ISO 12491:1997
(Main)Statistical methods for quality control of building materials and components
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
Standards Content (Sample)
IS0
INTERNATIONAL
12491
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)
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IS0 12491:1997(E)
Page
Contents
1
........................................................................................
Scope
1
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
9
Methods of statistical quality control .
......................................................... 9
5.1 Quality requirements
10
..................................................
5.2 Basic statistical methods
10
5.3 Bayesian approach .
12
...........................................................
5.4 Additional methods
........................................... 12
Estimation and tests of parameters
..................................... 12
6.1 Principles of estimation and tests
13
6.2 Estimation of the mean .
.................................................. 13
6.3 Estimation of the variance
14
.......................................................
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
27
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Bibliography
28
. . . . . . .~.~.
Alphabetical index
0 IS0 1997
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IS0 12491 :1997(E)
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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.
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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.
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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)
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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.
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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
2
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:
1
X=--CXi
n
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3.20 sample variance, s2: Sum of n squared deviations from the sample mean Z
divided by the sample size n minus 1:
1
2
=-
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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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
2
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
1
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
0
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
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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.
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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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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.
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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.
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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.
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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
...
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