of statistical
methods in the quality control of building materials and components in compliance with
the safety and serviceability requirements of IS0 2394.
This International Standard is applicable to all buildings and other civil engineering
work, existing or under construction, whatever the nature or combination of the
materials used, for example concrete, steel, wood, bricks.
2 Normative references
The following standards contain provisions which, through reference in this text,
constitute provisions of this International Standard. At the time of publication, the
editions indicated were valid. All standards are subject to revision, and parties to
agreements based on this International Standard are encouraged to investigate the
possibility of applying the most recent editions of the standards indicated below.
Members of IEC and IS0 maintain registers of currently valid International Standards.
General principles on reliability for structures.
IS0 2394:---l,
IS0 3534~19993, Statistics - Vocabulary and symbols - Part 1: Probability and
general statistical terms.
IS0 35342:1993, Statistics - Vocabulary and symbols - Part 2: Statistical quality
control.
3 Definitions
For the purposes of this International Standard, the definitions given in IS0 3534-1 and
IS0 3534-2, and the following definitions, apply.
NOTE - The terms and their definitions are listed in the order corresponding to their appearance
in the main text. An alphabetic list of these terms with numerical references to subclauses where
the terms appear is given in the index.
3.1 quality control: Operational techniques and activities that are used to fulfill
requirements for quality.
3.2 statistical quality control: That part of quality control in which statistical
methods are used (such as estimation and tests of parameters and sampling inspection).
’ To be published. (Revision of IS0 2394:1986)

IS0 12491:1997(E)
3.3 unit: Defined quantity of building material, component or element of a building or
other civil engineering work that can be individually considered and separately tested.
3.4 population: Totality of units under consideration.
3.5 (random) variable, X A variable which may take any of the values of a specified
set of values and with which is associated a probability distribution.
NOTE - A random variable that may take only isolated values is said to be “discrete ”. A random
variable which may take any value within a finite of infinite interval is said to be “continuous ”.
3.6 (probability) distribution: A function which gives the probability that a variable
X takes any given value (in the case of a discrete variable) or belongs to a given set of
values (in the case of a continuous variable).
3.7 distribution function, n(x): A function giving, for every value of x, the probability
that the variable X is less than or equal to x:
n(x) = Pr (XI x)
3.8 (probability) density function, f(x): The derivative (when it exists) of the
distribution function:
d=(x)
X=
f( >
dx
3.9 (population) parameter: Quantity used in describing the distribution of a random
variable in a population.
3.10 fractile, x : If X is a continuous variable and p is a real number between 0 and 1,
the p-fractile is the value of the variable X for which the distribution function equals p.
Thus x is a p-fractile if
P
PJX I x )=p
P
3.11 (population) mean, p: For a continuous variable X having the probability
density f(x), the mean, if it exists, is given by
the integral being extended over the interval(s) of variation of the variable X.
3.12 (population) variance, 02: For a continuous variable X having the probability
density function f(x), the variance, if it exists, is given by
02= ,(x-p)” f(x)&
the integral being extended over the interval(s) of variation of the variable X.
IS0 12491:1997(E)
3.13 (population) standard deviation, 0: Positive square root of the population
variance GL.
3J4 standardized variable: A random variable, the mean of which equals zero and
the standard deviation of which equals f. If the variable X has a mean equal to p and a
standard deviation equal to 0, the corresponding standardized variable is given as
NOTE - The distribution of the standardized variable is called “standardized distribution ”.
3.15 normal distribution: Probability distribution of a continuous variable X, the
probability density function of which is
oJ% 2 0
f( x > =- 1 exp H -- 1 - x-p 11
3.16 log-normal distribution: Probability distribution of a continuous variable X
which can take any value from x, to +m, or from - to x,.
In the former, more frequent, case the probability density function is given as
x 2 x0
pr and CJ, are, respectively, the mean and the standard deviation of the new variable;
Y = In (X-x0)
In the latter, less frequent, case the sign of the brackets (X-x,) and (x -x0) is to be
changed. Note that the variable Y has a normal distribution.
3.17 (random) sample: One or more sampling units taken from a population in such a
way that each unit of the population has the same probability of being taken.
3.18 (sample) size, n: Number of sampling units in the sample.
3.19 sample mean, T: Sum of n values Xi of sampling units divided by the sample size
n:
X=--CXi
n
IS0 12491:1997(E)
3.20 sample variance, s2: Sum of n squared deviations from the sample mean Z
divided by the sample size n minus 1:
=-
s2 Xi - -
c( x>
n-l
3.21 sample standard deviation, s: Positive square root of the sample variance s2.
3.22 estimation: Operation of assigning, from observations on a sample, numerical
values to the parameters of a distribution chosen as the statistical model of the
population from which this sample was taken.
3.23 estimator: Function of a set of the sample random variables used to estimate a
population parameter.
3.24 estimate: Value of an estimator obtained as a result of an estimation.
3.25 confidence level, y : Given value of the probability associated with a confidence
interval.
NOTE - In IS0 3534-1, it is designated (1 -OC ).
3.26 two-sided confidence interval: When Tl and T, are two functions of the
observed values such that, 8 being a parameter to be estimated, the probability Pr (T,
5 8 5 T,) is at least equal to the confidence level y (where y is a fixed number, positive
and less than 1), the interval between Tl and T, is a two-sided y confidence interval for 8.
3.27 one-sided confidence interval: When T is a function of the observed values
such that, 0 being a population parameter to be estimated, the probability Pr (T 2 8)
or the probability Pr (T 5 0) is at least equal to the confidence level y (where y is a fixed
number, positive and less than l), the interval from the smallest possible value of 8 up
to T (or the interval from the T up to the largest possible value of 0) is a one-sided y
confidence interval for 8.
3.28 outliers: Observations in a sample, so far separated in value from the remainder
as to suggest that they may be from a different population.
3.29 (statistical) test: Statistical procedure to decide whether a hypothesis about the
distribution of one or more populations should be accepted or rejected.
3.30 (statistical) hypothesis: Hypothesis, concerning the population, which is to be
accepted or rejected as the outcome of the test using sample observations.
3.31 significance level, a: Given value, which is the upper limit of the probability of
a statistical hypothesis being rejected when this hypothesis is true.
3.32 number of degrees of freedom, v : In general, the number of terms in a sum
minus the number of constraints on the terms of the sum.

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IS0 12491:1997(E)
3.33 x 2-distribution: Probability distribution of a continuous variable x 2 which can
take any value from 0 to = , the probability density function of which is
2 (v/2)-1
X
( 1
X
f(X2;v) =
exp -2
2(v'2) r(v/2)
c 1
where
x 2 - > 0 with a parameter (number of degrees of freedom) v = 1, 2, 3,. . .;
I? is the gamma function.
3.34 tdistribution: Probability distribution of a continuous variable t which can take
any value from - to +w, the probability density function of which is
1 qv + 1> 121
f(t;v)=-
r(v /2)
n:V
Al-
(l+ t2 lv)"'"'2
where
- c t c -em with a parameter (number of degrees of freedom) v = 1, 2, 3,. ;
r is the gamma function.
3.35 noncentral t-distribution: Probability distribution of a continuous variable t
which can take any value from - to +=, the probability density function of which is
1 1 1
K
f (t;v ’s)=F 2(v-1)/2
72 exp
where
.
I
- c t c += with two
parameters; 1.e. number of degrees of freedom v and
noncentrality parameter 6.
3.36 F-distribution: Probability distribution of a continuous variable F which can take
any value from 0 to 00, the probability density function of which is
+
q(vl+v,)q
f(W,J,) =
(vr2 (v2)v2'2 @ Fy;';;;,+v2),2
r(v,/2)r(v, 12)
1 2
@ IS0
IS0 12491: 1997(E)
where
F > - 0 with parameters (numbers of degrees of freedom) v1 ,v2 = 1, 2, 3,.;
r is the gamma function.
3.37 lot: Definite quantity of units, manufactured or produced under conditions which
are presumed uniform.
NOTE - In statistical quality control in building, a lot is usually equivalent to a “batch” and is
considered as a “population ”.
3.38 isolated lot: A lot separated from the sequence of lots in which it was produced
or collected, and not forming part of a current sequence of inspection lots.
NOTE - In statistical quality control in building, lots are usually considered as “isolated lots ”.
3.39 conforming unit: Unit which satisfies all the specified requirements.
3.40 nonconforming unit: Unit containing at least one nonconformity which causes
the unit not to satisfy specified requirements.
3.41 sampling inspection: Inspection in which decisions are made to accept or not
accept a lot, based on results of a sample selected from that lot.
Method of sampling inspection which
3.42 sampling inspection by variables:
consists of measuring a quantitative variable X for each unit of a sample.
3.43 sampling inspection by attributes: Method of sampling inspection which
consists of distinguishing between conforming and nonconforming units of a sample.
3.44 sampling plan: A plan according to which one or more samples are taken in
order to obtain information and the possibility of reaching a decision concerning the
acceptance of the lot.
NOTE - It includes the sample size n and the acceptance constants k, , k ( in sampling inspection
by variables), or the sample size n and the acceptance number AC (in sampling inspection by
attributes).
3.45 operating characteristic curve (OC curve): Curve showing, for a given
sampling plan, the probability that an acceptance criterion is satisfied, as a function of
the lot quality level.
3.46 producer: Any participant of the building process supplying a lot for further
procedure or use.
3.47 consumer: Any participant of the building process purchasing a lot for further
procedure or use.
3.48 producer ’s risk point (PRP): A point on the operating characteristic curve
corresponding to a predetermined and usually low probability of non-acceptance.
NOTE - This probability is the producer ’s risk (PR) when an isolated lot is considered.
@ IS0 IS0 12491:1997(E)
3.49 consumervs risk point (CRP): A point on the operating characteristic curve
corresponding to a predetermined and usually low probability of acceptance.
NOTE - This probability is the consumer ’s risk (CR) when an isolated lot is considered.
3.50 producer ’s risk (PR): For a given sampling plan, the probability of non-
acceptance of a lot when the lot quality has a value stated by the plan as acceptable.
NOTE - This quality is the producer% risk quality (PRQ) when an isolated lot is considered.
3.51 consumer ’s risk (CR): For a given sampling plan, the probability of acceptance
of a lot when the lot quality has a value stated by the plan as unsatisfactory.
NOTE - This quality is the consumer ’s risk quality (CRQ) when an isolated lot is considered.
3.52 producer ’s risk quality (PRQ): A lot quality level which, in the sampling plan
for an isolated lot, corresponds to a specified producer ’s risk (PR).
NOTE - When a continuing series of lots is considered, the acceptable quality level AQL is used
instead of PRQ.
3.53 consumer ’s risk quality (CRQ): A lot quality level which, in the sampling plan
for an isolated lot, corresponds to a specified consumer ’s risk (CR).
NOTE - When a continuing series of lots is considered, the limiting quality level LQL is used
instead of CRQ.
3.54 acceptance constants, k, , k : In sampling inspection by variables, constants
used in the criteria for accepting the lot, as given in the sampling plan.
NOTE 1 Both these constants are also used as coefficients in estimation of population fractiles.
NOTE 2 In IS0 3534-2, the acceptance constant is designated k.
3.55 acceptance number (AC): In sampling inspection by attributes, the largest
number of nonconforming units found in the sample that permits acceptance of the lot,
as given in the sampling plan.
3.56 lower specification limit, L: Specified value of the observed variable X giving
the lower boundary of the permissible value.
3.57 upper specification limit, U: Specified value of the observed variable X giving
the upper boundary of the permissible value.
3.58 number of nonconforming units, z: Actual number of nonconforming units
found in a sample.
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IS0 12491:1997(E)
4 Population and sample
4.1 General
Mechanical properties and dimensions of building materials and components are
described by random variables (called variables in this International Standard) with a
certain type of probability distribution. The popular normal distribution (Laplace-Gauss
distribution) may be used to approximate many actual symmetrical distributions. When
a remarkable asymmetry is observed, then another type of distribution reflecting this
asymmetry shall be considered. Often, three-parametric log-normal distribution is used
(see 4.3).
To simplify calculation procedures, standardized variables (see 3.14) are used, whose
means are equal to zero and whose variances are equal to one, and which have
standardized distributions for which numerical tables are available.
As a rule, only a limited number of observations constituting a random sample x,, x,
x, . .) xn of size n taken from a population (lot) is available. The aim of statistical
methods for quality control is to make a decision concerning the required quality of a
population using the information derived from one or more random samples.
4.2 Normal distribution
The well-known normal distribution of a continuous variable, X, is a fundamental type
of symmetrical distribution defined on an unlimited interval, which is fully described by
two parameters: the mean p and the variance 02. Any normal variable may be easily
transformed to a standardized variable U = (X - p)/o, for which tables of probability
density and distribution function are commonly available.
In quality control of building materials and components, the fractiles up are frequently
used, where the following values for the probability p are most often applied: p = 0,95;
0,975; 0,99; 0,995. The corresponding values of the fractiles up are given in table 1. It is
to be noted that for high ratios o/p there is a non-negligible probability of the occurrence
of negative values of the variable X. If X must be positive (which may follow from some
physical reasons), then other theoretical models for the probability distribution may be
more suitable.
All the information derived from a given random sample x,, x, . . .x of the size n, taken
from a normal population, is completely described by two samplencharacteristics only:
the sample mean Z and the sample variance s2. These characteristics are specific values
of the corresponding estimators of the population mean and variance, denoted by x
The mean estimator 1 is a random variable described by the normal
and S” .
distribution having the same mean p as the population and the variance equal to 02/n.
The variance estimator S” is a random variable described by transformed X2-distribution
with v = (n- 1) degrees of freedom as
S” = 02x21( n- 1)
This transformation allows any fractile of S” to be determined from the corresponding
fractile of x2. As the X2-distribution is asymmetrical, the lower fractiles & as well as
the upper fractiles xi2 are given in table 2. The recommended probabilities to be used
in building are as follows: p1 = 0,05; 0,025; 0,Ol; 0,005 andp, =0,95; 0,975; 0,99; 0,995.
IS0 12491:1997(E)
4.3 Log-normal distribution
defined on a semi-infinite interval, is
The asymmetrical log-normal distribution,
generally described by three parameters: the mean l.~, the variance 0 2 and, as the third
characteristic, the lower or upper limit value x, corresponding to a certain positive or
negative asymmetry may be used. In building, the log-normal distribution with the
lower limit x, (and positive skewness) is often considered. In this case, as indicated in
3.16, the distribution of a variable X may be easily transformed to the normal distribu-
tion of a variable Y given by the transformation
Y = In (X-x0)
The new variable Y is then treated in the same way as the variable X in the previous
case; this is valid also for the transformation to the standardized variable. (Generally, Y
and y should be used instead of X and x. )
Moreover, in building, it may often be assumed that x, = 0 and then only two parameters
( ~1 and CJ “> are involved. In this case, the normal variable Y is given as
Y=lnX
and the original variable X is assumed to have positive asymmetry, which is dependent
on the ratio CT/~, where 0; and /.L are the standard deviation and the mean of the variable
X, respectively.
4.4 Normality tests
The assumption of a normal distribution of the variable X (or variable Y when the
variable X has a log-normal distribution) may be tested using various normality tests: a
random sample is compared with the theoretical model of the normal distribution and
observed deviations are tested to determine whether they are significant or not. If the
deviations are insignificant, then the assumption of normal distribution is accepted,
otherwise it is rejected. Various normality tests, as established in IS0 5479, may be
used. The recommended significance level a to be used in building is 0,05 or 0,Ol (then
the risk of acceptance of a wrong hypothesis has a suitable value).
5 Methods of statistical quality control
5.1 Quality requirements
To control the quality of building materials and components, adequate requirements
should be specified for observed variables. These requirements usually involve
population parameters (the mean /.L and/or the variance 02> or a fractile x . The most
frequently applied quality requirements limit admissible values of the meanby specified
lower and upper boundary and/or limit the variance by a specified upper boundary, or
specify boundaries for a given fractile. Then, methods of estimation and tests of
fractiles have to be
population parameters and applied.

@ IS0
IS0 12491 :1997(E)
Special procedures of quality control use sampling inspection methods, the aim of which
is to decide directly on acceptance of a population using sample data and not to estimate
explicitly the population parameters.
A normal distribution of the variable X (or variable Y if the variable X has a log-normal
distribution) is assumed in most of the methods described in this International
Standard.
5.2 Basic statistical methods
Basic statistical methods used for the quality control of building materials and
components consist of estimation techniques, tests of statistical hypotheses, and sampl-
ing inspection.
Two estimation techniques for population parameters are generally applied in building:
- point estimation, and
- interval estimation.
These two basic techniques using a classical approach are described in 6.2 and 6.3. A
or prediction of population parameters using
modified approach to estimation
a Bayesian approach is introduced in 5.3 and 6.7.
Methods of tests of hypotheses concerning population parameters which are commonly
applied in building may be also divided into two groups:
- comparison of sample characteristics and the corresponding population parameters,
and
- comparison of the characteristics of two samples.
These methods are described in detail in 6.4 and 6.5.
An important statistical method, frequently used in quality control of building materials
and components, concerns the estimation or prediction of fractiles for normal
distribution; this technique is described in 6.6 and 6.7.
Methods of sampling inspection are used in those cases where a decision concerning
population quality is to be made without explicit determination of population
parameters. It is, however, recommended to combine methods of sampling inspection in
building with the systematic collection of data for the purpose of further evaluation.
A number of sampling plans and criteria are used in building to control the quality of
materials and components. It is, however, strongly recommended always to check the
power of a chosen plan using the operating characteristic curve (OC curve). In practical
cases, the operating characteristic curve may be substituted by two points only, the
producer ’s risk point (PRP) and the consumer ’s risk point (CRP) corresponding to a
specified producer ’s risk (PR) and consumer ’s risk (CR) respectively.
Recommended simple methods of sampling inspection, suitable for the purpose of
quality control of building materials and components, are described in clause 7.
5.3 Bayesian approach
An alterative technique to the basic methods of estimation and tests in quality control
procedures is provided by a Bayesian approach. This approach could be used effectively,
especially when a large continuous production of building materials and components is
to be checked.
@ IS0 IS0 12491:1997(E)
The fundamental principles of the Bayesian approach to quality control differ from the
principles of the classical statistical methods mentioned above. If the observed variable
Y is a function of a sample variable X and a vector of distribution parameters 0 (say p
and o), given as
Y=h (X,0)
the Bayesian approach considers 0 as a random variable and not as a deterministic
parameter vector, which is the case with classical methods. According to these
statistical methods, which are described in detail in clause 6, estimation of the
parameter vector 0 is made separately for each lot using information derived from
sample data. The Bayesian approach investigates the probability distribution of the
parameter vector 0 using its prior distribution as well as current sample data taken
from the lot under consideration.
Two kinds of distribution function of the parameter vector 0 are generally
distinguished: a prior distribution function n’ (O), based on prior information, and a
posterior distribution function II ”@ 1 x1,x2, . . . . , XJ derived from current data x,, x2, . . . . . xn
after sampling. The Bayesian ap roach provides methods to derive conjugate
distribution functions I-I ‘(0) and II ”@ x,x, , . . . . , xn), as well as the predictive distribution
P
function of the observed variable Y. The posterior distribution function n ”(O) may be
derived as
rI ”(O 1 x1,x2, . . . . . x,) = c n ’(o)f(x, IO) f where C denotes a normalizing factor and ~xi I o>, i = 1, 2, . . . . . n, denotes the probability
density function of the variable X if 0 is known.
The crucial point in the Bayesian procedure is the choice of the prior distribution
function l3’ (0). Engineering judgement is often needed. In some cases there may be
substantial information from previous tests of similar products which may be used to
construct II’ (0).
In a continuous production process, in which the units belong to sequence of lots, the
prior distribution for a new sample can often be taken as equal to the posterior
distribution derived using previous sample data. If no relevant information is available,
so-called “uninformative” or “vague” prior distributions should be used.
Furthermore, the predictive probability density function of the observed variable X
itself, given the prior distribution function n’ (0) and sample x,, x2 , . . . ., xn , may be
written as
* xxJ_,x2,.,xn )= Je f(xlo>n “(@~x1,x2,. •,x,)d@
f (I
where f*(xir,, x2 ,., xn) for the predictive probability density function of the
variable X given the sample
data ~1, ~2, l . . . , X, is used to distinguish it from fix I 0),
which denotes the probability density function of the variable X
if 0 is known.
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IS0 12491:1997(E)
Various effective and economical techniques of sampling inspection based on a
comparison of the prior and posterior distributions of the random vector 0 may be
derived from the above general principles. If, for example, doubt remains concerning the
decision to be taken as a result of the inspection of the lot, sample x1,x2,.,x,
may be increased to a new sample x1,x2 ,., x, ,., x, having a larger size and, using
Bayesian approach, the quality requirements may be checked again. Such a procedure
generally reduces sampling costs without loosing accuracy.
5.4 Additional methods
There are further statistical methods, besides those described above, which are applied
only exceptionally in building and, consequently, are not included in this International
Standard, but can be found in a general form in the other International Standards given
in annex A. These methods consist of:
a) determination of sample size guaranteeing the required accuracy of an estimate of
population parameters;
tests of outliers;
b)
comparison of characteristics of three or more samples;
c>
tests concerning accuracy, trueness and precision of measurements;
d)
statistical process control;
e>
determination of statistical tolerance intervals.
When applying the above methods using other International Standards, the confidence
and significance levels recommended in this International Standard should be accepted.
6 Estimation and tests of parameters
6.1 Principles of estimation and tests
A point estimate of a population parameter is given by one number, which is the value of
an estimator derived from a given sample. The best point estimate of a population
parameter is unbiased (the mean of the estimator is equal to the corresponding
population parameter) and efficient (the variance of the unbiased estimator is the
minimum).
An interval estimate of a population parameter is given by two numbers and contains
the parameter with a certain probability y called the confidence level. The following
selected values y = 0,90; 0,95 or 0,99, in some cases also y = 0,75, are recommended to be
used in quality control in building, depending on the type of variable and the possible
consequences of exceeding the estimated values. The interval estimates indicate the
accuracy of an estimate and are therefore preferable to point estimates.
A test of a statistical hypothesis is a procedure that is used to decide whether a
hypothesis about the distribution of one or more populations should be accepted or
rejected. If results derived from random samples do not differ markedly from those
expected under the assumption that the hypothesis is true, then the observed difference
is said to be insignificant and the hypothesis is accepted; otherwise the hypothesis is
rejected. The recommended significance level a = 0,Ol or 0,05 guarantees that the risk of
acceptance of a wrong hypothesis has a suitable value.
IS0 12491:1997(E)
Methods of estimation and tests of means and variances are covered in general in
IS0 2854 and IS0 2602. The most suitable methods, adjusted for quality control of
building materials and components, are described in 6.2 to 6.5. Also 6.6 describes the
classical approach to the estimation of fractiles and 6.7 indicates the Bay&an approach
to a prediction (a point estimate) of fractiles.
6.2 Estimation of the mean
The best point estimate of the population mean p is the sample mean X.
The interval estimate of the mean p depends on knowledge of the population standard
deviation 0.
If the standard deviation 0 is known, then the two-sided interval estimate at the
confidence level y = ( 2p- 1) is given by
3t-zp /J-?iqL~x+u,0 l&i
where up is the fractile of the standardized normal distribution corresponding to the
probabilityp (close to 1) given in table 1. (For further information see IS0 2854.)
If the population standard deviation CJ is unknown, then the two-sided interval estimate
at the confidence level y = ( 2p- 1) is given by
z- tpsl&qL~z+tpsl&
where
s is the sample standard deviation;
tp is the fractile of the t-distribution for v = ( n- 1) degrees of freedom;
p is the probability (close to 1) given in table 3.
(For further information see IS0 2854.)
In both the above cases only the one-sided interval estimate at the confidence level y = p
may be used when only the lower or only the upper limit of the above estimates is
considered. The value of p and corresponding fractiles up and tp should be specified in
accordance with chosen confidence level y = p (see 6.1).
6.3 Estimation of the variance
is the sample variance s2 .
The best point estimate of the population variance o2
The two-sided interval estimate for the variance o2 at the confidence level y = (p2 -PI)
is given as
(n-l)s2 /&Io2q4s2 l&
where $, and & are fractiles of the x 2 -distribution for v = ( YL- 1) degrees of freedom
corresponding to the probabilities p1 (close to 0) and p2 (close to 1) given in table 2. (For
further information see IS0 2854.)
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IS0 12491:1997(E)
Often the lower limit of the above interval estimate for the variance o2 is considered to
be 0 and then the confidence level y of the estimate equals (l- pl).
The estimate for the standard deviation (r may be obtained by square root of the
relationships derived for the variance 02.
6.4 Comparison of means
To test the difference between the sample mean 5 and the population mean p if the
population standard deviation o is known, the test value uO, given by
u. =lz-pIhi 10
is compared with the critical value up (table l), which is the fractile of the standardized
normal distribution corresponding to the significance level a = ( l- p) (close to 0). If
then the hypothesis that the sample is taken from the population with the
u() q)’
mean lo is accepted; otherwise it is rejected.
If the population standard deviation 0 is unknown, then the test value to, given by
to =l%ppi Is
is compared with the critical value tp (table 3), which is the fractile of the t-distribution
for the v = (n- 1) degrees of freedom corresponding to the significance level a = ( l- p)
(close to 0). If to L tp, then the hypothesis that the sample is taken from population
with the mean p is accepted; otherwise it is rejected.
To test the difference between the means g1 and Z2 of two samples of sizes n, and n,,
respectively, and which are taken from two populations having the same population
standard deviation 0, the test value u. , given by
240 =
13c1-3c2~ Jltln2 +JGq
is compared with the critical value up (table 1), which is the fractile of the standardized
normal distribution corresponding to the significance level CC = ( l- p) (close to 0). If
u, < up, then the hypothesis that the both samples are taken from the populations with
the same (though unknown) mean p is accepted; otherwise it is rejected.
If the standard deviation CJ of both populations is the same, but unknown, then it is
necessary to use sample standard deviations s, and s, *The test value t,, given by
=lq-x2( j/d #&F +(n2 -1)4](1+n2)
is compared with the critical value tp (table 3), which is the fractile of the t-distribution
- 2) degrees of freedom corresponding to the significance level a =
for the v = (n,+-
(l-p) (close to 0). If t, populations with the same (though unknown) mean p is accepted; otherwise it is
rejected.
IS0 12491:1997(E)
For two samples of the same size n, = n, = n, for which observed values may be coupled
(paired observations), the difference between the sample means may be tested using the
differences between coupled values w, = (XS -Q). First the mean u) and standard
deviation sW are determined and then the test value t,, given by
=u)
n Is,
t
I lJ--
is compared with the critical value tp (table 3), which is the fractile of the t-distribution
for the v = ( n- 1) degrees of freedom corresponding to the significance level a = ( l- p )
(close to 0). If to 5 t,, then the hypothesis that both samples are taken from populations
with the same (though unknown) mean p is accepted; otherwise it is rejected. (For
further information, see IS0 3301.)
6.5 Comparison of variances
To test the difference between the sample variance s2 and a population variance o2 the
test value x i, given by
x i = (n - 1) s2/cr2
is first determined.
If s2 I 02, then the test value xi is compared with the critical value & (table 2)
(n- 1) degrees of freedom and to the significance level a = pl. When
corresponding to v =
x i > - $, then the hypothesis that the sample is taken from the population with the
variance o2 is accepted; otherwise it is rejected.
If s2 > 02, then the test value x ”, is compared with the critical value x z2 (table 2)
corresponding to v = (n- 1) degrees of freedom and to the significance level a = (l- p2 ).
When x i 6 x 22, then the hypothesis that the sample is taken from the population with
P
the variance o2 is accepted, otherwise it is rejected.
For two samples of sizes n, and n,, the difference between the sample variances sf and
&the lower subscripts are chosen such that ~22 I sf > may be tested comparing the test
value F. , given by
with the critical value Fp, which is the fractile of the F-distribution given in table 4 (for
further information, see IS0 2854) for v1 = ( nl - 1) and v, = (3 - 1) degrees of
If F, < Fp, then the hypothesis that
freedom and for the significance level a = ( l- p ).
both samples are taken from populations with the same (though unknown) variance o2
is accepted; otherwise it is rejected.
6.6 Estimation of fractiles
Various methods for the estimation of fractiles are available for different assumptions
concerning the type of probability distribution and available data. The most efficient
and general methods of estimation of fractiles xP, independent of the type of distribution,
are based on the order statistic. According to the simplest procedure, a sample x1, x, . . . .
is first transformed into the ordered sample
xrl
IS0 12491:1997(E) @ IS0
x,’ I x2’ I) . .) Ix,’
and then the fractile estimate xPest is given as
>
X
= X6+1
best
where the integer k follows from the inequality
kInp The exact density function of the estimator Xpest of the p-fractile is given as
,
where n(x) denotes the distribution function and fcx) denotes the density function of the
population. With increasing values of n, the density g(xpest> tends to the normal density
function with the mean equal to xp and standard deviation equal to
(&GF) +J
For a population having a normal distribution, the following simple technique, which
depends on knowledge of the population standard deviation 0, is recommended.
If the population standard deviation 0 is known, then the p-fractile estimate xpest is
,
given
X =X+k,o
p, est
If CJ is unknown, then
X = Ix: +kSs
nest
The coefficients k, and KS depend on the sample size n, on the probability p
corresponding to the desired fractile xp, and, furthermore, on the confidence level y.
The coefficients k, and KS, derived from the normal and noncentral t-distribution
respectively (for further information, see IS0 3207), are given in tables 5 and 6 for the
probabilities p = 0,90; 0,95 or 0,99 (upper fractiles) and confidence levels y = 0,05; 0,lO;
0,25; 0,50; 0,75; 0,90 or 0,95. For the probabilities p
= OJO; 0,05; 0,Ol (lower fractiles)
tables 5 and 6 may be also used; in this case p shall be substituted by (l- p) and the
coefficients k,, and KS shall be taken with negative signs.
The confidence level y that the estimate xpest
will lie on the safe side of the correct value
xp should be greater than 0,50. In order to take into account statistical uncertainty, the
value y = 0,75 is recommended.
6.7 Prediction of fractiles using the Bayesian approach
The Bayesian approach described in 5.3 can be analytically elaborated for a normal
variable X and the prior distribution function II ’ (P,o) of /.L and 0, given as
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IS0 12491:1997(E)
“‘(p,(J)= co
where
C is the normalizing constant;
li(n ’)=Oforn ’=O;
8 (n ’) = 1 otherwise;
m ’, s ’, n ’, V’ are parameters asymptotically given as
E(p) = m’
E(o) = s’
S*
--
v(p) -
m’ J- n’
while the parameters n’ and V’ may be chosen arbitrarily. Here E(.) denotes the
expectation and V(.> the coefficient of variation of the variable in brackets.
The posterior distribution function n ”(~,~) of p and G is of the same type as the prior
distribution function, but with parameters m “, ~ ‘1, n ” and v ”, given as
v” =
v ’+v+iS(n ’)
m “n ‘I=
n ’m ’+ n Z
V “(s “)2 + n ‘I( m “)2 =
v ‘(s ‘)2 + n ‘(m ‘)2 + vs2 + n( F >”
where
Z and s are the sample mean and standard deviation;
n is the size of the sample;
v= n-l.
The predictive value xppred of a fractile xp is then
,
xp pd = m ”+ t, s ”Jl+ 1 In”
>
where tp is the fractile of the t-distribution (table 3) with v” degrees of freedom.
The values for tp given in table 3 should be there
...