ISO 16269-6:2005

Statistical interpretation of data - Part 6: Determination of statistical tolerance intervals

ISO 16269-6:2005 describes procedures for establishing tolerance intervals that include at least a specified proportion of the population with a specified confidence level. Both one-sided and two-sided statistical tolerance intervals are provided, a one-sided interval having either an upper or a lower limit while a two-sided interval has both upper and lower limits. Two methods are provided, a parametric method for the case where the characteristic being studied has a normal distribution and a distribution-free method for the case where nothing is known about the distribution except that it is continuous.

Interprétation statistique des données — Partie 6: Détermination des intervalles statistiques de tolérance

L'ISO 16269-6:2005 décrit des méthodes permettant d'établir les intervalles statistiques de tolérance qui comprennent au moins une proportion spécifiée de la population avec un niveau de confiance spécifié. Des intervalles statistiques de tolérance unilatéraux et bilatéraux sont fournis, l'intervalle statistique de tolérance unilatéral étant caractérisé par une limite supérieure ou par une limite inférieure, tandis que l'intervalle statistique bilatéral possède à la fois une limite supérieure et une limite inférieure. Deux méthodes sont exposées: une méthode paramétrique, lorsque la caractéristique étudiée a une distribution normale, et une méthode non paramétrique, lorsque rien n'est connu de la distribution si ce n'est qu'elle est continue.

Statistično tolmačenje podatkov – 6. del: Ugotavljanje statističnih tolerančnih intervalov

General Information

• Status: Withdrawn
• Publication Date: 11-Apr-2005
• Withdrawal Date: 11-Apr-2005
• ICS: 03.120.30 - Application of statistical methods
• Technical Committee: ISO/TC 69 - Applications of statistical methods
• Drafting Committee: ISO/TC 69/WG 3 - Statistical interpretation of data
• Current Stage: 9599 - Withdrawal of International Standard
• Start Date: 23-Jan-2014
• Completion Date: 13-Dec-2025
• Ref Project: SIST ISO 16269-6:2006 - Statistical interpretation of data -- Part 6: Determination of statistical tolerance intervals

Relations

• Revised: ISO 16269-6:2014 - Statistical interpretation of data - Part 6: Determination of statistical tolerance intervals
• Effective Date: 07-Aug-2010
• Revised: SIST ISO 3207:1996 - Statistical interpretation of data -- Determination of a statistical tolerance interval
• Effective Date: 12-May-2008
• Revises: ISO 3207:1975 - Statistical interpretation of data - Determination of a statistical tolerance interval
• Effective Date: 15-Apr-2008

Standards Content (Sample)

INTERNATIONAL ISO
STANDARD 16269-6
First edition
2005-04-01
Statistical interpretation of data —
Part 6:
Determination of statistical tolerance
intervals
Interprétation statistique des données —
Partie 6: Détermination des intervalles statistiques de tolérance

Reference number
©
ISO 2005
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ii © ISO 2005 – All rights reserved

Contents Page
Foreword. iv
Introduction . v
1 Scope. 1
2 Normative references . 1
3 Terms, definitions and symbols . 1
3.1 Terms and definitions. 1
3.2 Symbols . 2
4 Procedures . 3
4.1 Normal population with known variance and known mean. 3
4.2 Normal population with known variance and unknown mean . 3
4.3 Normal population with unknown variance and unknown mean. 3
4.4 Any continuous distribution of unknown type . 3
5 Examples. 3
5.1 Data. 3
5.2 Example 1: One-sided statistical tolerance interval under known variance. 4
5.3 Example 2: Two-sided statistical tolerance interval under known variance . 4
5.4 Example 3: One-sided statistical tolerance interval under unknown variance . 5
5.5 Example 4: Two-sided statistical tolerance interval under unknown variance. 6
5.6 Example 5: Distribution-free statistical tolerance interval for continuous distribution . 6
Annex A (informative) Forms for tolerance intervals. 8
Annex B (normative) One-sided statistical tolerance limit factors, k (n; p; 1 − α), for known σ . 14
Annex C (normative) Two-sided statistical tolerance limit factors, k (n; p; 1 − α), for known σ. 17
Annex D (normative) One-sided statistical tolerance limit factors, k (n; p; 1 − α), for unknown σ. 20
Annex E (normative) Two-sided statistical tolerance limit factors, k (n; p; 1 − α), for unknown σ. 23
Annex F (normative) One-sided distribution-free statistical tolerance intervals. 26
Annex G (normative) Two-sided distribution-free statistical tolerance intervals. 27
Annex H (informative) Construction of a distribution-free statistical tolerance interval for any type
of distribution . 28
Annex I (informative) Computation of factors for two-sided parametric statistical tolerance
intervals. 29
Bibliography . 30

Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO 16269-6 was prepared by Technical Committee ISO/TC 69, Applications of statistical methods.
This first edition of ISO 16269-6 cancels and replaces ISO 3207:1975, which has been technically revised.
ISO 16269 consists of the following parts, under the general title Statistical interpretation of data:
 Part 6: Determination of statistical tolerance intervals
 Part 7: Median — Estimation and confidence intervals
 Part 8: Determination of prediction intervals

iv © ISO 2005 – All rights reserved

Introduction
A statistical tolerance interval is an estimated interval, based on a sample, which can be asserted with
confidence 1 − α, for example 95 %, to contain at least a specified proportion p of the items in the population.
The limits of a statistical tolerance interval are called statistical tolerance limits. The confidence level 1 − α is
the probability that a statistical tolerance interval constructed in the prescribed manner will contain at least a
proportion p of the population. Conversely, the probability that this interval will contain less than the proportion
p of the population is α. This part of ISO 16269 describes both one-sided and two-sided statistical tolerance
intervals; a one-sided interval is constructed with an upper or a lower limit while a two-sided interval is
constructed with both an upper and a lower limit.
Tolerance intervals are functions of the observations of the sample, i.e. statistics, and they will generally take
different values for different samples. It is necessary that the observations be independent for the procedures
provided in this part of ISO 16269 to be valid.
Two types of tolerance interval are provided in this part of ISO 16269, parametric and distribution-free. The
parametric approach is based on the assumption that the characteristic being studied in the population has a
normal distribution; hence the confidence that the calculated statistical tolerance interval contains at least a
proportion p of the population can only be taken to be 1 − α if the normality assumption is true. For normally
distributed characteristics, the statistical tolerance interval is determined using one of the Forms A, B, C or D
given in Annex A.
Parametric methods for distributions other than the normal are not considered in this part of ISO 16269. If
departure from normality is suspected in the population, distribution-free statistical tolerance intervals may be
constructed. The procedure for the determination of a statistical tolerance interval for any continuous
distribution is provided in Forms E and F of Annex A.
The tolerance limits discussed in this part of ISO 16269 can be used to compare the natural capability of a
process with one or two given specification limits, either an upper one U or a lower one L or both in statistical
process management. An indication of this is the fact that these tolerance limits have also been called natural
process limits. See ISO 3534-2:1993, 3.2.4, and the general remarks in ISO 3207 which will be cancelled and
replaced by this part of ISO 16269.
Above the upper specification limit U there is the upper fraction nonconforming p (ISO 3534-2:—, 3.2.5.5 and
U
3.3.1.4) and below the lower specification limit L there is the lower fraction nonconforming p (ISO 3534-2:—,
L
3.2.5.6 and 3.3.1.5). The sum p + p = p is called the total fraction nonconforming. (ISO 3534-2:—, 3.2.5.7).
U L T
Between the specification limits U and L there is the fraction conforming 1 − p .
T
In statistical process management the limits U and L are fixed in advance and the fractions p , p and p are
U L T
either calculated, if the distribution is assumed to be known, or otherwise estimated. There are many
applications of statistical tolerance intervals, although the above shows an example to a quality control
problem. Wider applications and more statistical intervals are introduced in many textbooks such as Hahn and
[10]
Meeker .
In contrast, for the tolerance intervals considered in this part of ISO 16269, the confidence level for the interval
estimator and the proportion of the distribution within the interval (corresponding to the fraction conforming
mentioned above) are fixed in advance, and the limits are estimated. These limits may be compared with U
and L. Hence the appropriateness of the given specification limits U and L can be compared with the actual
properties of the process. The one-sided tolerance intervals are used when only either the upper specification
limit U or the lower specification limit L is relevant, while the two-sided intervals are used when both the upper
and the lower specification limits are considered simultaneously.
The terminology with regard to these different limits and intervals has been confusing as the “specification
limits” were earlier also called “tolerance limits” (see the terminology standard ISO 3534-2:1993, 1.4.3, where
both these terms as well as the term “limiting values” were all used as synonyms for this concept). In the latest
revision of ISO 3534-2:—, only the term specification limits have been kept for this concept. Furthermore, the
[5]
Guide for the expression of uncertainty in measurement uses the term “coverage factor” defined as a
“numerical factor used as a multiplier of the combined standard uncertainty in order to obtain an expanded
uncertainty”. This use of “coverage” differs from the use of the term in this part of ISO 16269.
vi © ISO 2005 – All rights reserved

INTERNATIONAL STANDARD ISO 16269-6:2005(E)

Statistical interpretation of data —
Part 6:
Determination of statistical tolerance intervals
1 Scope
This part of ISO 16269 describes procedures for establishing tolerance intervals that include at least a
specified proportion of the population with a specified confidence level. Both one-sided and two-sided
statistical tolerance intervals are provided, a one-sided interval having either an upper or a lower limit while a
two-sided interval has both upper and lower limits. Two methods are provided, a parametric method for the
case where the characteristic being studied has a normal distribution and a distribution-free method for the
case where nothing is known about the distribution except that it is continuous.
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated
references, only the edition cited applies. For undated references, the latest edition of the referenced
document (including any amendments) applies.
ISO 3534-1, Statistics — Vocabulary and symbols — Part 1: Probability and general statistical terms
1)
ISO 3534-2:— , Statistics — Vocabulary and symbols — Part 2: Applied statistics
3 Terms, definitions and symbols
3.1 Terms and definitions
For the purposes of this document, the terms and definition given in ISO 3534-1, ISO 3534-2 and the following
apply.
3.1.1
statistical tolerance interval
interval determined from a random sample in such a way that one may have a specified level of confidence
that the interval covers at least a specified proportion of the sampled population
NOTE The confidence level in this context is the long-run proportion of intervals constructed in this manner that will
include at least the specified proportion of the sampled population.
3.1.2
statistical tolerance limit
statistic representing an end point of a statistical tolerance interval
NOTE Statistical tolerance intervals can be either one-sided, in which case they have either an upper or a lower
statistical tolerance limit, or two-sided, in which case they have both.

1) To be published. (Revision of ISO 3534-2:1993)
3.1.3
coverage
proportion of items in a population lying within a statistical tolerance interval
NOTE This concept is not to be confused with the concept coverage factor used in the Guide for the expression of
[5]
uncertainty in measurement (GUM ) .
3.1.4
normal population
normally distributed population
3.2 Symbols
For the purposes of this part of ISO 16269, the following symbols apply.
i suffix of an observation
k (n; p; 1 − α) factor used to determine x or x when the value of σ is known for one-sided tolerance
L U
interval
k (n; p; 1 − α) factor used to determine x and x when the value of σ is known for two-sided tolerance
L U
interval
k (n; p; 1 − α) factor used to determine x or x when the value of σ is unknown for one-sided tolerance
L U
interval
k (n; p; 1 − α) factor used to determine x and x when the value of σ is unknown for two-sided tolerance

L U
interval
n number of observations in the sample
p minimum proportion of the population claimed to be lying in the statistical tolerance interval
u p-fractile of the standard normal distribution
p
x ith observed value (1in=,2,.,)
i
x maximum value of the observed values: x = max {x , x , …, x }
max max 1 2 n
x minimum value of the observed values: x = min {x , x , …, x }
min min 1 2 n
x lower limit of the statistical tolerance interval
L
x upper limit of the statistical tolerance interval
U
n
x sample mean, xx=
i
∑
n
i = 1
nn


nx − x
ii
∑∑
n 
ii==11
1 2

s sample standard deviation;sx=−x=
()
i
∑
nn−−11n
()
i = 1
1 − α confidence level for the claim that the proportion of the population lying within the tolerance
interval is greater than or equal to the specified level p
µ population mean
σ population standard deviation
2 © ISO 2005 – All rights reserved

4 Procedures
4.1 Normal population with known variance and known mean
When the values of the mean, µ, and the variance, σ , of a normally distributed population are known, the
distribution of the characteristic under investigation is fully determined. There is exactly a proportion p of the
population:
a) to the right of x = µ − u × σ (one-sided interval);
p
L
b) to the left of x = µ + u × σ (one-sided interval);
p
U
u u
c) between x = µ − × σ and x = µ + × σ (two-sided interval).
(1+ p)/ 2 (1+ p)/ 2
L U
NOTE As such statements are known to be true, they are made with 100 % confidence.
In the above equations, u is p-fractile of the standard normal distribution. Numerical values of u may be
p p
read from the bottom line of the Tables B.1 to B.6 and Tables C.1 to C.6.
4.2 Normal population with known variance and unknown mean
Forms A and B, given in Annex A, are applicable to the case where the variance of the normal population is
known while the mean is unknown. Form A applies to the one-sided case, while Form B applies to the
two-sided case.
4.3 Normal population with unknown variance and unknown mean
Forms C and D, given in Annex A, are applicable to the case where both the mean and the variance of the
normal population are unknown. Form C applies to the one-sided case, while Form D applies to the two-sided
case.
4.4 Any continuous distribution of unknown type
If the characteristic under investigation is a continuous variable from a population of unknown form, and if a
sample of n independent random observations of the characteristic has been taken, then a statistical tolerance
interval can be determined from the ranked observations. The procedure given in Forms E and F of Annex A
provide the determination of the coverage or sample size needed for tolerance intervals determined from the
extreme values x or x of the sample of observations with given confidence level 1 − α.
min max
NOTE Statistical tolerance intervals that do not depend on the shape of the sampled population are called
distribution-free tolerance intervals.
This part of ISO 16269 does not provide procedures for distributions of known type other than the normal
distribution. However, if the distribution is continuous, the distribution-free method may be used. Selected
references to scientific literature that may assist in determining tolerance intervals for other distributions are
also provided at the end of this document.
5 Examples
5.1 Data
Forms A to D, given in Annex A, are illustrated by examples using the numerical values of ISO 2854:1976,
Clause 2, paragraph 1 of the introductory remarks, Table X, yarn 2: 12 measures of the breaking load of
cotton yarn. It should be noted that the number of observations, n = 12, given here for these examples is
[1]
considerably lower than the one recommended in ISO 2602 . The numerical data and calculations in the
different examples are expressed in centi-newtons (see Table 1).
Table 1 — Data for Examples 1 to 4
Values in centi-newtons
228,6 232,7 238,8 317,2 315,8 275,1 222,2 236,7 224,7 251,2 210,4 270,7
x
These measurements were obtained from a batch of 12,000 bobbins, from one production job, packed in
120 boxes each containing 100 bobbins. Twelve boxes have been drawn at random from the batch and a
bobbin has been drawn at random from each of these boxes. Test pieces of 50 cm length have been cut from
the yarn on these bobbins, at about 5 m distance from the free end. The tests themselves have been carried
out on the central parts of these test pieces. Previous information makes it reasonable to assume that the
breaking loads measured in these conditions have virtually a normal distribution. It is demonstrated in
ISO 2954:1976 that the data do not contradict the assumption of a normal distribution.
These results yield the following:
Sample size: n = 12
Sample mean: x==3 024,1/12 252,01
nx − x
()
∑∑ 166 772,27
Sample standard deviation:
s== = 1263,426 3= 35,545
nn−×11211
()
The formal presentation of the calculations will be given only for Form C in Annex A (one-sided interval,
unknown variance).
5.2 Example 1: One-sided statistical tolerance interval under known variance
Suppose that previously obtained measurements have shown that the dispersion is constant from one batch
σ = 33,150
to another from the same supplier, and is represented by a standard deviation , although the mean
is not constant. A limit x is required such that it is possible to assert with confidence level 1 − α = 0,95 (95 %)
L
that at least 0,95 (95 %) of the breaking loads of the items in the batch, when measured under the same
conditions, are above x .
L
Table B.4 gives
k (12; 0,95; 0,95) = 2,120
whence
x = xk−−(n;p;1 ασ)×= 252,01− 2,120× 33,150= 181,732
L 1
A smaller value of the lower limit x would be obtained if a larger proportion of the population (for example
L
p = 0,99) and/or a higher confidence level (for example 1 − α = 0,99) were required.
5.3 Example 2: Two-sided statistical tolerance interval under known variance
Under the same conditions as in Example 1, suppose that limits x and x are required such that it is
L U
possible to assert with a confidence level 1 − α = 0,95 that at least a proportion of p = 0,90 (90 %) of the
breaking load of the batch falls between x and x .
L U
Table C.4 gives
k (12; 0,90; 0,95) = 1,889
4 © ISO 2005 – All rights reserved

whence
x = xk−−(n;p;1 ασ)×= 252,01−1,889× 33,150= 189,390
L 2
x = xk+−(n;p;1 ασ)×= 252,01+1,889× 33,150= 314,630
U 2
Comparison with Example 1 should make it clear that assuring that at least 90 % of a population lies between
the limits x and x is not the same thing as assuring that no more than 5 % lies beyond each limit.
L U
5.4 Example 3: One-sided statistical tolerance interval under unknown variance
Here, it is supposed that the standard deviation of the population is unknown and has to be estimated from the
sample. The same requirements will be assumed as for the case where the standard deviation is known
(Example 1), thus, p = 0,95 and 1 − α = 0,95. The presentation of the results is given in detail below.
Determination of the statistical tolerance interval of proportion p:
a) one-sided interval “to the right”
Determined values:
b) proportion of the population selected for the tolerance interval: p = 0,95
c) chosen confidence level: 1 − α = 0,95
d) sample size: n = 12
Value of tolerance factor from Table D.4:
k (n; p; 1 − α) = 2,737
Calculations:
xx==/n 252,01
∑
nx − x
()
∑∑
s = = 35,545
nn −1
()
kn( ;p;1−×α) s= 97,286 7
Results: one-sided interval “to the right”
The tolerance interval which will contain at least a proportion p of the population with confidence level 1 − α
has a lower limit
x =−xkn( ;p;1− α)×s= 154,723
L 3
5.5 Example 4: Two-sided statistical tolerance interval under unknown variance
Under the same conditions as in Example 2, suppose it is required to calculate the limits x and x such
L U
that it is possible to assert with a confidence level 1 − α = 0,95 that in a proportion of the batch at least equal
to p = 0,90 (90 %) the breaking load falls between x and x .
L U
Table E.4 gives
kn(;p;1−=α) 2,671
whence
xx=−k (n;p;1−α)×s= 252,01− 2,671× 35,545= 157,069
L 4
xx=+k (n;p;1−α)×s= 252,01+ 2,671× 35,545= 346,951
U 4
It will be noted that the value of x is smaller and the value of x higher than in Example 2 (known variance),
L U
because the use of s instead of σ requires a larger value of the tolerance factor to allow for the extra
uncertainty. It is necessary to have to pay a penalty for not knowing the population standard deviation σ and
the extension of the statistical tolerance interval takes this into account. Of course, it is not quite sure that the
value σ = 33,150 used in Examples 1 and 2 is correct. Therefore, it is wiser to use the estimate, s, together
with Tables D.4 or E.4.
5.6 Example 5: Distribution-free statistical tolerance interval for continuous distribution
In a fatigue test by rotational stress carried out on a component of an aeronautical engine, a sample of
15 items has given the results (measurement of endurance), shown in ascending order of values in Table 2.
Table 2 — Data for Example 5
x 0,200 0,330 0,450 0,490 0,780 0,920 0,950 0,970 1,040 1,710 2,220 2,275 3,650 7,000 8,800

A graphical examination of checking normality, such as probability plot, shows that the hypothesis of normality
for the population of components should almost certainly be rejected (see ISO 5479). The methods of Form E,
given in Annex A, for determination of a statistical tolerance interval are therefore applicable.
The extreme values from the sample of n = 15 measurements are:
x = 0,200, x = 8,800
min max
Suppose that the required confidence level 1 − α is 0,95.
a) What is the maximum proportion of the population of components that will fall below x = 0,200?
min
Table F.1, for 1 − α = 0,95, gives for the minimum proportion above x a value of p slightly higher than
min
0,75 (75 %). Hence, for the maximum proportion below x a value of 1 − p slightly lower than
min
0,25 (25 %).
b) What sample size is necessary for it to be possible to assert, at a confidence level 0,95, that a proportion
at least p = 0,90 (90 %) of the population of components will be found below the largest of the values from
that sample? Table F.1, for 1 − α = 0,95 and p = 0,90, gives n = 29.
6 © ISO 2005 – All rights reserved

c) At a confidence level of 0,95, what is the minimum proportion of the population of components that fall
between x = 0,200 and x = 8,800? Table G.1, for 1 − α = 0,95 and n = 15, gives p slightly below
min max
0,75 (75 %).
d) What sample size is necessary for it to be possible to assert at a confidence level 0,95 that a proportion
of at least p = 0,90 (90 %) of the population of components will be found to fall between the smallest and
the largest values from that sample? Table G.1, for 1 − α = 0,95 and p = 0,90, gives n = 46.
e) In general, if a check for normality (see ISO 5479) indicates a departure from the normal distribution,
some transformation will be recommended based on the knowledge of the collected data. For example,
fatigue data are often approximated lognormally distributed. In such cases, the data could be transformed
to normality. Tolerance intervals are then calculated and finally transformed back into the original units.
See Annex H for the construction of a statistical tolerance interval for distribution-free tolerance intervals for
any type of distribution. Annex I gives the computation of factors for two-sided parametric statistical tolerance
intervals.
Annex A
(informative)
Forms for tolerance intervals
Form A — One-sided statistical tolerance interval (known variance)
Determination of a one-sided statistical tolerance interval with coverage p at confidence level 1 − α
a) One-sided interval “to the left”
b) One-sided interval “to the right”
Known values:
c) the variance: σ =
d) the standard deviation: σ =
Determined values:
e) proportion of the population selected for the tolerance interval: p =
f) chosen confidence level: 1 − α =
g) sample size: n =
Tabulated factor:
k (n; p; 1 − α) =
This value can be read from the tables given in Annex B for a range of values of n, p and 1 − α.
Calculations:
xx==/n
∑
k (n; p; 1 − α) × σ =
Results:
a) One-sided interval “to the left”
The one-sided statistical tolerance interval with coverage p at confidence level 1 − α has upper limit
xx=+k (;np;1− α)×σ=
U 1
b) One-sided interval “to the right”
The one-sided statistical tolerance interval with coverage p at confidence level 1 − α has lower limit
x =−xkn(;p;1− α)×σ=
L 1
8 © ISO 2005 – All rights reserved

Form B — Two-sided statistical tolerance interval (known variance)
Determination of a two-sided statistical tolerance interval with coverage p at confidence level 1 − α
Known values:
a) the variance: σ =
b) the standard deviation: σ =
Determined values:
c) proportion of the population selected for the tolerance interval: p =
d) chosen confidence level: 1 − α =
e) sample size: n =
Tabulated factor:
k (n; p; 1 − α) =
This value can be read from the tables given in Annex C for a range of values of n, p and 1 − α.
Calculations:
xx==/n
∑
k (n; p; 1 − α) × σ =
Results:
The two-sided statistical tolerance interval with coverage p at confidence level 1 − α has limits
x =−xkn(;p;1− α)×σ=
L 2
xx=+k (;np;1− α)×σ=
U 2
Form C — One-sided statistical tolerance interval (unknown variance)
Determination of a one-sided statistical tolerance interval with coverage p at confidence level 1 − α
a) One-sided interval “to the left”
b) One-sided interval “to the right”
Determined values:
c) proportion of the population selected for the tolerance interval: p =
d) chosen confidence level: 1 − α =
e) sample size: n =
Tabulated factor:
k (n; p; 1 − α) =
This value can be read from the tables given in Annex D for a range of values of n, p and 1 − α.
Calculations:
xx==/n
∑
nx −()x
∑∑
s==
nn(1−)
k (n; p; 1 − α) × s =
Results:
a) One-sided interval “to the left”
The tolerance interval with coverage p at confidence level 1 − α has upper limit
xx=+k (;np;1−α)×s=
U 3
b) One-sided interval “to the right”
The tolerance interval with coverage p at confidence level 1 − α has lower limit
xx=−k (;np;1−α)×s=
L 3
10 © ISO 2005 – All rights reserved

Form D — Two-sided statistical tolerance interval (unknown variance)
Determination of a two-sided statistical tolerance interval with coverage p at confidence level 1 − α
Determined values:
a) proportion of the population selected for the tolerance interval: p =
b) chosen confidence level: 1 − α =
c) sample size: n =
Tabulated factor:
k (n; p; 1 − α) =
This value can be read from the tables given in Annex E for a range of values of n, p and 1 − α.
Calculations:
xx==/n
i
∑
nx −()x
∑∑
s==
nn(1−)
k (n; p; 1 − α) × s =
Results:
The two-sided statistical tolerance interval with coverage p at confidence level 1 − α has limits
xx=−k (;np;1−α)×s=
L 4
xx=+k (;np;1−α)×s=
U 4
Form E — One-sided statistical tolerance interval for any distribution
Determination of a one-sided distribution-free statistical tolerance interval with coverage p at confidence level
1 − α
a) One-sided interval “to the left”
b) One-sided interval “to the right”
Determined values:
c) proportion of the population selected for the tolerance interval: p =
d) chosen confidence level: 1 − α =
e) sample size: n =
(Either p or n is to be determined.)
Tabulated value
 p for given n and 1 − α.
 n for given p and 1 − α.
This value can be read from Table F.1 for a range of values of n, p and 1 − α.
Calculations and results
The one-sided statistical tolerance interval with coverage p at confidence level 1 − α has either
 lower limit xx==
L min
 or upper limit xx==
U max
12 © ISO 2005 – All rights reserved

Form F — Two-sided statistical tolerance interval for any distribution
Determination of a two-sided distribution-free statistical tolerance interval with coverage p at confidence level
1 − α
Determined values:
a) proportion of the population selected for the tolerance interval: p =
b) chosen confidence level: 1 − α =
c) sample size: n =
(Either p or n is to be determined.)
Tabulated value
 p for given n and 1 − α.
 n for given p and 1 − α.
This value can be read from Table G.1 for a range of values of n, p and 1 − α.
Calculations and results
The two-sided statistical tolerance interval with coverage p at confidence level 1 − α has
 lower limit xx= =
L min
 and upper limit xx==
U max
Annex B
(normative)
One-sided statistical tolerance limit factors, k (n; p; 1 − α), for known σ
Table B.1 — Confidence level 50,0 % Table B.2 — Confidence level 75,0 %
(1 − α = 0,50) (1 − α = 0,75)
p p
n n
0,50 0,75 0,90 0,95 0,99 0,999 0,50 0,75 0,90 0,95 0,99 0,999
2 0,000 0,675 1,282 1,645 2,327 3,091 2 0,477 1,152 1,759 2,122 2,804 3,568
3 0,000 0,675 1,282 1,645 2,327 3,091 3 0,390 1,064 1,671 2,035 2,716 3,480
4 0,000 0,675 1,282 1,645 2,327 3,091 4 0,338 1,012 1,619 1,983 2,664 3,428
5 0,000 0,675 1,282 1,645 2,327 3,091 5 0,302 0,977 1,584 1,947 2,628 3,392
6 0,000 0,675 1,282 1,645 2,327 3,091 6 0,276 0,950 1,557 1,921 2,602 3,366
7 0,000 0,675 1,282 1,645 2,327 3,091 7 0,255 0,930 1,537 1,900 2,582 3,346
8 0,000 0,675 1,282 1,645 2,327 3,091 8 0,239 0,913 1,521 1,884 2,565 3,329
9 0,000 0,675 1,282 1,645 2,327 3,091 9 0,225 0,900 1,507 1,870 2,552 3,316
10 0,000 0,675 1,282 1,645 2,327 3,091 10 0,214 0,888 1,495 1,859 2,540 3,304
11 0,000 0,675 1,282 1,645 2,327 3,091 11 0,204 0,878 1,485 1,849 2,530 3,294
12 0,000 0,675 1,282 1,645 2,327 3,091 12 0,195 0,870 1,477 1,840 2,522 3,285
13 0,000 0,675 1,282 1,645 2,327 3,091 13 0,188 0,862 1,469 1,832 2,514 3,278
14 0,000 0,675 1,282 1,645 2,327 3,091 14 0,181 0,855 1,462 1,826 2,507 3,271
15 0,000 0,675 1,282 1,645 2,327 3,091 15 0,175 0,849 1,456 1,820 2,501 3,265
16 0,000 0,675 1,282 1,645 2,327 3,091 16 0,169 0,844 1,451 1,814 2,495 3,259
17 0,000 0,675 1,282 1,645 2,327 3,091 17 0,164 0,839 1,446 1,809 2,490 3,254
18 0,000 0,675 1,282 1,645 2,327 3,091 18 0,159 0,834 1,441 1,804 2,486 3,250
19 0,000 0,675 1,282 1,645 2,327 3,091 19 0,155 0,830 1,437 1,800 2,482 3,245

20 0,000 0,675 1,282 1,645 2,327 3,091 20 0,151 0,826 1,433 1,796 2,478 3,242
22 0,000 0,675 1,282 1,645 2,327 3,091 22 0,144 0,819 1,426 1,789 2,471 3,235
24 0,000 0,675 1,282 1,645 2,327 3,091 24 0,138 0,813 1,420 1,783 2,465 3,228
26 0,000 0,675 1,282 1,645 2,327 3,091 26 0,133 0,807 1,414 1,778 2,459 3,223
28 0,000 0,675 1,282 1,645 2,327 3,091 28 0,128 0,802 1,410 1,773 2,454 3,218
30 0,000 0,675 1,282 1,645 2,327 3,091 30 0,124 0,798 1,405 1,768 2,450 3,214
35 0,000 0,675 1,282 1,645 2,327 3,091 35 0,115 0,789 1,396 1,759 2,441 3,205
40 0,000 0,675 1,282 1,645 2,327 3,091 40 0,107 0,782 1,389 1,752 2,433 3,197
45 0,000 0,675 1,282 1,645 2,327 3,091 45 0,101 0,776 1,383 1,746 2,427 3,191
50 0,000 0,675 1,282 1,645 2,327 3,091 50 0,096 0,770 1,377 1,741 2,422 3,186
60 0,000 0,675 1,282 1,645 2,327 3,091 60 0,088 0,762 1,369 1,732 2,414 3,178
70 0,000 0,675 1,282 1,645 2,327 3,091 70 0,081 0,756 1,363 1,726 2,407 3,171
80 0,000 0,675 1,282 1,645 2,327 3,091 80 0,076 0,750 1,357 1,721 2,402 3,166
90 0,000 0,675 1,282 1,645 2,327 3,091 90 0,072 0,746 1,353 1,716 2,398 3,162
100 0,000 0,675 1,282 1,645 2,327 3,091 100 0,068 0,742 1,350 1,713 2,394 3,158
150 0,000 0,675 1,282 1,645 2,327 3,091 150 0,056 0,730 1,337 1,700 2,382 3,146
200 0,000 0,675 1,282 1,645 2,327 3,091 200 0,048 0,723 1,330 1,693 2,375 3,138
250 0,000 0,675 1,282 1,645 2,327 3,091 250 0,043 0,718 1,325 1,688 2,370 3,133
300 0,000 0,675 1,282 1,645 2,327 3,091 300 0,039 0,714 1,321 1,684 2,366 3,130
400 0,000 0,675 1,282 1,645 2,327 3,091 400 0,034 0,709 1,316 1,679 2,361 3,124
500 0,000 0,675 1,282 1,645 2,327 3,091 500 0,031 0,705 1,312 1,676 2,357 3,121
1 000 0,000 0,675 1,282 1,645 2,327 3,091 1 000 0,022 0,696 1,303 1,667 2,348 3,112
∞ 0,000 0,675 1,282 1,645 2,327 3,091 ∞ 0,000 0,675 1,282 1,645 2,327 3,091
14 © ISO 2005 – All rights reserved

Table B.3 — Confidence level 90,0 % Table B.4 — Confidence level 95,0 %
(1 − α = 0,90) (1 − α = 0,95)
p p
n n
0,50 0,75 0,90 0,95 0,99 0,999 0,50 0,75 0,90 0,95 0,99 0,999
2 0,907 1,581 2,188 2,552 3,233 3,997 2 1,164 1,838 2,445 2,808 3,490 4,254
3 0,740 1,415 2,022 2,385 3,067 3,831 3 0,950 1,625 2,232 2,595 3,277 4,040
4 0,641 1,316 1,923 2,286 2,968 3,732 4 0,823 1,497 2,104 2,468 3,149 3,913
5 0,574 1,248 1,855 2,218 2,900 3,664 5 0,736 1,411 2,018 2,381 3,062 3,826
6 0,524 1,198 1,805 2,169 2,850 3,614 6 0,672 1,346 1,954 2,317 2,998 3,762
7 0,485 1,159 1,766 2,130 2,811 3,575 7 0,622 1,297 1,904 2,267 2,949 3,712
8 0,454 1,128 1,735 2,098 2,780 3,544 8 0,582 1,257 1,864 2,227 2,908 3,672
9 0,428 1,102 1,709 2,073 2,754 3,518 9 0,549 1,223 1,830 2,194 2,875 3,639
10 0,406 1,080 1,687 2,051 2,732 3,496 10 0,521 1,195 1,802 2,166 2,847 3,611
11 0,387 1,061 1,668 2,032 2,713 3,477 11 0,496 1,171 1,778 2,141 2,823 3,587
12 0,370 1,045 1,652 2,015 2,697 3,461 12 0,475 1,150 1,757 2,120 2,802 3,566
13 0,356 1,030 1,637 2,001 2,682 3,446 13 0,457 1,131 1,738 2,102 2,783 3,547
14 0,343 1,017 1,625 1,988 2,669 3,433 14 0,440 1,115 1,722 2,085 2,766 3,530
15 0,331 1,006 1,613 1,976 2,658 3,422 15 0,425 1,100 1,707 2,070 2,752 3,515
16 0,321 0,995 1,602 1,966 2,647 3,411 16 0,412 1,086 1,693 2,057 2,738 3,502
17 0,311 0,986 1,593 1,956 2,638 3,402 17 0,399 1,074 1,681 2,044 2,726 3,490
18 0,303 0,977 1,584 1,947 2,629 3,393 18 0,388 1,063 1,670 2,033 2,715 3,478
19 0,295 0,969 1,576 1,939 2,621 3,385 19 0,378 1,052 1,659 2,023 2,704 3,468

20 0,287 0,962 1,569 1,932 2,613 3,377 20 0,368 1,043 1,650 2,013 2,695 3,459
22 0,274 0,948 1,555 1,919 2,600 3,364 22 0,351 1,026 1,633 1,996 2,678 3,441
24 0,262 0,937 1,544 1,907 2,588 3,352 24 0,336 1,011 1,618 1,981 2,663 3,426
26 0,252 0,926 1,533 1,897 2,578 3,342 26 0,323 0,998 1,605 1,968 2,649 3,413
28 0,243 0,917 1,524 1,888 2,569 3,333 28 0,311 0,986 1,593 1,956 2,638 3,402
30 0,234 0,909 1,516 1,879 2,561 3,325 30 0,301 0,975 1,582 1,946 2,627 3,391
35 0,217 0,892 1,499 1,862 2,543 3,307 35 0,279 0,953 1,560 1,923 2,605 3,369
40 0,203 0,878 1,485 1,848 2,529 3,293 40 0,261 0,935 1,542 1,905 2,587 3,351
45 0,192 0,866 1,473 1,836 2,518 3,282 45 0,246 0,920 1,527 1,891 2,572 3,336
50 0,182 0,856 1,463 1,827 2,508 3,272 50 0,233 0,908 1,515 1,878 2,559 3,323
60 0,166 0,840 1,447 1,811 2,492 3,256 60 0,213 0,887 1,494 1,858 2,539 3,303
70 0,154 0,828 1,435 1,799 2,480 3,244 70 0,197 0,872 1,479 1,842 2,523 3,287
80 0,144 0,818 1,425 1,789 2,470 3,234 80 0,184 0,859 1,466 1,829 2,511 3,275
90 0,136 0,810 1,417 1,780 2,462 3,226 90 0,174 0,848 1,455 1,819 2,500 3,264
100 0,129 0,803 1,410 1,774 2,455 3,219 100 0,165 0,839 1,447 1,810 2,491 3,255
150 0,105 0,780 1,387 1,750 2,431 3,195 150 0,135 0,809 1,416 1,780 2,461 3,225
200 0,091 0,766 1,373 1,736 2,417 3,181 200 0,117 0,791 1,398 1,762 2,443 3,207
250 0,082 0,756 1,363 1,726 2,408 3,172 250 0,105 0,779 1,386 1,749 2,431 3,195
300 0,074 0,749 1,356 1,719 2,401 3,165 300 0,095 0,770 1,377 1,740 2,422 3,186
400 0,065 0,739 1,346 1,709 2,391 3,155 400 0,083 0,757 1,364 1,728 2,409 3,173
500 0,058 0,732 1,339 1,703 2,384 3,148 500 0,074 0,749 1,356 1,719 2,400 3,164
1 000 0,041 0,716 1,323 1,686 2,367 3,131 1 000 0,053 0,727 1,334 1,697 2,379 3,143
∞ 0,000 0,675 1,282 1,645 2,327 3,091 ∞ 0,000 0,675 1,282 1,645 2,327 3,091
Table B.5 — Confidence level 99,0 % Table B.6 — Confidence level 99,9 %
(1 − α = 0,99) (1 − α = 0,999)
p p
n n
0,50 0,75 0,90 0,95 0,99 0,999 0,50 0,75 0,90 0,95 0,99 0,999
2 1,645 2,320 2,927 3,290 3,972 4,736 2 2,186 2,860 3,467 3,830 4,512 5,276
3 1,344 2,018 2,625 2,988 3,670 4,434 3 1,785 2,459 3,066 3,430 4,111 4,875
4 1,164 1,838 2,445 2,809 3,490 4,254 4 1,546 2,220 2,827 3,190 3,872 4,636
5 1,041 1,715 2,322 2,686 3,367 4,131 5 1,382 2,057 2,664 3,027 3,709 4,473
6 0,950 1,625 2,232 2,595 3,277 4,040 6 1,262 1,937 2,544 2,907 3,588 4,352
7 0,880 1,554 2,161 2,525 3,206 3,970 7 1,168 1,843 2,450 2,813 3,495 4,259
8 0,823 1,497 2,105 2,468 3,149 3,913 8 1,093 1,768 2,375 2,738 3,419 4,183
9 0,776 1,450 2,058 2,421 3,102 3,866 9 1,031 1,705 2,312 2,675 3,357 4,121
10 0,736 1,411 2,018 2,381 3,063 3,826 10 0,978 1,652 2,259 2,623 3,304 4,068
11 0,702 1,376 1,983 2,347 3,028 3,792 11 0,932 1,607 2,214 2,577 3,259 4,022
12 0,672 1,347 1,954 2,317 2,998 3,762 12 0,893 1,567 2,174 2,537 3,219 3,983
13 0,646 1,320 1,927 2,291 2,972 3,736 13 0,858 1,532 2,139 2,502 3,184 3,948
14 0,622 1,297 1,904 2,267 2,949 3,712 14 0,826 1,501 2,108 2,471 3,153 3,917
15 0,601 1,276 1,883 2,246 2,928 3,691 15 0,798 1,473 2,080 2,443 3,125 3,889
16 0,582 1,257 1,864 2,227 2,908 3,672 16 0,773 1,448 2,055 2,418 3,099 3,863
17 0,565 1,239 1,846 2,210 2,891 3,655 17 0,750 1,424 2,032 2,395 3,076 3,840
18 0,549 1,223 1,830 2,194 2,875 3,639 18 0,729 1,403 2,010 2,374 3,055 3,819
19 0,534 1,209 1,816 2,179 2,861 3,624 19 0,709 1,384 1,991 2,354 3,036 3,800

20 0,521 1,195 1,802 2,166 2,847 3,611 20 0,691 1,366 1,973 2,336 3,018 3,782
22 0,496 1,171 1,778 2,141 2,823 3,587 22 0,659 1,334 1,941 2,304 2,986 3,750
24 0,475 1,150 1,757 2,120 2,802 3,566 24 0,631 1,306 1,913 2,276 2,958 3,722
26 0,457 1,131 1,738 2,102 2,783 3,547 26 0,607 1,281 1,888 2,251 2,933 3,697
28 0,440 1,115 1,722 2,085 2,766 3,530 28 0,584 1,259 1,866 2,229 2,911 3,675
30 0,425 1,100 1,707 2,070 2,752 3,515 30 0,565 1,239 1,846 2,210 2,891 3,655
35 0,394 1,068 1,675 2,039 2,720 3,484 35 0,523 1,197 1,804 2,168 2,849 3,613
40 0,368 1,043 1,650 2,013 2,695 3,459 40 0,489 1,164 1,771 2,134 2,815 3,579
45 0,347 1,022 1,629 1,992 2,674 3,438 45 0,461 1,136 1,743 2,106 2,788 3,551
50 0,329 1,004 1,611 1,974 2,656 3,420 50 0,438 1,112 1,719 2,082 2,764 3,528
60 0,301 0,975 1,582 1,946 2,627 3,391 60 0,399 1,074 1,681 2,044 2,726 3,490
70 0,279 0,953 1,560 1,923 2,605 3,369 70 0,370 1,044 1,651 2,015 2,696 3,460
80 0,261 0,935 1,542 1,905 2,587 3,351 80 0,346 1,020 1,628 1,991 2,672 3,436
90 0,246 0,920 1,527 1,891 2,572 3,336 90 0,326 1,001 1,608 1,971 2,653 3,416
100 0,233 0,908 1,515 1,878 2,559 3,323 100 0,310 0,984 1,591 1,954 2,636 3,400
150 0,190 0,865 1,472 1,835 2,517 3,281 150 0,253 0,927 1,534 1,898 2,579 3,343
200 0,165 0,839 1,447 1,810 2,491 3,255 200 0,219 0,894 1,501 1,864 2,545 3,309
250 0,148 0,822 1,429 1,792 2,474 3,238 250 0,196 0,870 1,477 1,841 2,522 3,286
300 0,135 0,809 1,416 1,780 2,461 3,225 300 0,179 0,853 1,460 1,824 2,505 3,269
400 0,117 0,791 1,398 1,762 2,443 3,207 400 0,155 0,830 1,437 1,800 2,481 3,245
500 0,105 0,779 1,386 1,749 2,431 3,195 500 0,139 0,813 1,420 1,784 2,465 3,229
1 000 0,074 0,749 1,356 1,719 2,400 3,164 1 000 0,098 0,773 1,380 1,743 2,425 3,188
∞ 0,000 0,675 1,282 1,645 2,327 3,091 ∞ 0,000 0,675 1,282 1,645 2,327 3,091

16 © ISO 2005 – All rights reserved
...

INTERNATIONAL ISO
STANDARD 16269-6
First edition
2005-04-01
Statistical interpretation of data —
Part 6:
Determination of statistical tolerance
intervals
Interprétation statistique des données —
Partie 6: Détermination des intervalles statistiques de tolérance

Reference number
©
ISO 2005
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ii © ISO 2005 – All rights reserved

Contents Page
Foreword. iv
Introduction . v
1 Scope. 1
2 Normative references . 1
3 Terms, definitions and symbols . 1
3.1 Terms and definitions. 1
3.2 Symbols . 2
4 Procedures . 3
4.1 Normal population with known variance and known mean. 3
4.2 Normal population with known variance and unknown mean . 3
4.3 Normal population with unknown variance and unknown mean. 3
4.4 Any continuous distribution of unknown type . 3
5 Examples. 3
5.1 Data. 3
5.2 Example 1: One-sided statistical tolerance interval under known variance. 4
5.3 Example 2: Two-sided statistical tolerance interval under known variance . 4
5.4 Example 3: One-sided statistical tolerance interval under unknown variance . 5
5.5 Example 4: Two-sided statistical tolerance interval under unknown variance. 6
5.6 Example 5: Distribution-free statistical tolerance interval for continuous distribution . 6
Annex A (informative) Forms for tolerance intervals. 8
Annex B (normative) One-sided statistical tolerance limit factors, k (n; p; 1 − α), for known σ . 14
Annex C (normative) Two-sided statistical tolerance limit factors, k (n; p; 1 − α), for known σ. 17
Annex D (normative) One-sided statistical tolerance limit factors, k (n; p; 1 − α), for unknown σ. 20
Annex E (normative) Two-sided statistical tolerance limit factors, k (n; p; 1 − α), for unknown σ. 23
Annex F (normative) One-sided distribution-free statistical tolerance intervals. 26
Annex G (normative) Two-sided distribution-free statistical tolerance intervals. 27
Annex H (informative) Construction of a distribution-free statistical tolerance interval for any type
of distribution . 28
Annex I (informative) Computation of factors for two-sided parametric statistical tolerance
intervals. 29
Bibliography . 30

Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO 16269-6 was prepared by Technical Committee ISO/TC 69, Applications of statistical methods.
This first edition of ISO 16269-6 cancels and replaces ISO 3207:1975, which has been technically revised.
ISO 16269 consists of the following parts, under the general title Statistical interpretation of data:
 Part 6: Determination of statistical tolerance intervals
 Part 7: Median — Estimation and confidence intervals
 Part 8: Determination of prediction intervals

iv © ISO 2005 – All rights reserved

Introduction
A statistical tolerance interval is an estimated interval, based on a sample, which can be asserted with
confidence 1 − α, for example 95 %, to contain at least a specified proportion p of the items in the population.
The limits of a statistical tolerance interval are called statistical tolerance limits. The confidence level 1 − α is
the probability that a statistical tolerance interval constructed in the prescribed manner will contain at least a
proportion p of the population. Conversely, the probability that this interval will contain less than the proportion
p of the population is α. This part of ISO 16269 describes both one-sided and two-sided statistical tolerance
intervals; a one-sided interval is constructed with an upper or a lower limit while a two-sided interval is
constructed with both an upper and a lower limit.
Tolerance intervals are functions of the observations of the sample, i.e. statistics, and they will generally take
different values for different samples. It is necessary that the observations be independent for the procedures
provided in this part of ISO 16269 to be valid.
Two types of tolerance interval are provided in this part of ISO 16269, parametric and distribution-free. The
parametric approach is based on the assumption that the characteristic being studied in the population has a
normal distribution; hence the confidence that the calculated statistical tolerance interval contains at least a
proportion p of the population can only be taken to be 1 − α if the normality assumption is true. For normally
distributed characteristics, the statistical tolerance interval is determined using one of the Forms A, B, C or D
given in Annex A.
Parametric methods for distributions other than the normal are not considered in this part of ISO 16269. If
departure from normality is suspected in the population, distribution-free statistical tolerance intervals may be
constructed. The procedure for the determination of a statistical tolerance interval for any continuous
distribution is provided in Forms E and F of Annex A.
The tolerance limits discussed in this part of ISO 16269 can be used to compare the natural capability of a
process with one or two given specification limits, either an upper one U or a lower one L or both in statistical
process management. An indication of this is the fact that these tolerance limits have also been called natural
process limits. See ISO 3534-2:1993, 3.2.4, and the general remarks in ISO 3207 which will be cancelled and
replaced by this part of ISO 16269.
Above the upper specification limit U there is the upper fraction nonconforming p (ISO 3534-2:—, 3.2.5.5 and
U
3.3.1.4) and below the lower specification limit L there is the lower fraction nonconforming p (ISO 3534-2:—,
L
3.2.5.6 and 3.3.1.5). The sum p + p = p is called the total fraction nonconforming. (ISO 3534-2:—, 3.2.5.7).
U L T
Between the specification limits U and L there is the fraction conforming 1 − p .
T
In statistical process management the limits U and L are fixed in advance and the fractions p , p and p are
U L T
either calculated, if the distribution is assumed to be known, or otherwise estimated. There are many
applications of statistical tolerance intervals, although the above shows an example to a quality control
problem. Wider applications and more statistical intervals are introduced in many textbooks such as Hahn and
[10]
Meeker .
In contrast, for the tolerance intervals considered in this part of ISO 16269, the confidence level for the interval
estimator and the proportion of the distribution within the interval (corresponding to the fraction conforming
mentioned above) are fixed in advance, and the limits are estimated. These limits may be compared with U
and L. Hence the appropriateness of the given specification limits U and L can be compared with the actual
properties of the process. The one-sided tolerance intervals are used when only either the upper specification
limit U or the lower specification limit L is relevant, while the two-sided intervals are used when both the upper
and the lower specification limits are considered simultaneously.
The terminology with regard to these different limits and intervals has been confusing as the “specification
limits” were earlier also called “tolerance limits” (see the terminology standard ISO 3534-2:1993, 1.4.3, where
both these terms as well as the term “limiting values” were all used as synonyms for this concept). In the latest
revision of ISO 3534-2:—, only the term specification limits have been kept for this concept. Furthermore, the
[5]
Guide for the expression of uncertainty in measurement uses the term “coverage factor” defined as a
“numerical factor used as a multiplier of the combined standard uncertainty in order to obtain an expanded
uncertainty”. This use of “coverage” differs from the use of the term in this part of ISO 16269.
vi © ISO 2005 – All rights reserved

INTERNATIONAL STANDARD ISO 16269-6:2005(E)

Statistical interpretation of data —
Part 6:
Determination of statistical tolerance intervals
1 Scope
This part of ISO 16269 describes procedures for establishing tolerance intervals that include at least a
specified proportion of the population with a specified confidence level. Both one-sided and two-sided
statistical tolerance intervals are provided, a one-sided interval having either an upper or a lower limit while a
two-sided interval has both upper and lower limits. Two methods are provided, a parametric method for the
case where the characteristic being studied has a normal distribution and a distribution-free method for the
case where nothing is known about the distribution except that it is continuous.
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated
references, only the edition cited applies. For undated references, the latest edition of the referenced
document (including any amendments) applies.
ISO 3534-1, Statistics — Vocabulary and symbols — Part 1: Probability and general statistical terms
1)
ISO 3534-2:— , Statistics — Vocabulary and symbols — Part 2: Applied statistics
3 Terms, definitions and symbols
3.1 Terms and definitions
For the purposes of this document, the terms and definition given in ISO 3534-1, ISO 3534-2 and the following
apply.
3.1.1
statistical tolerance interval
interval determined from a random sample in such a way that one may have a specified level of confidence
that the interval covers at least a specified proportion of the sampled population
NOTE The confidence level in this context is the long-run proportion of intervals constructed in this manner that will
include at least the specified proportion of the sampled population.
3.1.2
statistical tolerance limit
statistic representing an end point of a statistical tolerance interval
NOTE Statistical tolerance intervals can be either one-sided, in which case they have either an upper or a lower
statistical tolerance limit, or two-sided, in which case they have both.

1) To be published. (Revision of ISO 3534-2:1993)
3.1.3
coverage
proportion of items in a population lying within a statistical tolerance interval
NOTE This concept is not to be confused with the concept coverage factor used in the Guide for the expression of
[5]
uncertainty in measurement (GUM ) .
3.1.4
normal population
normally distributed population
3.2 Symbols
For the purposes of this part of ISO 16269, the following symbols apply.
i suffix of an observation
k (n; p; 1 − α) factor used to determine x or x when the value of σ is known for one-sided tolerance
L U
interval
k (n; p; 1 − α) factor used to determine x and x when the value of σ is known for two-sided tolerance
L U
interval
k (n; p; 1 − α) factor used to determine x or x when the value of σ is unknown for one-sided tolerance
L U
interval
k (n; p; 1 − α) factor used to determine x and x when the value of σ is unknown for two-sided tolerance

L U
interval
n number of observations in the sample
p minimum proportion of the population claimed to be lying in the statistical tolerance interval
u p-fractile of the standard normal distribution
p
x ith observed value (1in=,2,.,)
i
x maximum value of the observed values: x = max {x , x , …, x }
max max 1 2 n
x minimum value of the observed values: x = min {x , x , …, x }
min min 1 2 n
x lower limit of the statistical tolerance interval
L
x upper limit of the statistical tolerance interval
U
n
x sample mean, xx=
i
∑
n
i = 1
nn


nx − x
ii
∑∑
n 
ii==11
1 2

s sample standard deviation;sx=−x=
()
i
∑
nn−−11n
()
i = 1
1 − α confidence level for the claim that the proportion of the population lying within the tolerance
interval is greater than or equal to the specified level p
µ population mean
σ population standard deviation
2 © ISO 2005 – All rights reserved

4 Procedures
4.1 Normal population with known variance and known mean
When the values of the mean, µ, and the variance, σ , of a normally distributed population are known, the
distribution of the characteristic under investigation is fully determined. There is exactly a proportion p of the
population:
a) to the right of x = µ − u × σ (one-sided interval);
p
L
b) to the left of x = µ + u × σ (one-sided interval);
p
U
u u
c) between x = µ − × σ and x = µ + × σ (two-sided interval).
(1+ p)/ 2 (1+ p)/ 2
L U
NOTE As such statements are known to be true, they are made with 100 % confidence.
In the above equations, u is p-fractile of the standard normal distribution. Numerical values of u may be
p p
read from the bottom line of the Tables B.1 to B.6 and Tables C.1 to C.6.
4.2 Normal population with known variance and unknown mean
Forms A and B, given in Annex A, are applicable to the case where the variance of the normal population is
known while the mean is unknown. Form A applies to the one-sided case, while Form B applies to the
two-sided case.
4.3 Normal population with unknown variance and unknown mean
Forms C and D, given in Annex A, are applicable to the case where both the mean and the variance of the
normal population are unknown. Form C applies to the one-sided case, while Form D applies to the two-sided
case.
4.4 Any continuous distribution of unknown type
If the characteristic under investigation is a continuous variable from a population of unknown form, and if a
sample of n independent random observations of the characteristic has been taken, then a statistical tolerance
interval can be determined from the ranked observations. The procedure given in Forms E and F of Annex A
provide the determination of the coverage or sample size needed for tolerance intervals determined from the
extreme values x or x of the sample of observations with given confidence level 1 − α.
min max
NOTE Statistical tolerance intervals that do not depend on the shape of the sampled population are called
distribution-free tolerance intervals.
This part of ISO 16269 does not provide procedures for distributions of known type other than the normal
distribution. However, if the distribution is continuous, the distribution-free method may be used. Selected
references to scientific literature that may assist in determining tolerance intervals for other distributions are
also provided at the end of this document.
5 Examples
5.1 Data
Forms A to D, given in Annex A, are illustrated by examples using the numerical values of ISO 2854:1976,
Clause 2, paragraph 1 of the introductory remarks, Table X, yarn 2: 12 measures of the breaking load of
cotton yarn. It should be noted that the number of observations, n = 12, given here for these examples is
[1]
considerably lower than the one recommended in ISO 2602 . The numerical data and calculations in the
different examples are expressed in centi-newtons (see Table 1).
Table 1 — Data for Examples 1 to 4
Values in centi-newtons
228,6 232,7 238,8 317,2 315,8 275,1 222,2 236,7 224,7 251,2 210,4 270,7
x
These measurements were obtained from a batch of 12,000 bobbins, from one production job, packed in
120 boxes each containing 100 bobbins. Twelve boxes have been drawn at random from the batch and a
bobbin has been drawn at random from each of these boxes. Test pieces of 50 cm length have been cut from
the yarn on these bobbins, at about 5 m distance from the free end. The tests themselves have been carried
out on the central parts of these test pieces. Previous information makes it reasonable to assume that the
breaking loads measured in these conditions have virtually a normal distribution. It is demonstrated in
ISO 2954:1976 that the data do not contradict the assumption of a normal distribution.
These results yield the following:
Sample size: n = 12
Sample mean: x==3 024,1/12 252,01
nx − x
()
∑∑ 166 772,27
Sample standard deviation:
s== = 1263,426 3= 35,545
nn−×11211
()
The formal presentation of the calculations will be given only for Form C in Annex A (one-sided interval,
unknown variance).
5.2 Example 1: One-sided statistical tolerance interval under known variance
Suppose that previously obtained measurements have shown that the dispersion is constant from one batch
σ = 33,150
to another from the same supplier, and is represented by a standard deviation , although the mean
is not constant. A limit x is required such that it is possible to assert with confidence level 1 − α = 0,95 (95 %)
L
that at least 0,95 (95 %) of the breaking loads of the items in the batch, when measured under the same
conditions, are above x .
L
Table B.4 gives
k (12; 0,95; 0,95) = 2,120
whence
x = xk−−(n;p;1 ασ)×= 252,01− 2,120× 33,150= 181,732
L 1
A smaller value of the lower limit x would be obtained if a larger proportion of the population (for example
L
p = 0,99) and/or a higher confidence level (for example 1 − α = 0,99) were required.
5.3 Example 2: Two-sided statistical tolerance interval under known variance
Under the same conditions as in Example 1, suppose that limits x and x are required such that it is
L U
possible to assert with a confidence level 1 − α = 0,95 that at least a proportion of p = 0,90 (90 %) of the
breaking load of the batch falls between x and x .
L U
Table C.4 gives
k (12; 0,90; 0,95) = 1,889
4 © ISO 2005 – All rights reserved

whence
x = xk−−(n;p;1 ασ)×= 252,01−1,889× 33,150= 189,390
L 2
x = xk+−(n;p;1 ασ)×= 252,01+1,889× 33,150= 314,630
U 2
Comparison with Example 1 should make it clear that assuring that at least 90 % of a population lies between
the limits x and x is not the same thing as assuring that no more than 5 % lies beyond each limit.
L U
5.4 Example 3: One-sided statistical tolerance interval under unknown variance
Here, it is supposed that the standard deviation of the population is unknown and has to be estimated from the
sample. The same requirements will be assumed as for the case where the standard deviation is known
(Example 1), thus, p = 0,95 and 1 − α = 0,95. The presentation of the results is given in detail below.
Determination of the statistical tolerance interval of proportion p:
a) one-sided interval “to the right”
Determined values:
b) proportion of the population selected for the tolerance interval: p = 0,95
c) chosen confidence level: 1 − α = 0,95
d) sample size: n = 12
Value of tolerance factor from Table D.4:
k (n; p; 1 − α) = 2,737
Calculations:
xx==/n 252,01
∑
nx − x
()
∑∑
s = = 35,545
nn −1
()
kn( ;p;1−×α) s= 97,286 7
Results: one-sided interval “to the right”
The tolerance interval which will contain at least a proportion p of the population with confidence level 1 − α
has a lower limit
x =−xkn( ;p;1− α)×s= 154,723
L 3
5.5 Example 4: Two-sided statistical tolerance interval under unknown variance
Under the same conditions as in Example 2, suppose it is required to calculate the limits x and x such
L U
that it is possible to assert with a confidence level 1 − α = 0,95 that in a proportion of the batch at least equal
to p = 0,90 (90 %) the breaking load falls between x and x .
L U
Table E.4 gives
kn(;p;1−=α) 2,671
whence
xx=−k (n;p;1−α)×s= 252,01− 2,671× 35,545= 157,069
L 4
xx=+k (n;p;1−α)×s= 252,01+ 2,671× 35,545= 346,951
U 4
It will be noted that the value of x is smaller and the value of x higher than in Example 2 (known variance),
L U
because the use of s instead of σ requires a larger value of the tolerance factor to allow for the extra
uncertainty. It is necessary to have to pay a penalty for not knowing the population standard deviation σ and
the extension of the statistical tolerance interval takes this into account. Of course, it is not quite sure that the
value σ = 33,150 used in Examples 1 and 2 is correct. Therefore, it is wiser to use the estimate, s, together
with Tables D.4 or E.4.
5.6 Example 5: Distribution-free statistical tolerance interval for continuous distribution
In a fatigue test by rotational stress carried out on a component of an aeronautical engine, a sample of
15 items has given the results (measurement of endurance), shown in ascending order of values in Table 2.
Table 2 — Data for Example 5
x 0,200 0,330 0,450 0,490 0,780 0,920 0,950 0,970 1,040 1,710 2,220 2,275 3,650 7,000 8,800

A graphical examination of checking normality, such as probability plot, shows that the hypothesis of normality
for the population of components should almost certainly be rejected (see ISO 5479). The methods of Form E,
given in Annex A, for determination of a statistical tolerance interval are therefore applicable.
The extreme values from the sample of n = 15 measurements are:
x = 0,200, x = 8,800
min max
Suppose that the required confidence level 1 − α is 0,95.
a) What is the maximum proportion of the population of components that will fall below x = 0,200?
min
Table F.1, for 1 − α = 0,95, gives for the minimum proportion above x a value of p slightly higher than
min
0,75 (75 %). Hence, for the maximum proportion below x a value of 1 − p slightly lower than
min
0,25 (25 %).
b) What sample size is necessary for it to be possible to assert, at a confidence level 0,95, that a proportion
at least p = 0,90 (90 %) of the population of components will be found below the largest of the values from
that sample? Table F.1, for 1 − α = 0,95 and p = 0,90, gives n = 29.
6 © ISO 2005 – All rights reserved

c) At a confidence level of 0,95, what is the minimum proportion of the population of components that fall
between x = 0,200 and x = 8,800? Table G.1, for 1 − α = 0,95 and n = 15, gives p slightly below
min max
0,75 (75 %).
d) What sample size is necessary for it to be possible to assert at a confidence level 0,95 that a proportion
of at least p = 0,90 (90 %) of the population of components will be found to fall between the smallest and
the largest values from that sample? Table G.1, for 1 − α = 0,95 and p = 0,90, gives n = 46.
e) In general, if a check for normality (see ISO 5479) indicates a departure from the normal distribution,
some transformation will be recommended based on the knowledge of the collected data. For example,
fatigue data are often approximated lognormally distributed. In such cases, the data could be transformed
to normality. Tolerance intervals are then calculated and finally transformed back into the original units.
See Annex H for the construction of a statistical tolerance interval for distribution-free tolerance intervals for
any type of distribution. Annex I gives the computation of factors for two-sided parametric statistical tolerance
intervals.
Annex A
(informative)
Forms for tolerance intervals
Form A — One-sided statistical tolerance interval (known variance)
Determination of a one-sided statistical tolerance interval with coverage p at confidence level 1 − α
a) One-sided interval “to the left”
b) One-sided interval “to the right”
Known values:
c) the variance: σ =
d) the standard deviation: σ =
Determined values:
e) proportion of the population selected for the tolerance interval: p =
f) chosen confidence level: 1 − α =
g) sample size: n =
Tabulated factor:
k (n; p; 1 − α) =
This value can be read from the tables given in Annex B for a range of values of n, p and 1 − α.
Calculations:
xx==/n
∑
k (n; p; 1 − α) × σ =
Results:
a) One-sided interval “to the left”
The one-sided statistical tolerance interval with coverage p at confidence level 1 − α has upper limit
xx=+k (;np;1− α)×σ=
U 1
b) One-sided interval “to the right”
The one-sided statistical tolerance interval with coverage p at confidence level 1 − α has lower limit
x =−xkn(;p;1− α)×σ=
L 1
8 © ISO 2005 – All rights reserved

Form B — Two-sided statistical tolerance interval (known variance)
Determination of a two-sided statistical tolerance interval with coverage p at confidence level 1 − α
Known values:
a) the variance: σ =
b) the standard deviation: σ =
Determined values:
c) proportion of the population selected for the tolerance interval: p =
d) chosen confidence level: 1 − α =
e) sample size: n =
Tabulated factor:
k (n; p; 1 − α) =
This value can be read from the tables given in Annex C for a range of values of n, p and 1 − α.
Calculations:
xx==/n
∑
k (n; p; 1 − α) × σ =
Results:
The two-sided statistical tolerance interval with coverage p at confidence level 1 − α has limits
x =−xkn(;p;1− α)×σ=
L 2
xx=+k (;np;1− α)×σ=
U 2
Form C — One-sided statistical tolerance interval (unknown variance)
Determination of a one-sided statistical tolerance interval with coverage p at confidence level 1 − α
a) One-sided interval “to the left”
b) One-sided interval “to the right”
Determined values:
c) proportion of the population selected for the tolerance interval: p =
d) chosen confidence level: 1 − α =
e) sample size: n =
Tabulated factor:
k (n; p; 1 − α) =
This value can be read from the tables given in Annex D for a range of values of n, p and 1 − α.
Calculations:
xx==/n
∑
nx −()x
∑∑
s==
nn(1−)
k (n; p; 1 − α) × s =
Results:
a) One-sided interval “to the left”
The tolerance interval with coverage p at confidence level 1 − α has upper limit
xx=+k (;np;1−α)×s=
U 3
b) One-sided interval “to the right”
The tolerance interval with coverage p at confidence level 1 − α has lower limit
xx=−k (;np;1−α)×s=
L 3
10 © ISO 2005 – All rights reserved

Form D — Two-sided statistical tolerance interval (unknown variance)
Determination of a two-sided statistical tolerance interval with coverage p at confidence level 1 − α
Determined values:
a) proportion of the population selected for the tolerance interval: p =
b) chosen confidence level: 1 − α =
c) sample size: n =
Tabulated factor:
k (n; p; 1 − α) =
This value can be read from the tables given in Annex E for a range of values of n, p and 1 − α.
Calculations:
xx==/n
i
∑
nx −()x
∑∑
s==
nn(1−)
k (n; p; 1 − α) × s =
Results:
The two-sided statistical tolerance interval with coverage p at confidence level 1 − α has limits
xx=−k (;np;1−α)×s=
L 4
xx=+k (;np;1−α)×s=
U 4
Form E — One-sided statistical tolerance interval for any distribution
Determination of a one-sided distribution-free statistical tolerance interval with coverage p at confidence level
1 − α
a) One-sided interval “to the left”
b) One-sided interval “to the right”
Determined values:
c) proportion of the population selected for the tolerance interval: p =
d) chosen confidence level: 1 − α =
e) sample size: n =
(Either p or n is to be determined.)
Tabulated value
 p for given n and 1 − α.
 n for given p and 1 − α.
This value can be read from Table F.1 for a range of values of n, p and 1 − α.
Calculations and results
The one-sided statistical tolerance interval with coverage p at confidence level 1 − α has either
 lower limit xx==
L min
 or upper limit xx==
U max
12 © ISO 2005 – All rights reserved

Form F — Two-sided statistical tolerance interval for any distribution
Determination of a two-sided distribution-free statistical tolerance interval with coverage p at confidence level
1 − α
Determined values:
a) proportion of the population selected for the tolerance interval: p =
b) chosen confidence level: 1 − α =
c) sample size: n =
(Either p or n is to be determined.)
Tabulated value
 p for given n and 1 − α.
 n for given p and 1 − α.
This value can be read from Table G.1 for a range of values of n, p and 1 − α.
Calculations and results
The two-sided statistical tolerance interval with coverage p at confidence level 1 − α has
 lower limit xx= =
L min
 and upper limit xx==
U max
Annex B
(normative)
One-sided statistical tolerance limit factors, k (n; p; 1 − α), for known σ
Table B.1 — Confidence level 50,0 % Table B.2 — Confidence level 75,0 %
(1 − α = 0,50) (1 − α = 0,75)
p p
n n
0,50 0,75 0,90 0,95 0,99 0,999 0,50 0,75 0,90 0,95 0,99 0,999
2 0,000 0,675 1,282 1,645 2,327 3,091 2 0,477 1,152 1,759 2,122 2,804 3,568
3 0,000 0,675 1,282 1,645 2,327 3,091 3 0,390 1,064 1,671 2,035 2,716 3,480
4 0,000 0,675 1,282 1,645 2,327 3,091 4 0,338 1,012 1,619 1,983 2,664 3,428
5 0,000 0,675 1,282 1,645 2,327 3,091 5 0,302 0,977 1,584 1,947 2,628 3,392
6 0,000 0,675 1,282 1,645 2,327 3,091 6 0,276 0,950 1,557 1,921 2,602 3,366
7 0,000 0,675 1,282 1,645 2,327 3,091 7 0,255 0,930 1,537 1,900 2,582 3,346
8 0,000 0,675 1,282 1,645 2,327 3,091 8 0,239 0,913 1,521 1,884 2,565 3,329
9 0,000 0,675 1,282 1,645 2,327 3,091 9 0,225 0,900 1,507 1,870 2,552 3,316
10 0,000 0,675 1,282 1,645 2,327 3,091 10 0,214 0,888 1,495 1,859 2,540 3,304
11 0,000 0,675 1,282 1,645 2,327 3,091 11 0,204 0,878 1,485 1,849 2,530 3,294
12 0,000 0,675 1,282 1,645 2,327 3,091 12 0,195 0,870 1,477 1,840 2,522 3,285
13 0,000 0,675 1,282 1,645 2,327 3,091 13 0,188 0,862 1,469 1,832 2,514 3,278
14 0,000 0,675 1,282 1,645 2,327 3,091 14 0,181 0,855 1,462 1,826 2,507 3,271
15 0,000 0,675 1,282 1,645 2,327 3,091 15 0,175 0,849 1,456 1,820 2,501 3,265
16 0,000 0,675 1,282 1,645 2,327 3,091 16 0,169 0,844 1,451 1,814 2,495 3,259
17 0,000 0,675 1,282 1,645 2,327 3,091 17 0,164 0,839 1,446 1,809 2,490 3,254
18 0,000 0,675 1,282 1,645 2,327 3,091 18 0,159 0,834 1,441 1,804 2,486 3,250
19 0,000 0,675 1,282 1,645 2,327 3,091 19 0,155 0,830 1,437 1,800 2,482 3,245

20 0,000 0,675 1,282 1,645 2,327 3,091 20 0,151 0,826 1,433 1,796 2,478 3,242
22 0,000 0,675 1,282 1,645 2,327 3,091 22 0,144 0,819 1,426 1,789 2,471 3,235
24 0,000 0,675 1,282 1,645 2,327 3,091 24 0,138 0,813 1,420 1,783 2,465 3,228
26 0,000 0,675 1,282 1,645 2,327 3,091 26 0,133 0,807 1,414 1,778 2,459 3,223
28 0,000 0,675 1,282 1,645 2,327 3,091 28 0,128 0,802 1,410 1,773 2,454 3,218
30 0,000 0,675 1,282 1,645 2,327 3,091 30 0,124 0,798 1,405 1,768 2,450 3,214
35 0,000 0,675 1,282 1,645 2,327 3,091 35 0,115 0,789 1,396 1,759 2,441 3,205
40 0,000 0,675 1,282 1,645 2,327 3,091 40 0,107 0,782 1,389 1,752 2,433 3,197
45 0,000 0,675 1,282 1,645 2,327 3,091 45 0,101 0,776 1,383 1,746 2,427 3,191
50 0,000 0,675 1,282 1,645 2,327 3,091 50 0,096 0,770 1,377 1,741 2,422 3,186
60 0,000 0,675 1,282 1,645 2,327 3,091 60 0,088 0,762 1,369 1,732 2,414 3,178
70 0,000 0,675 1,282 1,645 2,327 3,091 70 0,081 0,756 1,363 1,726 2,407 3,171
80 0,000 0,675 1,282 1,645 2,327 3,091 80 0,076 0,750 1,357 1,721 2,402 3,166
90 0,000 0,675 1,282 1,645 2,327 3,091 90 0,072 0,746 1,353 1,716 2,398 3,162
100 0,000 0,675 1,282 1,645 2,327 3,091 100 0,068 0,742 1,350 1,713 2,394 3,158
150 0,000 0,675 1,282 1,645 2,327 3,091 150 0,056 0,730 1,337 1,700 2,382 3,146
200 0,000 0,675 1,282 1,645 2,327 3,091 200 0,048 0,723 1,330 1,693 2,375 3,138
250 0,000 0,675 1,282 1,645 2,327 3,091 250 0,043 0,718 1,325 1,688 2,370 3,133
300 0,000 0,675 1,282 1,645 2,327 3,091 300 0,039 0,714 1,321 1,684 2,366 3,130
400 0,000 0,675 1,282 1,645 2,327 3,091 400 0,034 0,709 1,316 1,679 2,361 3,124
500 0,000 0,675 1,282 1,645 2,327 3,091 500 0,031 0,705 1,312 1,676 2,357 3,121
1 000 0,000 0,675 1,282 1,645 2,327 3,091 1 000 0,022 0,696 1,303 1,667 2,348 3,112
∞ 0,000 0,675 1,282 1,645 2,327 3,091 ∞ 0,000 0,675 1,282 1,645 2,327 3,091
14 © ISO 2005 – All rights reserved

Table B.3 — Confidence level 90,0 % Table B.4 — Confidence level 95,0 %
(1 − α = 0,90) (1 − α = 0,95)
p p
n n
0,50 0,75 0,90 0,95 0,99 0,999 0,50 0,75 0,90 0,95 0,99 0,999
2 0,907 1,581 2,188 2,552 3,233 3,997 2 1,164 1,838 2,445 2,808 3,490 4,254
3 0,740 1,415 2,022 2,385 3,067 3,831 3 0,950 1,625 2,232 2,595 3,277 4,040
4 0,641 1,316 1,923 2,286 2,968 3,732 4 0,823 1,497 2,104 2,468 3,149 3,913
5 0,574 1,248 1,855 2,218 2,900 3,664 5 0,736 1,411 2,018 2,381 3,062 3,826
6 0,524 1,198 1,805 2,169 2,850 3,614 6 0,672 1,346 1,954 2,317 2,998 3,762
7 0,485 1,159 1,766 2,130 2,811 3,575 7 0,622 1,297 1,904 2,267 2,949 3,712
8 0,454 1,128 1,735 2,098 2,780 3,544 8 0,582 1,257 1,864 2,227 2,908 3,672
9 0,428 1,102 1,709 2,073 2,754 3,518 9 0,549 1,223 1,830 2,194 2,875 3,639
10 0,406 1,080 1,687 2,051 2,732 3,496 10 0,521 1,195 1,802 2,166 2,847 3,611
11 0,387 1,061 1,668 2,032 2,713 3,477 11 0,496 1,171 1,778 2,141 2,823 3,587
12 0,370 1,045 1,652 2,015 2,697 3,461 12 0,475 1,150 1,757 2,120 2,802 3,566
13 0,356 1,030 1,637 2,001 2,682 3,446 13 0,457 1,131 1,738 2,102 2,783 3,547
14 0,343 1,017 1,625 1,988 2,669 3,433 14 0,440 1,115 1,722 2,085 2,766 3,530
15 0,331 1,006 1,613 1,976 2,658 3,422 15 0,425 1,100 1,707 2,070 2,752 3,515
16 0,321 0,995 1,602 1,966 2,647 3,411 16 0,412 1,086 1,693 2,057 2,738 3,502
17 0,311 0,986 1,593 1,956 2,638 3,402 17 0,399 1,074 1,681 2,044 2,726 3,490
18 0,303 0,977 1,584 1,947 2,629 3,393 18 0,388 1,063 1,670 2,033 2,715 3,478
19 0,295 0,969 1,576 1,939 2,621 3,385 19 0,378 1,052 1,659 2,023 2,704 3,468

20 0,287 0,962 1,569 1,932 2,613 3,377 20 0,368 1,043 1,650 2,013 2,695 3,459
22 0,274 0,948 1,555 1,919 2,600 3,364 22 0,351 1,026 1,633 1,996 2,678 3,441
24 0,262 0,937 1,544 1,907 2,588 3,352 24 0,336 1,011 1,618 1,981 2,663 3,426
26 0,252 0,926 1,533 1,897 2,578 3,342 26 0,323 0,998 1,605 1,968 2,649 3,413
28 0,243 0,917 1,524 1,888 2,569 3,333 28 0,311 0,986 1,593 1,956 2,638 3,402
30 0,234 0,909 1,516 1,879 2,561 3,325 30 0,301 0,975 1,582 1,946 2,627 3,391
35 0,217 0,892 1,499 1,862 2,543 3,307 35 0,279 0,953 1,560 1,923 2,605 3,369
40 0,203 0,878 1,485 1,848 2,529 3,293 40 0,261 0,935 1,542 1,905 2,587 3,351
45 0,192 0,866 1,473 1,836 2,518 3,282 45 0,246 0,920 1,527 1,891 2,572 3,336
50 0,182 0,856 1,463 1,827 2,508 3,272 50 0,233 0,908 1,515 1,878 2,559 3,323
60 0,166 0,840 1,447 1,811 2,492 3,256 60 0,213 0,887 1,494 1,858 2,539 3,303
70 0,154 0,828 1,435 1,799 2,480 3,244 70 0,197 0,872 1,479 1,842 2,523 3,287
80 0,144 0,818 1,425 1,789 2,470 3,234 80 0,184 0,859 1,466 1,829 2,511 3,275
90 0,136 0,810 1,417 1,780 2,462 3,226 90 0,174 0,848 1,455 1,819 2,500 3,264
100 0,129 0,803 1,410 1,774 2,455 3,219 100 0,165 0,839 1,447 1,810 2,491 3,255
150 0,105 0,780 1,387 1,750 2,431 3,195 150 0,135 0,809 1,416 1,780 2,461 3,225
200 0,091 0,766 1,373 1,736 2,417 3,181 200 0,117 0,791 1,398 1,762 2,443 3,207
250 0,082 0,756 1,363 1,726 2,408 3,172 250 0,105 0,779 1,386 1,749 2,431 3,195
300 0,074 0,749 1,356 1,719 2,401 3,165 300 0,095 0,770 1,377 1,740 2,422 3,186
400 0,065 0,739 1,346 1,709 2,391 3,155 400 0,083 0,757 1,364 1,728 2,409 3,173
500 0,058 0,732 1,339 1,703 2,384 3,148 500 0,074 0,749 1,356 1,719 2,400 3,164
1 000 0,041 0,716 1,323 1,686 2,367 3,131 1 000 0,053 0,727 1,334 1,697 2,379 3,143
∞ 0,000 0,675 1,282 1,645 2,327 3,091 ∞ 0,000 0,675 1,282 1,645 2,327 3,091
Table B.5 — Confidence level 99,0 % Table B.6 — Confidence level 99,9 %
(1 − α = 0,99) (1 − α = 0,999)
p p
n n
0,50 0,75 0,90 0,95 0,99 0,999 0,50 0,75 0,90 0,95 0,99 0,999
2 1,645 2,320 2,927 3,290 3,972 4,736 2 2,186 2,860 3,467 3,830 4,512 5,276
3 1,344 2,018 2,625 2,988 3,670 4,434 3 1,785 2,459 3,066 3,430 4,111 4,875
4 1,164 1,838 2,445 2,809 3,490 4,254 4 1,546 2,220 2,827 3,190 3,872 4,636
5 1,041 1,715 2,322 2,686 3,367 4,131 5 1,382 2,057 2,664 3,027 3,709 4,473
6 0,950 1,625 2,232 2,595 3,277 4,040 6 1,262 1,937 2,544 2,907 3,588 4,352
7 0,880 1,554 2,161 2,525 3,206 3,970 7 1,168 1,843 2,450 2,813 3,495 4,259
8 0,823 1,497 2,105 2,468 3,149 3,913 8 1,093 1,768 2,375 2,738 3,419 4,183
9 0,776 1,450 2,058 2,421 3,102 3,866 9 1,031 1,705 2,312 2,675 3,357 4,121
10 0,736 1,411 2,018 2,381 3,063 3,826 10 0,978 1,652 2,259 2,623 3,304 4,068
11 0,702 1,376 1,983 2,347 3,028 3,792 11 0,932 1,607 2,214 2,577 3,259 4,022
12 0,672 1,347 1,954 2,317 2,998 3,762 12 0,893 1,567 2,174 2,537 3,219 3,983
13 0,646 1,320 1,927 2,291 2,972 3,736 13 0,858 1,532 2,139 2,502 3,184 3,948
14 0,622 1,297 1,904 2,267 2,949 3,712 14 0,826 1,501 2,108 2,471 3,153 3,917
15 0,601 1,276 1,883 2,246 2,928 3,691 15 0,798 1,473 2,080 2,443 3,125 3,889
16 0,582 1,257 1,864 2,227 2,908 3,672 16 0,773 1,448 2,055 2,418 3,099 3,863
17 0,565 1,239 1,846 2,210 2,891 3,655 17 0,750 1,424 2,032 2,395 3,076 3,840
18 0,549 1,223 1,830 2,194 2,875 3,639 18 0,729 1,403 2,010 2,374 3,055 3,819
19 0,534 1,209 1,816 2,179 2,861 3,624 19 0,709 1,384 1,991 2,354 3,036 3,800

20 0,521 1,195 1,802 2,166 2,847 3,611 20 0,691 1,366 1,973 2,336 3,018 3,782
22 0,496 1,171 1,778 2,141 2,823 3,587 22 0,659 1,334 1,941 2,304 2,986 3,750
24 0,475 1,150 1,757 2,120 2,802 3,566 24 0,631 1,306 1,913 2,276 2,958 3,722
26 0,457 1,131 1,738 2,102 2,783 3,547 26 0,607 1,281 1,888 2,251 2,933 3,697
28 0,440 1,115 1,722 2,085 2,766 3,530 28 0,584 1,259 1,866 2,229 2,911 3,675
30 0,425 1,100 1,707 2,070 2,752 3,515 30 0,565 1,239 1,846 2,210 2,891 3,655
35 0,394 1,068 1,675 2,039 2,720 3,484 35 0,523 1,197 1,804 2,168 2,849 3,613
40 0,368 1,043 1,650 2,013 2,695 3,459 40 0,489 1,164 1,771 2,134 2,815 3,579
45 0,347 1,022 1,629 1,992 2,674 3,438 45 0,461 1,136 1,743 2,106 2,788 3,551
50 0,329 1,004 1,611 1,974 2,656 3,420 50 0,438 1,112 1,719 2,082 2,764 3,528
60 0,301 0,975 1,582 1,946 2,627 3,391 60 0,399 1,074 1,681 2,044 2,726 3,490
70 0,279 0,953 1,560 1,923 2,605 3,369 70 0,370 1,044 1,651 2,015 2,696 3,460
80 0,261 0,935 1,542 1,905 2,587 3,351 80 0,346 1,020 1,628 1,991 2,672 3,436
90 0,246 0,920 1,527 1,891 2,572 3,336 90 0,326 1,001 1,608 1,971 2,653 3,416
100 0,233 0,908 1,515 1,878 2,559 3,323 100 0,310 0,984 1,591 1,954 2,636 3,400
150 0,190 0,865 1,472 1,835 2,517 3,281 150 0,253 0,927 1,534 1,898 2,579 3,343
200 0,165 0,839 1,447 1,810 2,491 3,255 200 0,219 0,894 1,501 1,864 2,545 3,309
250 0,148 0,822 1,429 1,792 2,474 3,238 250 0,196 0,870 1,477 1,841 2,522 3,286
300 0,135 0,809 1,416 1,780 2,461 3,225 300 0,179 0,853 1,460 1,824 2,505 3,269
400 0,117 0,791 1,398 1,762 2,443 3,207 400 0,155 0,830 1,437 1,800 2,481 3,245
500 0,105 0,779 1,386 1,749 2,431 3,195 500 0,139 0,813 1,420 1,784 2,465 3,229
1 000 0,074 0,749 1,356 1,719 2,400 3,164 1 000 0,098 0,773 1,380 1,743 2,425 3,188
∞ 0,000 0,675 1,282 1,645 2,327 3,091 ∞ 0,000 0,675 1,282 1,645 2,327 3,091

16 © ISO 2005 – All rights reser
...

SLOVENSKI STANDARD
01-april-2006
6WDWLVWLþQRWROPDþHQMHSRGDWNRY±GHO8JRWDYOMDQMHVWDWLVWLþQLKWROHUDQþQLK
LQWHUYDORY
Statistical interpretation of data -- Part 6: Determination of statistical tolerance intervals
Interprétation statistique des données -- Partie 6: Détermination des intervalles
statistiques de tolérance
Ta slovenski standard je istoveten z: ISO 16269-6:2005
ICS:
03.120.30 8SRUDEDVWDWLVWLþQLKPHWRG Application of statistical
methods
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

INTERNATIONAL ISO
STANDARD 16269-6
First edition
2005-04-01
Statistical interpretation of data —
Part 6:
Determination of statistical tolerance
intervals
Interprétation statistique des données —
Partie 6: Détermination des intervalles statistiques de tolérance

Reference number
©
ISO 2005
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ii © ISO 2005 – All rights reserved

Contents Page
Foreword. iv
Introduction . v
1 Scope. 1
2 Normative references . 1
3 Terms, definitions and symbols . 1
3.1 Terms and definitions. 1
3.2 Symbols . 2
4 Procedures . 3
4.1 Normal population with known variance and known mean. 3
4.2 Normal population with known variance and unknown mean . 3
4.3 Normal population with unknown variance and unknown mean. 3
4.4 Any continuous distribution of unknown type . 3
5 Examples. 3
5.1 Data. 3
5.2 Example 1: One-sided statistical tolerance interval under known variance. 4
5.3 Example 2: Two-sided statistical tolerance interval under known variance . 4
5.4 Example 3: One-sided statistical tolerance interval under unknown variance . 5
5.5 Example 4: Two-sided statistical tolerance interval under unknown variance. 6
5.6 Example 5: Distribution-free statistical tolerance interval for continuous distribution . 6
Annex A (informative) Forms for tolerance intervals. 8
Annex B (normative) One-sided statistical tolerance limit factors, k (n; p; 1 − α), for known σ . 14
Annex C (normative) Two-sided statistical tolerance limit factors, k (n; p; 1 − α), for known σ. 17
Annex D (normative) One-sided statistical tolerance limit factors, k (n; p; 1 − α), for unknown σ. 20
Annex E (normative) Two-sided statistical tolerance limit factors, k (n; p; 1 − α), for unknown σ. 23
Annex F (normative) One-sided distribution-free statistical tolerance intervals. 26
Annex G (normative) Two-sided distribution-free statistical tolerance intervals. 27
Annex H (informative) Construction of a distribution-free statistical tolerance interval for any type
of distribution . 28
Annex I (informative) Computation of factors for two-sided parametric statistical tolerance
intervals. 29
Bibliography . 30

Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO 16269-6 was prepared by Technical Committee ISO/TC 69, Applications of statistical methods.
This first edition of ISO 16269-6 cancels and replaces ISO 3207:1975, which has been technically revised.
ISO 16269 consists of the following parts, under the general title Statistical interpretation of data:
 Part 6: Determination of statistical tolerance intervals
 Part 7: Median — Estimation and confidence intervals
 Part 8: Determination of prediction intervals

iv © ISO 2005 – All rights reserved

Introduction
A statistical tolerance interval is an estimated interval, based on a sample, which can be asserted with
confidence 1 − α, for example 95 %, to contain at least a specified proportion p of the items in the population.
The limits of a statistical tolerance interval are called statistical tolerance limits. The confidence level 1 − α is
the probability that a statistical tolerance interval constructed in the prescribed manner will contain at least a
proportion p of the population. Conversely, the probability that this interval will contain less than the proportion
p of the population is α. This part of ISO 16269 describes both one-sided and two-sided statistical tolerance
intervals; a one-sided interval is constructed with an upper or a lower limit while a two-sided interval is
constructed with both an upper and a lower limit.
Tolerance intervals are functions of the observations of the sample, i.e. statistics, and they will generally take
different values for different samples. It is necessary that the observations be independent for the procedures
provided in this part of ISO 16269 to be valid.
Two types of tolerance interval are provided in this part of ISO 16269, parametric and distribution-free. The
parametric approach is based on the assumption that the characteristic being studied in the population has a
normal distribution; hence the confidence that the calculated statistical tolerance interval contains at least a
proportion p of the population can only be taken to be 1 − α if the normality assumption is true. For normally
distributed characteristics, the statistical tolerance interval is determined using one of the Forms A, B, C or D
given in Annex A.
Parametric methods for distributions other than the normal are not considered in this part of ISO 16269. If
departure from normality is suspected in the population, distribution-free statistical tolerance intervals may be
constructed. The procedure for the determination of a statistical tolerance interval for any continuous
distribution is provided in Forms E and F of Annex A.
The tolerance limits discussed in this part of ISO 16269 can be used to compare the natural capability of a
process with one or two given specification limits, either an upper one U or a lower one L or both in statistical
process management. An indication of this is the fact that these tolerance limits have also been called natural
process limits. See ISO 3534-2:1993, 3.2.4, and the general remarks in ISO 3207 which will be cancelled and
replaced by this part of ISO 16269.
Above the upper specification limit U there is the upper fraction nonconforming p (ISO 3534-2:—, 3.2.5.5 and
U
3.3.1.4) and below the lower specification limit L there is the lower fraction nonconforming p (ISO 3534-2:—,
L
3.2.5.6 and 3.3.1.5). The sum p + p = p is called the total fraction nonconforming. (ISO 3534-2:—, 3.2.5.7).
U L T
Between the specification limits U and L there is the fraction conforming 1 − p .
T
In statistical process management the limits U and L are fixed in advance and the fractions p , p and p are
U L T
either calculated, if the distribution is assumed to be known, or otherwise estimated. There are many
applications of statistical tolerance intervals, although the above shows an example to a quality control
problem. Wider applications and more statistical intervals are introduced in many textbooks such as Hahn and
[10]
Meeker .
In contrast, for the tolerance intervals considered in this part of ISO 16269, the confidence level for the interval
estimator and the proportion of the distribution within the interval (corresponding to the fraction conforming
mentioned above) are fixed in advance, and the limits are estimated. These limits may be compared with U
and L. Hence the appropriateness of the given specification limits U and L can be compared with the actual
properties of the process. The one-sided tolerance intervals are used when only either the upper specification
limit U or the lower specification limit L is relevant, while the two-sided intervals are used when both the upper
and the lower specification limits are considered simultaneously.
The terminology with regard to these different limits and intervals has been confusing as the “specification
limits” were earlier also called “tolerance limits” (see the terminology standard ISO 3534-2:1993, 1.4.3, where
both these terms as well as the term “limiting values” were all used as synonyms for this concept). In the latest
revision of ISO 3534-2:—, only the term specification limits have been kept for this concept. Furthermore, the
[5]
Guide for the expression of uncertainty in measurement uses the term “coverage factor” defined as a
“numerical factor used as a multiplier of the combined standard uncertainty in order to obtain an expanded
uncertainty”. This use of “coverage” differs from the use of the term in this part of ISO 16269.
vi © ISO 2005 – All rights reserved

INTERNATIONAL STANDARD ISO 16269-6:2005(E)

Statistical interpretation of data —
Part 6:
Determination of statistical tolerance intervals
1 Scope
This part of ISO 16269 describes procedures for establishing tolerance intervals that include at least a
specified proportion of the population with a specified confidence level. Both one-sided and two-sided
statistical tolerance intervals are provided, a one-sided interval having either an upper or a lower limit while a
two-sided interval has both upper and lower limits. Two methods are provided, a parametric method for the
case where the characteristic being studied has a normal distribution and a distribution-free method for the
case where nothing is known about the distribution except that it is continuous.
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated
references, only the edition cited applies. For undated references, the latest edition of the referenced
document (including any amendments) applies.
ISO 3534-1, Statistics — Vocabulary and symbols — Part 1: Probability and general statistical terms
1)
ISO 3534-2:— , Statistics — Vocabulary and symbols — Part 2: Applied statistics
3 Terms, definitions and symbols
3.1 Terms and definitions
For the purposes of this document, the terms and definition given in ISO 3534-1, ISO 3534-2 and the following
apply.
3.1.1
statistical tolerance interval
interval determined from a random sample in such a way that one may have a specified level of confidence
that the interval covers at least a specified proportion of the sampled population
NOTE The confidence level in this context is the long-run proportion of intervals constructed in this manner that will
include at least the specified proportion of the sampled population.
3.1.2
statistical tolerance limit
statistic representing an end point of a statistical tolerance interval
NOTE Statistical tolerance intervals can be either one-sided, in which case they have either an upper or a lower
statistical tolerance limit, or two-sided, in which case they have both.

1) To be published. (Revision of ISO 3534-2:1993)
3.1.3
coverage
proportion of items in a population lying within a statistical tolerance interval
NOTE This concept is not to be confused with the concept coverage factor used in the Guide for the expression of
[5]
uncertainty in measurement (GUM ) .
3.1.4
normal population
normally distributed population
3.2 Symbols
For the purposes of this part of ISO 16269, the following symbols apply.
i suffix of an observation
k (n; p; 1 − α) factor used to determine x or x when the value of σ is known for one-sided tolerance
L U
interval
k (n; p; 1 − α) factor used to determine x and x when the value of σ is known for two-sided tolerance
L U
interval
k (n; p; 1 − α) factor used to determine x or x when the value of σ is unknown for one-sided tolerance
L U
interval
k (n; p; 1 − α) factor used to determine x and x when the value of σ is unknown for two-sided tolerance

L U
interval
n number of observations in the sample
p minimum proportion of the population claimed to be lying in the statistical tolerance interval
u p-fractile of the standard normal distribution
p
x ith observed value (1in=,2,.,)
i
x maximum value of the observed values: x = max {x , x , …, x }
max max 1 2 n
x minimum value of the observed values: x = min {x , x , …, x }
min min 1 2 n
x lower limit of the statistical tolerance interval
L
x upper limit of the statistical tolerance interval
U
n
x sample mean, xx=
i
∑
n
i = 1
nn


nx − x
ii
∑∑
n 
ii==11
1 2

s sample standard deviation;sx=−x=
()
i
∑
nn−−11n
()
i = 1
1 − α confidence level for the claim that the proportion of the population lying within the tolerance
interval is greater than or equal to the specified level p
µ population mean
σ population standard deviation
2 © ISO 2005 – All rights reserved

4 Procedures
4.1 Normal population with known variance and known mean
When the values of the mean, µ, and the variance, σ , of a normally distributed population are known, the
distribution of the characteristic under investigation is fully determined. There is exactly a proportion p of the
population:
a) to the right of x = µ − u × σ (one-sided interval);
p
L
b) to the left of x = µ + u × σ (one-sided interval);
p
U
u u
c) between x = µ − × σ and x = µ + × σ (two-sided interval).
(1+ p)/ 2 (1+ p)/ 2
L U
NOTE As such statements are known to be true, they are made with 100 % confidence.
In the above equations, u is p-fractile of the standard normal distribution. Numerical values of u may be
p p
read from the bottom line of the Tables B.1 to B.6 and Tables C.1 to C.6.
4.2 Normal population with known variance and unknown mean
Forms A and B, given in Annex A, are applicable to the case where the variance of the normal population is
known while the mean is unknown. Form A applies to the one-sided case, while Form B applies to the
two-sided case.
4.3 Normal population with unknown variance and unknown mean
Forms C and D, given in Annex A, are applicable to the case where both the mean and the variance of the
normal population are unknown. Form C applies to the one-sided case, while Form D applies to the two-sided
case.
4.4 Any continuous distribution of unknown type
If the characteristic under investigation is a continuous variable from a population of unknown form, and if a
sample of n independent random observations of the characteristic has been taken, then a statistical tolerance
interval can be determined from the ranked observations. The procedure given in Forms E and F of Annex A
provide the determination of the coverage or sample size needed for tolerance intervals determined from the
extreme values x or x of the sample of observations with given confidence level 1 − α.
min max
NOTE Statistical tolerance intervals that do not depend on the shape of the sampled population are called
distribution-free tolerance intervals.
This part of ISO 16269 does not provide procedures for distributions of known type other than the normal
distribution. However, if the distribution is continuous, the distribution-free method may be used. Selected
references to scientific literature that may assist in determining tolerance intervals for other distributions are
also provided at the end of this document.
5 Examples
5.1 Data
Forms A to D, given in Annex A, are illustrated by examples using the numerical values of ISO 2854:1976,
Clause 2, paragraph 1 of the introductory remarks, Table X, yarn 2: 12 measures of the breaking load of
cotton yarn. It should be noted that the number of observations, n = 12, given here for these examples is
[1]
considerably lower than the one recommended in ISO 2602 . The numerical data and calculations in the
different examples are expressed in centi-newtons (see Table 1).
Table 1 — Data for Examples 1 to 4
Values in centi-newtons
228,6 232,7 238,8 317,2 315,8 275,1 222,2 236,7 224,7 251,2 210,4 270,7
x
These measurements were obtained from a batch of 12,000 bobbins, from one production job, packed in
120 boxes each containing 100 bobbins. Twelve boxes have been drawn at random from the batch and a
bobbin has been drawn at random from each of these boxes. Test pieces of 50 cm length have been cut from
the yarn on these bobbins, at about 5 m distance from the free end. The tests themselves have been carried
out on the central parts of these test pieces. Previous information makes it reasonable to assume that the
breaking loads measured in these conditions have virtually a normal distribution. It is demonstrated in
ISO 2954:1976 that the data do not contradict the assumption of a normal distribution.
These results yield the following:
Sample size: n = 12
Sample mean: x==3 024,1/12 252,01
nx − x
()
∑∑ 166 772,27
Sample standard deviation:
s== = 1263,426 3= 35,545
nn−×11211
()
The formal presentation of the calculations will be given only for Form C in Annex A (one-sided interval,
unknown variance).
5.2 Example 1: One-sided statistical tolerance interval under known variance
Suppose that previously obtained measurements have shown that the dispersion is constant from one batch
σ = 33,150
to another from the same supplier, and is represented by a standard deviation , although the mean
is not constant. A limit x is required such that it is possible to assert with confidence level 1 − α = 0,95 (95 %)
L
that at least 0,95 (95 %) of the breaking loads of the items in the batch, when measured under the same
conditions, are above x .
L
Table B.4 gives
k (12; 0,95; 0,95) = 2,120
whence
x = xk−−(n;p;1 ασ)×= 252,01− 2,120× 33,150= 181,732
L 1
A smaller value of the lower limit x would be obtained if a larger proportion of the population (for example
L
p = 0,99) and/or a higher confidence level (for example 1 − α = 0,99) were required.
5.3 Example 2: Two-sided statistical tolerance interval under known variance
Under the same conditions as in Example 1, suppose that limits x and x are required such that it is
L U
possible to assert with a confidence level 1 − α = 0,95 that at least a proportion of p = 0,90 (90 %) of the
breaking load of the batch falls between x and x .
L U
Table C.4 gives
k (12; 0,90; 0,95) = 1,889
4 © ISO 2005 – All rights reserved

whence
x = xk−−(n;p;1 ασ)×= 252,01−1,889× 33,150= 189,390
L 2
x = xk+−(n;p;1 ασ)×= 252,01+1,889× 33,150= 314,630
U 2
Comparison with Example 1 should make it clear that assuring that at least 90 % of a population lies between
the limits x and x is not the same thing as assuring that no more than 5 % lies beyond each limit.
L U
5.4 Example 3: One-sided statistical tolerance interval under unknown variance
Here, it is supposed that the standard deviation of the population is unknown and has to be estimated from the
sample. The same requirements will be assumed as for the case where the standard deviation is known
(Example 1), thus, p = 0,95 and 1 − α = 0,95. The presentation of the results is given in detail below.
Determination of the statistical tolerance interval of proportion p:
a) one-sided interval “to the right”
Determined values:
b) proportion of the population selected for the tolerance interval: p = 0,95
c) chosen confidence level: 1 − α = 0,95
d) sample size: n = 12
Value of tolerance factor from Table D.4:
k (n; p; 1 − α) = 2,737
Calculations:
xx==/n 252,01
∑
nx − x
()
∑∑
s = = 35,545
nn −1
()
kn( ;p;1−×α) s= 97,286 7
Results: one-sided interval “to the right”
The tolerance interval which will contain at least a proportion p of the population with confidence level 1 − α
has a lower limit
x =−xkn( ;p;1− α)×s= 154,723
L 3
5.5 Example 4: Two-sided statistical tolerance interval under unknown variance
Under the same conditions as in Example 2, suppose it is required to calculate the limits x and x such
L U
that it is possible to assert with a confidence level 1 − α = 0,95 that in a proportion of the batch at least equal
to p = 0,90 (90 %) the breaking load falls between x and x .
L U
Table E.4 gives
kn(;p;1−=α) 2,671
whence
xx=−k (n;p;1−α)×s= 252,01− 2,671× 35,545= 157,069
L 4
xx=+k (n;p;1−α)×s= 252,01+ 2,671× 35,545= 346,951
U 4
It will be noted that the value of x is smaller and the value of x higher than in Example 2 (known variance),
L U
because the use of s instead of σ requires a larger value of the tolerance factor to allow for the extra
uncertainty. It is necessary to have to pay a penalty for not knowing the population standard deviation σ and
the extension of the statistical tolerance interval takes this into account. Of course, it is not quite sure that the
value σ = 33,150 used in Examples 1 and 2 is correct. Therefore, it is wiser to use the estimate, s, together
with Tables D.4 or E.4.
5.6 Example 5: Distribution-free statistical tolerance interval for continuous distribution
In a fatigue test by rotational stress carried out on a component of an aeronautical engine, a sample of
15 items has given the results (measurement of endurance), shown in ascending order of values in Table 2.
Table 2 — Data for Example 5
x 0,200 0,330 0,450 0,490 0,780 0,920 0,950 0,970 1,040 1,710 2,220 2,275 3,650 7,000 8,800

A graphical examination of checking normality, such as probability plot, shows that the hypothesis of normality
for the population of components should almost certainly be rejected (see ISO 5479). The methods of Form E,
given in Annex A, for determination of a statistical tolerance interval are therefore applicable.
The extreme values from the sample of n = 15 measurements are:
x = 0,200, x = 8,800
min max
Suppose that the required confidence level 1 − α is 0,95.
a) What is the maximum proportion of the population of components that will fall below x = 0,200?
min
Table F.1, for 1 − α = 0,95, gives for the minimum proportion above x a value of p slightly higher than
min
0,75 (75 %). Hence, for the maximum proportion below x a value of 1 − p slightly lower than
min
0,25 (25 %).
b) What sample size is necessary for it to be possible to assert, at a confidence level 0,95, that a proportion
at least p = 0,90 (90 %) of the population of components will be found below the largest of the values from
that sample? Table F.1, for 1 − α = 0,95 and p = 0,90, gives n = 29.
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c) At a confidence level of 0,95, what is the minimum proportion of the population of components that fall
between x = 0,200 and x = 8,800? Table G.1, for 1 − α = 0,95 and n = 15, gives p slightly below
min max
0,75 (75 %).
d) What sample size is necessary for it to be possible to assert at a confidence level 0,95 that a proportion
of at least p = 0,90 (90 %) of the population of components will be found to fall between the smallest and
the largest values from that sample? Table G.1, for 1 − α = 0,95 and p = 0,90, gives n = 46.
e) In general, if a check for normality (see ISO 5479) indicates a departure from the normal distribution,
some transformation will be recommended based on the knowledge of the collected data. For example,
fatigue data are often approximated lognormally distributed. In such cases, the data could be transformed
to normality. Tolerance intervals are then calculated and finally transformed back into the original units.
See Annex H for the construction of a statistical tolerance interval for distribution-free tolerance intervals for
any type of distribution. Annex I gives the computation of factors for two-sided parametric statistical tolerance
intervals.
Annex A
(informative)
Forms for tolerance intervals
Form A — One-sided statistical tolerance interval (known variance)
Determination of a one-sided statistical tolerance interval with coverage p at confidence level 1 − α
a) One-sided interval “to the left”
b) One-sided interval “to the right”
Known values:
c) the variance: σ =
d) the standard deviation: σ =
Determined values:
e) proportion of the population selected for the tolerance interval: p =
f) chosen confidence level: 1 − α =
g) sample size: n =
Tabulated factor:
k (n; p; 1 − α) =
This value can be read from the tables given in Annex B for a range of values of n, p and 1 − α.
Calculations:
xx==/n
∑
k (n; p; 1 − α) × σ =
Results:
a) One-sided interval “to the left”
The one-sided statistical tolerance interval with coverage p at confidence level 1 − α has upper limit
xx=+k (;np;1− α)×σ=
U 1
b) One-sided interval “to the right”
The one-sided statistical tolerance interval with coverage p at confidence level 1 − α has lower limit
x =−xkn(;p;1− α)×σ=
L 1
8 © ISO 2005 – All rights reserved

Form B — Two-sided statistical tolerance interval (known variance)
Determination of a two-sided statistical tolerance interval with coverage p at confidence level 1 − α
Known values:
a) the variance: σ =
b) the standard deviation: σ =
Determined values:
c) proportion of the population selected for the tolerance interval: p =
d) chosen confidence level: 1 − α =
e) sample size: n =
Tabulated factor:
k (n; p; 1 − α) =
This value can be read from the tables given in Annex C for a range of values of n, p and 1 − α.
Calculations:
xx==/n
∑
k (n; p; 1 − α) × σ =
Results:
The two-sided statistical tolerance interval with coverage p at confidence level 1 − α has limits
x =−xkn(;p;1− α)×σ=
L 2
xx=+k (;np;1− α)×σ=
U 2
Form C — One-sided statistical tolerance interval (unknown variance)
Determination of a one-sided statistical tolerance interval with coverage p at confidence level 1 − α
a) One-sided interval “to the left”
b) One-sided interval “to the right”
Determined values:
c) proportion of the population selected for the tolerance interval: p =
d) chosen confidence level: 1 − α =
e) sample size: n =
Tabulated factor:
k (n; p; 1 − α) =
This value can be read from the tables given in Annex D for a range of values of n, p and 1 − α.
Calculations:
xx==/n
∑
nx −()x
∑∑
s==
nn(1−)
k (n; p; 1 − α) × s =
Results:
a) One-sided interval “to the left”
The tolerance interval with coverage p at confidence level 1 − α has upper limit
xx=+k (;np;1−α)×s=
U 3
b) One-sided interval “to the right”
The tolerance interval with coverage p at confidence level 1 − α has lower limit
xx=−k (;np;1−α)×s=
L 3
10 © ISO 2005 – All rights reserved

Form D — Two-sided statistical tolerance interval (unknown variance)
Determination of a two-sided statistical tolerance interval with coverage p at confidence level 1 − α
Determined values:
a) proportion of the population selected for the tolerance interval: p =
b) chosen confidence level: 1 − α =
c) sample size: n =
Tabulated factor:
k (n; p; 1 − α) =
This value can be read from the tables given in Annex E for a range of values of n, p and 1 − α.
Calculations:
xx==/n
i
∑
nx −()x
∑∑
s==
nn(1−)
k (n; p; 1 − α) × s =
Results:
The two-sided statistical tolerance interval with coverage p at confidence level 1 − α has limits
xx=−k (;np;1−α)×s=
L 4
xx=+k (;np;1−α)×s=
U 4
Form E — One-sided statistical tolerance interval for any distribution
Determination of a one-sided distribution-free statistical tolerance interval with coverage p at confidence level
1 − α
a) One-sided interval “to the left”
b) One-sided interval “to the right”
Determined values:
c) proportion of the population selected for the tolerance interval: p =
d) chosen confidence level: 1 − α =
e) sample size: n =
(Either p or n is to be determined.)
Tabulated value
 p for given n and 1 − α.
 n for given p and 1 − α.
This value can be read from Table F.1 for a range of values of n, p and 1 − α.
Calculations and results
The one-sided statistical tolerance interval with coverage p at confidence level 1 − α has either
 lower limit xx==
L min
 or upper limit xx==
U max
12 © ISO 2005 – All rights reserved

Form F — Two-sided statistical tolerance interval for any distribution
Determination of a two-sided distribution-free statistical tolerance interval with coverage p at confidence level
1 − α
Determined values:
a) proportion of the population selected for the tolerance interval: p =
b) chosen confidence level: 1 − α =
c) sample size: n =
(Either p or n is to be determined.)
Tabulated value
 p for given n and 1 − α.
 n for given p and 1 − α.
This value can be read from Table G.1 for a range of values of n, p and 1 − α.
Calculations and results
The two-sided statistical tolerance interval with coverage p at confidence level 1 − α has
 lower limit xx= =
L min
 and upper limit xx==
U max
Annex B
(normative)
One-sided statistical tolerance limit factors, k (n; p; 1 − α), for known σ
Table B.1 — Confidence level 50,0 % Table B.2 — Confidence level 75,0 %
(1 − α = 0,50) (1 − α = 0,75)
p p
n n
0,50 0,75 0,90 0,95 0,99 0,999 0,50 0,75 0,90 0,95 0,99 0,999
2 0,000 0,675 1,282 1,645 2,327 3,091 2 0,477 1,152 1,759 2,122 2,804 3,568
3 0,000 0,675 1,282 1,645 2,327 3,091 3 0,390 1,064 1,671 2,035 2,716 3,480
4 0,000 0,675 1,282 1,645 2,327 3,091 4 0,338 1,012 1,619 1,983 2,664 3,428
5 0,000 0,675 1,282 1,645 2,327 3,091 5 0,302 0,977 1,584 1,947 2,628 3,392
6 0,000 0,675 1,282 1,645 2,327 3,091 6 0,276 0,950 1,557 1,921 2,602 3,366
7 0,000 0,675 1,282 1,645 2,327 3,091 7 0,255 0,930 1,537 1,900 2,582 3,346
8 0,000 0,675 1,282 1,645 2,327 3,091 8 0,239 0,913 1,521 1,884 2,565 3,329
9 0,000 0,675 1,282 1,645 2,327 3,091 9 0,225 0,900 1,507 1,870 2,552 3,316
10 0,000 0,675 1,282 1,645 2,327 3,091 10 0,214 0,888 1,495 1,859 2,540 3,304
11 0,000 0,675 1,282 1,645 2,327 3,091 11 0,204 0,878 1,485 1,849 2,530 3,294
12 0,000 0,675 1,282 1,645 2,327 3,091 12 0,195 0,870 1,477 1,840 2,522 3,285
13 0,000 0,675 1,282 1,645 2,327 3,091 13 0,188 0,862 1,469 1,832 2,514 3,278
14 0,000 0,675 1,282 1,645 2,327 3,091 14 0,181 0,855 1,462 1,826 2,507 3,271
15 0,000 0,675 1,282 1,645 2,327 3,091 15 0,175 0,849 1,456 1,820 2,501 3,265
16 0,000 0,675 1,282 1,645 2,327 3,091 16 0,169 0,844 1,451 1,814 2,495 3,259
17 0,000 0,675 1,282 1,645 2,327 3,091 17 0,164 0,839 1,446 1,809 2,490 3,254
18 0,000 0,675 1,282 1,645 2,327 3,091 18 0,159 0,834 1,441 1,804 2,486 3,250
19 0,000 0,675 1,282 1,645 2,327 3,091 19 0,155 0,830 1,437 1,800 2,482 3,245

20 0,000 0,675 1,282 1,645 2,327 3,091 20 0,151 0,826 1,433 1,796 2,478 3,242
22 0,000 0,675 1,282 1,645 2,327 3,091 22 0,144 0,819 1,426 1,789 2,471 3,235
24 0,000 0,675 1,282 1,645 2,327 3,091 24 0,138 0,813 1,420 1,783 2,465 3,228
26 0,000 0,675 1,282 1,645 2,327 3,091 26 0,133 0,807 1,414 1,778 2,459 3,223
28 0,000 0,675 1,282 1,645 2,327 3,091 28 0,128 0,802 1,410 1,773 2,454 3,218
30 0,000 0,675 1,282 1,645 2,327 3,091 30 0,124 0,798 1,405 1,768 2,450 3,214
35 0,000 0,675 1,282 1,645 2,327 3,091 35 0,115 0,789 1,396 1,759 2,441 3,205
40 0,000 0,675 1,282 1,645 2,327 3,091 40 0,107 0,782 1,389 1,752 2,433 3,197
45 0,000 0,675 1,282 1,645 2,327 3,091 45 0,101 0,776 1,383 1,746 2,427 3,191
50 0,000 0,675 1,282 1,645 2,327 3,091 50 0,096 0,770 1,377 1,741 2,422 3,186
60 0,000 0,675 1,282 1,645 2,327 3,091 60 0,088 0,762 1,369 1,732 2,414 3,178
70 0,000 0,675 1,282 1,645 2,327 3,091 70 0,081 0,756 1,363 1,726 2,407 3,171
80 0,000 0,675 1,282 1,645 2,327 3,091 80 0,076 0,750 1,357 1,721 2,402 3,166
90 0,000 0,675 1,282 1,645 2,327 3,091 90 0,072 0,746 1,353 1,716 2,398 3,162
100 0,000 0,675 1,282 1,645 2,327 3,091 100 0,068 0,742 1,350 1,713 2,394 3,158
150 0,000 0,675 1,282 1,645 2,327 3,091 150 0,056 0,730 1,337 1,700 2,382 3,146
200 0,000 0,675 1,282 1,645 2,327 3,091 200 0,048 0,723 1,330 1,693 2,375 3,138
250 0,000 0,675 1,282 1,645 2,327 3,091 250 0,043 0,718 1,325 1,688 2,370 3,133
300 0,000 0,675 1,282 1,645 2,327 3,091 300 0,039 0,714 1,321 1,684 2,366 3,130
400 0,000 0,675 1,282 1,645 2,327 3,091 400 0,034 0,709 1,316 1,679 2,361 3,124
500 0,000 0,675 1,282 1,645 2,327 3,091 500 0,031 0,705 1,312 1,676 2,357 3,121
1 000 0,000 0,675 1,282 1,645 2,327 3,091 1 000 0,022 0,696 1,303 1,667 2,348 3,112
∞ 0,000 0,675 1,282 1,645 2,327 3,091 ∞ 0,000 0,675 1,282 1,645 2,327 3,091
14 © ISO 2005 – All rights reserved

Table B.3 — Confidence level 90,0 % Table B.4 — Confidence level 95,0 %
(1 − α = 0,90) (1 − α = 0,95)
p p
n n
0,50 0,75 0,90 0,95 0,99 0,999 0,50 0,75 0,90 0,95 0,99 0,999
2 0,907 1,581 2,188 2,552 3,233 3,997 2 1,164 1,838 2,445 2,808 3,490 4,254
3 0,740 1,415 2,022 2,385 3,067 3,831 3 0,950 1,625 2,232 2,595 3,277 4,040
4 0,641 1,316 1,923 2,286 2,968 3,732 4 0,823 1,497 2,104 2,468 3,149 3,913
5 0,574 1,248 1,855 2,218 2,900 3,664 5 0,736 1,411 2,018 2,381 3,062 3,826
6 0,524 1,198 1,805 2,169 2,850 3,614 6 0,672 1,346 1,954 2,317 2,998 3,762
7 0,485 1,159 1,766 2,130 2,811 3,575 7 0,622 1,297 1,904 2,267 2,949 3,712
8 0,454 1,128 1,735 2,098 2,780 3,544 8 0,582 1,257 1,864 2,227 2,908 3,672
9 0,428 1,102 1,709 2,073 2,754 3,518 9 0,549 1,223 1,830 2,194 2,875 3,639
10 0,406 1,080 1,687 2,051 2,732 3,496 10 0,521 1,195 1,802 2,166 2,847 3,611
11 0,387 1,061 1,668 2,032 2,713 3,477 11 0,496 1,171 1,778 2,141 2,823 3,587
12 0,370 1,045 1,652 2,015 2,697 3,461 12 0,475 1,150 1,757 2,120 2,802 3,566
13 0,356 1,030 1,637 2,001 2,682 3,446 13 0,457 1,131 1,738 2,102 2,783 3,547
14 0,343 1,017 1,625 1,988 2,669 3,433 14 0,440 1,115 1,722 2,085 2,766 3,530
15 0,331 1,006 1,613 1,976 2,658 3,422 15 0,425 1,100 1,707 2,070 2,752 3,515
16 0,321 0,995 1,602 1,966 2,647 3,411 16 0,412 1,086 1,693 2,057 2,738 3,502
17 0,311 0,986 1,593 1,956 2,638 3,402 17 0,399 1,074 1,681 2,044 2,726 3,490
18 0,303 0,977 1,584 1,947 2,629 3,393 18 0,388 1,063 1,670 2,033 2,715 3,478
19 0,295 0,969 1,576 1,939 2,621 3,385 19 0,378 1,052 1,659 2,023 2,704 3,468

20 0,287 0,962 1,569 1,932 2,613 3,377 20 0,368 1,043 1,650 2,013 2,695 3,459
22 0,274 0,948 1,555 1,919 2,600 3,364 22 0,351 1,026 1,633 1,996 2,678 3,441
24 0,262 0,937 1,544 1,907 2,588 3,352 24 0,336 1,011 1,618 1,981 2,663 3,426
26 0,252 0,926 1,533 1,897 2,578 3,342 26 0,323 0,998 1,605 1,968 2,649 3,413
28 0,243 0,917 1,524 1,888 2,569 3,333 28 0,311 0,986 1,593 1,956 2,638 3,402
30 0,234 0,909 1,516 1,879 2,561 3,325 30 0,301 0,975 1,582 1,946 2,627 3,391
35 0,217 0,892 1,499 1,862 2,543 3,307 35 0,279 0,953 1,560 1,923 2,605 3,369
40 0,203 0,878 1,485 1,848 2,529 3,293 40 0,261 0,935 1,542 1,905 2,587 3,351
45 0,192 0,866 1,473 1,836 2,518 3,282 45 0,246 0,920 1,527 1,891 2,572 3,336
50 0,182 0,856 1,463 1,827 2,508 3,272 50 0,233 0,908 1,515 1,878 2,559 3,323
60 0,166 0,840 1,447 1,811 2,492 3,256 60 0,213 0,887 1,494 1,858 2,539 3,303
70 0,154 0,828 1,435 1,799 2,480 3,244 70 0,197 0,872 1,479 1,842 2,523 3,287
80 0,144 0,818 1,425 1,789 2,470 3,234 80 0,184 0,859 1,466 1,829 2,511 3,275
90 0,136 0,810 1,417 1,780 2,462 3,226 90 0,174 0,848 1,455 1,819 2,500 3,264
100 0,129 0,803 1,410 1,774 2,455 3,219 100 0,165 0,839 1,447 1,810 2,491 3,255
150 0,105 0,780 1,387 1,750 2,431 3,195 150 0,135 0,809 1,416 1,780 2,461 3,225
200 0,091 0,766 1,373 1,736 2,417 3,181 200 0,117 0,791 1,398 1,762 2,443 3,207
250 0,082 0,756 1,363 1,726 2,408 3,172 250 0,105 0,779 1,386 1,749 2,431 3,195
300 0,074 0,749 1,356 1,719 2,401 3,165 300 0,095 0,770 1,377 1,740 2,422 3,186
400 0,065 0,739 1,346 1,709 2,391 3,155 400 0,083 0,757 1,364 1,728 2,409 3,173
500 0,058 0,732 1,339 1,703 2,384 3,148 500 0,074 0,749 1,356 1,719 2,400 3,164
1 000 0,041 0,716 1,323 1,686 2,367 3,131 1 000 0,053 0,727 1,334 1,697 2,379 3,143
∞ 0,000 0,675 1,282 1,645 2,327 3,091 ∞ 0,000 0,675 1,282 1,645 2,327 3,091
Table B.5 — Confidence level 99,0 % Table B.6 — Confidence level 99,9 %
(1 − α = 0,99) (1 − α = 0,999)
p p
n n
0,50 0,75 0,90 0,95 0,99 0,999 0,50 0,75 0,90 0,95 0,99 0,999
2 1,645 2,320 2,927 3,290 3,972 4,736 2 2,186 2,860 3,467 3,830 4,512 5,276
3 1,344 2,018 2,625 2,988 3,670 4,434 3 1,785 2,459 3,066 3,430 4,111 4,875
4 1,164 1,838 2,445 2,809 3,490 4,254 4 1,546 2,220 2,827 3,190 3,872 4,636
5 1,041 1,715 2,322 2,686 3,367 4,131 5 1,382 2,057 2,664 3,027 3,709 4,473
6 0,950 1,625 2,232 2,595 3,277 4,040 6 1,262 1,937 2,544 2,907 3,588 4,352
7 0,880 1,554 2,161 2,525 3,206 3,970 7 1,168 1,843 2,450 2,813 3,495 4,259
8 0,823 1,497 2,105 2,468 3,149 3,913 8 1,093 1,768 2,375 2,738 3,419 4,183
9 0,776 1,450 2,058 2,421 3,102 3,866 9 1,031 1,705 2,312 2,675 3,357 4,121
10 0,736 1,411 2,018 2,381 3,063 3,826 10 0,978 1,652 2,259 2,623 3,304 4,068
11 0,702 1,376 1,983 2,347 3,028 3,792 11 0,932 1,607 2,214 2,577 3,259 4,022
12 0,672 1,347 1,954 2,317 2,998 3,762 12 0,893 1,567 2,174 2,537 3,219 3,983
13 0,646 1,320 1,927 2,291 2,972 3,736 13 0,858 1,532 2,139 2,502 3,184 3,948
14 0,622 1,297 1,904 2,267 2,949 3,712 14 0,826 1,501 2,108 2,471 3,153 3,917
15 0,601 1,276 1,883 2,246 2,928 3,691 15 0,798 1,473 2,080 2,443 3,125 3,889
16 0,582 1,257 1,864 2,227 2,908 3,672 16 0,773 1,448 2,055 2,418 3,099 3,863
17 0,565 1,239 1,846 2,210 2,891 3,655 17 0,750 1,424 2,032 2,395 3,076 3,840
18 0,549 1,223 1,830 2,194 2,875 3,639 18 0,729 1,403 2,010 2,374 3,055 3,819
19 0,534 1,209 1,816 2,179 2,861 3,624 19 0,709 1,384 1,991 2,354 3,036 3,800

20 0,521 1,195 1,802 2,166 2,847 3,611 20 0,691 1,366 1,973 2,336 3,018 3,782
22 0,496 1,171 1,778 2,141 2,823 3,587 22 0,659 1,334 1,941 2,304 2,986 3,750
24 0,475 1,150 1,757 2,120 2,802 3,566 24 0,631 1,306 1,913 2,276 2,958 3,722
26 0,457 1,131 1,738 2,102 2,783 3,547 26 0,607 1,281 1,888 2,251 2,933 3,697
28 0,440 1,115 1,722 2,085 2,766 3,530 28 0,584 1,259 1,866 2,229 2,911 3,675
30 0,425 1,100 1,707 2,070 2,752 3,515 30 0,565 1,239 1,846 2,210 2,891 3,655
35 0,394 1,068 1,675 2,039 2,720 3,484 35 0,523 1,197 1,804 2,168 2,849 3,613
40 0,368 1,043 1,650 2,013 2,695 3,459 40 0,489 1,164 1,771 2,134 2,
...

NORME ISO
INTERNATIONALE 16269-6
Première édition
2005-04-01
Version corrigée
2006-01-01
Interprétation statistique des données —
Partie 6:
Détermination des intervalles statistiques
de dispersion
Statistical interpretation of data —
Part 6: Determination of statistical tolerance intervals

Numéro de référence
©
ISO 2005
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ii © ISO 2005 – Tous droits réservés

Sommaire Page
Avant-propos. iv
Introduction . v
1 Domaine d'application. 1
2 Références normatives . 1
3 Termes, définitions et symboles . 1
3.1 Termes et définitions. 1
3.2 Symboles . 2
4 Méthodes . 3
4.1 Population normale avec une variance et une moyenne connues. 3
4.2 Population normale avec une variance connue et une moyenne inconnue. 3
4.3 Population normale avec une variance et une moyenne inconnues . 3
4.4 Distribution continue quelconque de type inconnu. 3
5 Exemples . 4
5.1 Données. 4
5.2 Exemple 1: intervalle statistique de dispersion unilatéral sous variance connue . 4
5.3 Exemple 2: intervalle statistique de dispersion bilatéral sous variance connue. 5
5.4 Exemple 3: intervalle statistique de dispersion unilatéral sous variance inconnue . 5
5.5 Exemple 4: intervalle statistique de dispersion bilatéral sous variance inconnue . 6
5.6 Exemple 5: intervalle statistique de dispersion non paramétrique pour une distribution
continue . 7
Annexe A (informative) Formulaires pour les intervalles de dispersion . 8
Annexe B (normative) Facteurs de la limite statistique de dispersion unilatérale, k (n; p; 1 − α), pour
un écart-type de la population, σ, connu. 14
Annexe C (normative) Facteurs de la limite statistique de dispersion bilatérale, k (n; p; 1 − α), pour
un écart-type de la population, σ, connu. 17
Annexe D (normative) Facteurs de la limite statistique de dispersion unilatérale, k (n; p; 1 − α), pour
un écart-type de la population, σ, inconnu . 20
Annexe E (normative) Facteurs de la limite statistique de dispersion bilatérale, k (n; p; 1 − α), pour
un écart-type de la population, σ, inconnu . 23
Annexe F (normative) Intervalles statistiques de dispersion unilatéraux non paramétriques . 26
Annexe G (normative) Intervalles statistiques de dispersion bilatéraux non paramétriques. 27
Annexe H (informative) Construction d'un intervalle statistique de dispersion non paramétrique
pour un type de distribution quelconque.28
Annexe I (informative) Calculs des facteurs des intervalles statistiques de dispersion bilatéraux
paramétriques . 29
Bibliographie . 30

Avant-propos
L'ISO (Organisation internationale de normalisation) est une fédération mondiale d'organismes nationaux de
normalisation (comités membres de l'ISO). L'élaboration des Normes internationales est en général confiée
aux comités techniques de l'ISO. Chaque comité membre intéressé par une étude a le droit de faire partie du
comité technique créé à cet effet. Les organisations internationales, gouvernementales et non
gouvernementales, en liaison avec l'ISO participent également aux travaux. L'ISO collabore étroitement avec
la Commission électrotechnique internationale (CEI) en ce qui concerne la normalisation électrotechnique.
Les Normes internationales sont rédigées conformément aux règles données dans les Directives ISO/CEI,
Partie 2.
La tâche principale des comités techniques est d'élaborer les Normes internationales. Les projets de Normes
internationales adoptés par les comités techniques sont soumis aux comités membres pour vote. Leur
publication comme Normes internationales requiert l'approbation de 75 % au moins des comités membres
votants.
L'attention est appelée sur le fait que certains des éléments du présent document peuvent faire l'objet de
droits de propriété intellectuelle ou de droits analogues. L'ISO ne saurait être tenue pour responsable de ne
pas avoir identifié de tels droits de propriété et averti de leur existence.
L'ISO 16269-6 a été élaborée par le comité technique ISO/TC 69, Application des méthodes statistiques.
Cette première édition de l'ISO 16269-6 annule et remplace l'ISO 3207:1975, qui a fait l'objet d'une révision
technique.
L'ISO 16269 comprend les parties suivantes, présentées sous le titre général Interprétation statistique des
données:
 Partie 6: Détermination des intervalles statistiques de dispersion
 Partie 7: Médiane — Estimation et intervalles de confiance
 Partie 8: Détermination des intervalles de prédiction
Dans la présente version corrigée, le terme «tolérance» a été remplacé par «dispersion» et les termes «limite
de spécification» ont été remplacés par «limite de tolérance» dans l'ensemble du document.
iv © ISO 2005 – Tous droits réservés

Introduction
Un intervalle statistique de dispersion est un intervalle estimé, d'après un échantillon, pour lequel il est
possible d'affirmer avec un niveau de confiance 1 − α, par exemple 95 %, qu'il contient au moins une
proportion donnée p d'individus de la population. Les limites d'un intervalle statistique de dispersion sont
appelées limites statistiques de dispersion. Le niveau de confiance 1 − α est la probabilité selon laquelle un
intervalle statistique de dispersion construit de la manière spécifiée contiendra au moins une proportion p
d'individus de la population. Inversement, la probabilité que cet intervalle contiendra moins que la proportion p
d'individus de la population est α. La présente partie de l'ISO 16269 décrit les intervalles statistiques de
dispersion unilatéraux et les intervalles statistiques de dispersion bilatéraux; un intervalle unilatéral est
construit avec une limite inférieure ou une limite supérieure tandis qu'un intervalle bilatéral est construit avec
une limite supérieure et une limite inférieure.
Les intervalles de dispersion sont fonction des observations de l'échantillon, c'est-à-dire des statistiques, et
leurs valeurs seront généralement différentes pour des échantillons différents. Il est nécessaire que les
observations soient indépendantes pour que les méthodes indiquées dans la présente partie de l'ISO 16269
soient valables.
La présente partie de l'ISO 16269 stipule deux types d'intervalles statistiques de dispersion: l'intervalle
paramétrique et l'intervalle non paramétrique. L'approche paramétrique se fonde sur l'hypothèse selon
laquelle la caractéristique étudiée dans la population a une distribution normale; ainsi, si l'hypothèse de
normalité est avérée, le niveau de confiance avec lequel l'intervalle statistique de dispersion contient au moins
une proportion p d'individus de la population ne peut être que de 1 − α. Pour les caractéristiques distribuées
normalement, l'intervalle statistique de dispersion est déterminé à l'aide des formulaires A, B, C et D donnés
dans l'Annexe A.
La présente partie de l'ISO 16269 ne traite pas des méthodes paramétriques s'appliquant à des distributions
autres que les distributions normales. Si des écarts par rapport à la normalité sont suspectés dans la
population, des intervalles statistiques de dispersion non paramétriques peuvent être construits. La procédure
de détermination d'un intervalle statistique de dispersion pour une distribution continue quelconque est
indiquée aux formulaires E et F de l'Annexe A.
Les limites de dispersion abordées dans la présente partie de l'ISO 16269 peuvent être utilisées pour
comparer l'aptitude naturelle d'un processus avec une ou deux limites de tolérance données, soit une limite
supérieure, U, soit une limite inférieure, L, ou encore les deux, dans la gestion d'un processus statistique.
Cela est indiqué par le fait que ces limites de dispersion ont également été appelées limites naturelles du
processus. Voir l'ISO 3534-2:1993, 3.2.4, ainsi que les remarques générales de l'ISO 3207, qui sera annulée
et remplacée par la présente partie de l'ISO 16269.
Au-dessus de la limite de tolérance supérieure, U, il y a la fraction supérieure non conforme, p
U
(ISO 3534-2:—, 3.2.5.5 et 3.3.1.4), et en dessous de la limite de tolérance inférieure, L, il y a la fraction
inférieure non conforme, p (ISO 3534-2:—, 3.2.5.6 et 3.3.1.5). La somme p + p = p est appelée fraction
L U L T
totale non conforme (ISO 3534-2:—, 3.2.5.7). Entre les limites de tolérance U et L, il y a la fraction conforme
1 – p .
T
Dans la gestion du processus statistique, les limites U et L sont fixées à l'avance et les fractions p , p et p
U L T
sont soit calculées, lorsque la distribution est supposée connue, soit estimées. Il existe beaucoup
d'applications d'intervalles statistiques de dispersion, bien que l'exemple ci-dessus montre un exemple d'un
problème de contrôle qualité. Des applications plus importantes et plus d'intervalles statistiques sont introduits
[10]
dans de nombreux ouvrages tels que Hahn et Meeker .
Par contraste, pour les intervalles de dispersion dont il est question dans la présente partie de l'ISO 16269, le
niveau de confiance pour l'estimateur d'intervalle et la proportion de distribution dans l'intervalle
(correspondant à la fraction conforme mentionnée ci-dessus) sont fixés à l'avance, et les limites sont estimées.
Ces limites peuvent être comparées à U et à L. Ainsi la justesse des limites de tolérance données U et L peut
être comparée aux propriétés réelles du processus. Les intervalles de dispersion unilatéraux sont utilisés
uniquement lorsque la limite de tolérance supérieure, U, ou la limite de tolérance inférieure, L, est appropriée,
tandis que les intervalles bilatéraux sont utilisés lorsque les limites supérieure et inférieure sont prises en
compte simultanément.
La terminologie relative à ces limites et intervalles différents est confuse car les «limites de tolérance» étaient
également autrefois appelées «limites de dispersion» (voir la Norme de terminologie ISO 3534-2:1993, 1.4.3,
où ces deux termes, mais aussi le terme «valeurs limites», étaient utilisés comme synonymes pour désigner
ce concept). Dans la dernière révision de l'ISO 3534-2:—, seul le terme «limites de tolérance» a été conservé
[5]
pour désigner ce concept. En outre, le Guide pour l'expression de l'incertitude de mesure (GUM) utilise le
terme «facteur d'élargissement», défini comme un «facteur numérique utilisé comme multiplicateur de
l'incertitude-type composée pour obtenir l'incertitude élargie». Cette utilisation du terme «élargissement» est
différente de celle de la présente partie de l'ISO 16269.

vi © ISO 2005 – Tous droits réservés

NORME INTERNATIONALE ISO 16269-6:2005(F)

Interprétation statistique des données —
Partie 6:
Détermination des intervalles statistiques de dispersion
1 Domaine d'application
La présente partie de l'ISO 16269 décrit des méthodes permettant d'établir les intervalles statistiques de
dispersion qui comprennent au moins une proportion spécifiée de la population avec un niveau de confiance
spécifié. Des intervalles statistiques de dispersion unilatéraux et bilatéraux sont fournis, l'intervalle statistique
de dispersion unilatéral étant caractérisé par une limite supérieure ou par une limite inférieure, tandis que
l'intervalle statistique bilatéral possède à la fois une limite supérieure et une limite inférieure. Deux méthodes
sont exposées: une méthode paramétrique, lorsque la caractéristique étudiée a une distribution normale, et
une méthode non paramétrique, lorsque rien n'est connu de la distribution si ce n'est qu'elle est continue.
2 Références normatives
Les documents de référence suivants sont indispensables pour l'application du présent document. Pour les
références datées, seule l'édition citée s'applique. Pour les références non datées, la dernière édition du
document de référence s'applique (y compris les éventuels amendements).
ISO 3534-1, Statistique — Vocabulaire et symboles — Partie 1: Probabilité et termes statistiques généraux
1)
ISO 3534-2:— , Statistique — Vocabulaire et symboles — Partie 2: Statistique appliquée
3 Termes, définitions et symboles
3.1 Termes et définitions
Pour les besoins du présent document, les termes et définitions donnés dans l'ISO 3534-1 et l'ISO 3534-2
ainsi que les suivants s'appliquent.
3.1.1
intervalle statistique de dispersion
intervalle déterminé à partir d'un échantillon prélevé au hasard de manière qu'à un niveau de confiance
spécifié, l'intervalle couvre au moins une proportion spécifiée de la population échantillonnée
NOTE Dans ce contexte, le niveau de confiance est la proportion à long terme d'intervalles construits de cette
manière qui comprendra au moins la proportion également donnée de la population échantillonnée.
3.1.2
limite statistique de dispersion
statistique représentant un point limite d'un intervalle statistique de dispersion
NOTE Les intervalles statistiques de dispersion peuvent être soit unilatéraux, auquel cas ils ont une limite statistique
de dispersion supérieure ou inférieure, soit bilatéraux, auquel cas ils possèdent les deux limites.

1) À publier. (Révision de l'ISO 3534-2:1993)
3.1.3
élargissement
proportion des individus d'une population se trouvant dans un intervalle statistique de dispersion
NOTE Ce concept est à ne pas confondre avec le concept de facteur d'élargissement utilisé dans le Guide pour
[5]
l'expression de l'incertitude de mesure (GUM) .
3.1.4
population normale
population distribuée normalement
3.2 Symboles
Pour les besoins de la présente partie de l'ISO 16269, les symboles suivants s'appliquent.
i suffixe d'une observation
k (n; p; 1 − α) facteur utilisé pour déterminer x ou x lorsque la valeur de σ est connue pour un intervalle
1 L U
de dispersion unilatéral
k (n; p; 1 − α) facteur utilisé pour déterminer x et x lorsque la valeur de σ est connue pour un intervalle de
2 L U
dispersion bilatéral
k (n; p; 1 − α) facteur utilisé pour déterminer x ou x lorsque la valeur de σ est inconnue pour un intervalle
3 L U
de dispersion unilatéral
k (n; p; 1 − α) facteur utilisé pour déterminer x et x lorsque la valeur de σ est inconnue pour un intervalle
4 L U
de dispersion bilatéral
n nombre d'observations dans l'échantillon
p proportion minimale de la population déclarée comme se trouvant dans l'intervalle statistique
de dispersion
u fractile d'ordre p de la distribution normale réduite
p
ème
x i valeur observée (i = 1, 2, …, n)
i
x valeur maximale des valeurs observées, x = max {x , x , …, x }
max max 1 2 n
x valeur minimale des valeurs observées, x = min {x , x , …, x }
min min 1 2 n
x limite inférieure de l'intervalle statistique de dispersion
L
x limite supérieure de l'intervalle statistique de dispersion
U
n
x moyenne de l'échantillon, xx=
∑ i
n
i = 1
nn

nx − x
∑∑ii
n 
1 ii==11

s écart-type de l'échantillon; sx x
=−()=
∑ i
nn−−11n
()
i = 1
2 © ISO 2005 – Tous droits réservés

1 − α niveau de confiance de la déclaration selon laquelle la proportion de la population se
trouvant dans l'intervalle de dispersion est supérieure ou égale au niveau spécifié p
µ moyenne de la population
σ écart-type de la population
4 Méthodes
4.1 Population normale avec une variance et une moyenne connues
Lorsque les valeurs de la moyenne, µ, et de la variance, σ , d'une population normalement distribuée sont
connues, la distribution de la caractéristique étudiée est complètement déterminée. Il y a exactement une
proportion p de la population:
a) à la droite de x = µ − u × σ (intervalle unilatéral);
L p
b) à la gauche de x = µ + u × σ (intervalle unilatéral);
U p
c) entre x = µ − u × σ et x = µ + u × σ (intervalle bilatéral).
L (1 + p)/2 U (1 + p)/2
NOTE Dans la mesure où l'on sait que ces déclarations sont justes, elles sont faites avec un niveau de confiance de
100 %.
Dans les équations ci-dessus, u est le fractile d'ordre p de la distribution normale réduite. Les valeurs
p
numériques de u sont indiquées aux dernières lignes des Tableaux B.1 à B.6 et C.1 à C.6.
p
4.2 Population normale avec une variance connue et une moyenne inconnue
Les formulaires A et B, donnés dans l'Annexe A, sont applicables lorsque la variance de la population
normale est connue alors que la moyenne est inconnue. Le formulaire A s'applique aux cas unilatéraux tandis
que le formulaire B s'applique aux cas bilatéraux.
4.3 Population normale avec une variance et une moyenne inconnues
Les formulaires C et D, donnés dans l'Annexe A, sont applicables lorsque la moyenne et la variance de la
population normale sont inconnues. Le formulaire C s'applique aux cas unilatéraux tandis que le formulaire D
s'applique aux cas bilatéraux.
4.4 Distribution continue quelconque de type inconnu
Si la caractéristique à l'étude est une variable continue provenant d'une population de forme inconnue et si un
échantillon de n observations aléatoires et indépendantes de la caractéristique a été prélevé, alors un
intervalle statistique de dispersion peut être déterminé à partir des observations ordonnées. La méthode
indiquée aux formulaires E et F de l'Annexe A permet de déterminer l'élargissement ou l'effectif de
l'échantillon nécessaires aux intervalles de dispersion déterminés à partir des valeurs extrêmes x et x
min max
de l'échantillon d'observations avec un niveau de confiance de 1 − α.
NOTE Les intervalles statistiques de dispersion qui ne sont pas fonction de la forme de la population échantillonnée
sont appelés intervalles de dispersion non paramétriques.
La présente partie de l'ISO 16269 ne préconise pas de méthode pour les distributions d'un type connu autre
que la distribution normale. Toutefois, si la distribution est continue, la méthode non paramétrique peut être
utilisée. Une sélection de références à de la littérature scientifique pouvant aider à déterminer les intervalles
de dispersion pour d'autres distributions est aussi fournie à la fin de ce document.
5 Exemples
5.1 Données
Les formulaires A à D, donnés dans l'Annexe A, sont illustrés par des exemples à l'aide des valeurs
numériques de l'ISO 2854:1976, Article 2, paragraphe 1 des remarques introductives, Tableau X, fil 2:
12 mesures de la charge de rupture du fil en coton. Il convient de noter que le nombre d'observations, n = 12,
[1]
indiqué pour ces exemples, est considérablement inférieur à celui recommandé dans l'ISO 2602 . L'unité de
mesure pour exprimer les données numériques et les calculs dans les différents exemples est le centinewton
(voir Tableau 1).
Tableau 1 — Données pour les Exemples 1 à 4
Valeurs en centinewtons
x 228,6 232,7 238,8 317,2 315,8 275,1 222,2 236,7 224,7 251,2 210,4 270,7
Ces mesures proviennent d'un lot de 12 000 bobines, d'une même série de fabrication, emballées dans
120 boîtes contenant chacune 100 bobines. Douze boîtes de ce lot ont été prélevées au hasard et une bobine
a été prise au hasard dans chacune de ces boîtes. Des éprouvettes de 50 cm de long ont été découpées
dans le fil de ces bobines, à environ 5 m de l'extrémité libre. Les essais proprement dits ont été réalisés sur
les parties centrales de ces éprouvettes. Des informations antérieures permettent de penser raisonnablement
que les charges de rupture mesurées dans ces conditions ont une distribution pratiquement normale. Il est
démontré, dans l'ISO 2954:1976, que les données ne contredisent pas l'hypothèse d'une distribution normale.
Les résultats suivants sont produits:
Effectif de l'échantillon: n = 12
Moyenne
de l'échantillon: x/==3 024,1 12 252,01
nx −()x
166 772,27
∑∑
Écart-type de l'échantillon: s== 1= 263,4263=35,545
nn (1−×) 12 11
La présentation formelle des calculs sera donnée uniquement pour le formulaire C de l'Annexe A (intervalle
unilatéral, variance inconnue).
5.2 Exemple 1: intervalle statistique de dispersion unilatéral sous variance connue
Supposer que les mesures obtenues précédemment montrent que la dispersion est constante d'un lot à
l'autre, provenant du même fournisseur, et qu'elle est représentée par un écart-type σ = 33,150, bien que la
moyenne ne soit pas constante. Une limite x est requise de manière à ce qu'il soit possible d'affirmer avec un
L
niveau de confiance 1 − α = 0,95 (95 %) qu'au moins 0,95 (95 %) des charges de rupture des individus du lot,
mesurées dans les mêmes conditions, sont supérieures à x .
L
La Tableau B.4 donne
k (12; 0,95; 0,95) = 2,120
d'où
xx=−k (n; ; p 1−ασ)× =252,01−2,120×33,150=181,732
L 1
4 © ISO 2005 – Tous droits réservés

Une plus petite valeur de la limite inférieure, x , serait obtenue si une proportion plus importante de la
L
population (par exemple, p = 0,99) et/ou un niveau de confiance supérieur (par exemple 1 − α = 0,99) étaient
requis.
5.3 Exemple 2: intervalle statistique de dispersion bilatéral sous variance connue
Dans des conditions identiques à celles de l'Exemple 1, supposer que les limites x et x sont requises de
L U
manière qu'il soit possible d'affirmer avec un niveau de confiance 1 − α = 0,95 qu'au moins une proportion de
p = 0,90 (90 %) des charges de rupture du lot se situe entre x et x .
L U
Le Tableau C.4 donne
k (12; 0,90; 0,95) = 1,889
d'où
xx=−k (n;p;1−ασ)× = 252,01−1,889× 33,150= 189,390
L 2
xx=+k (n;p;1−ασ)× = 252,01+1,889× 33,150= 314,630
U 2
Il convient que la comparaison de cet exemple avec l'Exemple 1 fasse bien comprendre la différence entre le
fait de garantir qu'au moins 90 % d'une population se situe entre les limites x et x et le fait de garantir qu'au
L U
plus 5 % se trouve au-delà de ces limites.
5.4 Exemple 3: intervalle statistique de dispersion unilatéral sous variance inconnue
Il est ici supposé que l'écart-type de la population est inconnu et doit être estimé à partir de l'échantillon. Les
mêmes exigences seront supposées que pour le cas où l'écart-type est connu (Exemple 1); ainsi, p = 0,95 et
1 − α = 0,95. La présentation détaillée des résultats est donnée ci-après.
Détermination de l'intervalle statistique de dispersion de la proportion p
a) Intervalle unilatéral «à droite»
Valeurs déterminées:
b) proportion de la population choisie pour l'intervalle de dispersion: p = 0,95
c) niveau de confiance choisi: 1 − α = 0,95
d) effectif de l'échantillon: n = 12
Valeur du facteur de dispersion, provenant du Tableau D.4:
k (n; p; 1 − α) = 2,737
Calculs:
xx==/n 252,01
∑
nx −()x
∑∑
s== 35,545
nn(1−)
k (n; p; 1 − α) × s = 97,286 7
Résultats: intervalle unilatéral «à droite»
L'intervalle de dispersion qui contiendra au moins une proportion p de la population avec un niveau de
confiance 1 − α a une limite inférieure:
xx=−k n; ; p 1−α×s=154,723
( )
L 3
5.5 Exemple 4: intervalle statistique de dispersion bilatéral sous variance inconnue
Dans les mêmes conditions que dans l'Exemple 2, supposer qu'il est requis de calculer les limites x et x de
L U
manière qu'il soit possible d'affirmer avec un niveau de confiance 1 − α = 0,95 que dans une proportion du lot
au moins égale à p = 0,90 (90 %), la charge de rupture est comprise entre x et x .
L U
La Tableau E.4 donne
kn( ; ;p 1−=α) 2,671
d'où
xx=−k (n; ; p 1− α)×s=252,01−2,671×35,545=157,069
L 4
xx=+k n; ; p 1−α×s=252,01+2,671×35,545=346,951
( )
U 4
Noter que la valeur de x est inférieure et que la valeur de x est supérieure à celles de l'Exemple 2 (variance
L U
connue) parce que l'utilisation de s à la place de σ nécessite une valeur de la constante de dispersion plus
importante pour tenir compte de l'incertitude supplémentaire. Il est nécessaire d'assumer le fait de ne pas
connaître l'écart-type de la population, σ; et l'extension de l'intervalle statistique de dispersion prend cela en
6 © ISO 2005 – Tous droits réservés

compte. Bien sûr, il n'est pas certain que la valeur σ = 33,150 utilisée dans les Exemples 1 et 2 soit correcte.
Donc, il est plus sage d'utiliser l'estimation, s, avec les Tableaux D.4 ou E.4.
5.6 Exemple 5: intervalle statistique de dispersion non paramétrique pour une distribution
continue
Lors d'un essai de fatigue par flexions rotatives, réalisé sur un composant d'un engin aéronautique, un
échantillon de 15 individus a donné les résultats du Tableau 2 (mesurage de l'endurance), classés par ordre
croissant.
Tableau 2 — Données de l'Exemple 5
x 0,200 0,330 0,450 0,490 0,780 0,920 0,950 0,970 1,040 1,710 2,220 2,275 3,650 7,000 8,800
Un examen graphique permettant de vérifier la normalité, comme une courbe de probabilité, montre qu'il
convient quasi certainement de rejeter l'hypothèse de normalité de la population de composants (voir
l'ISO 5479). Les méthodes du formulaire E, donné dans l'Annexe A, pour la détermination de l'intervalle
statistique de dispersion sont par conséquent applicables.
Les valeurs extrêmes de l'échantillon n = 15 mesures sont:
x = 0,200, x = 8,800.
min max
Supposer que le niveau de confiance 1 − α requis est égal à 0,95.
a) Quelle proportion maximale de la population des composants sera inférieure à x = 0,200?
min
Le Tableau F.1, pour 1 − α = 0,95, donne pour la proportion minimale supérieure à x une valeur de p
min
légèrement supérieure à 0,75 (75 %) et donc, pour la proportion maximale inférieure à x , une valeur de
min
1 − p légèrement inférieure à 0,25 (25 %).
b) Quel effectif d'échantillon est nécessaire pour pouvoir affirmer, à un niveau de confiance de 0,95, qu'une
proportion d'au moins p = 0,90 (90 %) de la population des composants sera inférieure à la plus grande
des valeurs de cet échantillon?
Le Tableau F.1, pour 1 − α = 0,95 et p = 0,90, donne n = 29.
c) À un niveau de confiance de 0,95, quelle est la proportion minimale de la population de composants
comprise entre x = 0,200 et x = 8,800?
min max
Le Tableau G.1, pour 1 − α = 0,95 et n = 15, donne p légèrement inférieur à 0,75 (75 %).
d) Quel effectif d'échantillon est nécessaire pour pouvoir affirmer, à un niveau de confiance de 0,95, qu'une
proportion d'au moins p = 0,90 (90 %) de la population des composants sera comprise entre la plus petite
et la plus grande des valeurs obtenues à partir de cet échantillon?
Le Tableau G.1, pour 1 − α = 0,95 et p = 0,90, donne n = 46.
e) En général, si une vérification de la normalité (voir l'ISO 5479) indique une déviation par rapport à la
distribution normale, il est recommandé d'effectuer des transformations sur la base de la connaissance
des données collectées. Par exemple, les données de fatigue sont souvent approximées selon une
distribution lognormale. Dans de tels cas, les données peuvent être transformées à la normalité. Les
intervalles de dispersion sont alors calculés et finalement retransformés dans les unités d'origine.
Voir l'Annexe H pour la construction d'un intervalle statistique de dispersion non paramétrique pour un type de
distribution quelconque. L'Annexe I donne les calculs des facteurs des intervalles statistiques de dispersion
bilatéraux paramétriques.
Annexe A
(informative)
Formulaires pour les intervalles de dispersion
Formulaire A — Intervalle statistique de dispersion unilatéral (variance connue)
Détermination d'un intervalle statistique de dispersion unilatéral avec un élargissement p à un niveau de
confiance 1 − α
a) Intervalle unilatéral «à gauche»
b) Intervalle unilatéral «à droite»
Valeurs connues:
c) la variance: σ =
d) l'écart-type: σ =
Valeurs déterminées:
e) proportion de la population choisie pour l'intervalle de dispersion: p =
f) niveau de confiance choisi: 1 − α =
g) effectif de l'échantillon: n =
Facteur tabulé:
k (n; p; 1 − α) =
Cette valeur peut être obtenue à partir des tableaux de l'Annexe B pour différentes valeurs de n, de p et de
1 − α.
Calculs:
xx==/n
∑
k (n; p; 1 − α) × σ =
Résultats:
a) Intervalle unilatéral «à gauche»
L'intervalle statistique de dispersion unilatéral avec un élargissement p à un niveau de confiance 1 − α
a une limite supérieure:
xx=+k (;n ;p 1−α)×σ=
U 1
b) Intervalle unilatéral «à droite»
L'intervalle statistique de dispersion unilatéral avec un élargissement p à un niveau de confiance 1 − α
a une limite inférieure:
xx=−k (;n ;p 1− α)×σ=
L 1
8 © ISO 2005 – Tous droits réservés

Formulaire B — Intervalle statistique de dispersion bilatéral (variance connue)
Détermination d'un intervalle statistique de dispersion bilatéral avec un élargissement p à un niveau de
confiance 1 − α
Valeurs connues:
a) la variance: σ =
b) l'écart-type: σ =
Valeurs déterminées:
c) proportion de la population choisie pour l'intervalle de dispersion: p =
d) niveau de confiance choisi: 1 − α =
e) effectif de l'échantillon: n =
Facteur tabulé:
k (n; p; 1 − α) =
Cette valeur peut être obtenue à partir des tableaux de l'Annexe C pour différentes valeurs de n, de p et de
1 − α.
Calculs:
xx==/n
∑
k (n; p; 1 − α) × σ =
Résultats:
L'intervalle de dispersion statistique bilatéral avec un élargissement p à un niveau de confiance 1 − α a les
limites:
xx=−k (;n ;p 1−α)×σ=
L 2
xx=+k (;n ;p 1−α)×σ=
U 2
Formulaire C — Intervalle statistique de dispersion unilatéral (variance inconnue)
Détermination d'un intervalle statistique de dispersion unilatéral avec un élargissement p à un niveau de
confiance 1 − α
a) Intervalle unilatéral «à gauche»
b) Intervalle unilatéral «à droite»
Valeurs déterminées:
c) proportion de la population choisie pour l'intervalle de dispersion: p =
d) niveau de confiance choisi: 1 − α =
e) effectif de l'échantillon: n =
Facteur tabulé:
k (n; p; 1 − α) =
Cette valeur peut être obtenue à partir des tableaux de l'Annexe D pour différentes valeurs de n, de p et de
1 − α.
Calculs:
xx==/n
∑
nx −()x
∑∑
s==
nn −1
()
k (n; p; 1 − α) × s =
Résultats:
a) Intervalle unilatéral «à gauche»
L'intervalle de dispersion avec un élargissement p à un niveau de confiance 1 − α a une limite
supérieure:
xx=+k (;n ;p 1−α)×s=
U 3
b) Intervalle unilatéral «à droite»
L'intervalle de dispersion avec un élargissement p à un niveau de confiance 1 − α a une limite
inférieure:
xx=−k (;n ;p 1− α)×s=
L 3
10 © ISO 2005 – Tous droits réservés

Formulaire D — Intervalle statistique de dispersion bilatéral (variance inconnue)
Détermination d'un intervalle statistique de dispersion bilatéral avec un élargissement p à un niveau de
confiance 1 − α
Valeurs déterminées:
a) proportion de la population choisie pour l'intervalle de dispersion: p =
b) niveau de confiance choisi: 1 − α =
c) effectif de l'échantillon: n =
Facteur tabulé:
k (n; p; 1 − α) =
Cette valeur peut être obtenue à partir des tableaux donnés à l'Annexe E pour différentes valeurs de n, de p
et de 1 − α.
Calculs:
xx==/n
∑ i
nx −()x
∑∑
s==
nn(1−)
k (n; p; 1 − α) × s =
Résultats:
L'intervalle statistique de dispersion bilatéral avec un élargissement p à un niveau de confiance 1 − α a les
limites:
xx=−k n; ; p 1− α×s=
( )
L 4
xx=+k n; ; p 1− α×s=
( )
U 4
Formulaire E — Intervalle statistique de dispersion unilatéral pour une distribution quelconque
Détermination d'un intervalle statistique de dispersion unilatéral non paramétrique avec un élargissement p
à un niveau de confiance 1 − α
a) Intervalle unilatéral «à gauche»
b) Intervalle unilatéral «à droite»
Valeurs déterminées:
c) proportion de la population choisie pour l'intervalle de dispersion: p =
d) niveau de confiance choisi: 1 − α =
e) effectif de l'échantillon: n =
(Déterminer soit p soit n)
Valeur tabulée:
 p pour un n donné et 1 − α;
 n pour un p donné et 1 − α.
Cette valeur peut être obtenue à partir du Tableau F.1 pour différentes valeurs de n, de p et de 1 − α.
Calculs et résultats
L'intervalle statistique de dispersion unilatéral avec un élargissement p à un niveau de confiance 1 − α a:
 soit une limite inférieure x = x =
L min
 soit une limite supérieure x = x =
U max
12 © ISO 2005 – Tous droits réservés

Formulaire F — Intervalle statistique de dispersion bilatéral pour une distribution quelconque
Détermination d'un intervalle statistique de dispersion bilatéral non paramétrique avec un élargissement p
à un niveau de confiance 1 − α
Valeurs déterminées:
a) proportion de la population choisie pour l'intervalle de dispersion: p =
b) niveau de confiance choisi: 1 − α =
c) effectif de l'échantillon: n =
(Déterminer soit p soit n)
Valeur tabulée:
 p pour un n donné et 1 − α;
 n pour un p donné et 1 − α.
Cette valeur peut être obtenue à partir du Tableau G.1 pour différentes valeurs de n, de p et de 1 − α.
Calculs et résultats
L'intervalle statistique de dispersion bilatéral avec un élargissement p à un niveau de confiance 1 − α a:
 une limite inférieure x = x =
L min
 et une limite supérieure x = x =
U max
Annexe B
(normative)
Facteurs de la limite statistique de dispersion unilatérale, k (n; p; 1 −−−− α),
pour un écart-type de la population, σ, connu
Tableau B.1 — Niveau de confiance Tableau B.2 — Niveau de confiance
de 50,0 % (1 − α = 0,50) de 75,0 % (1 − α = 0,75)
p p
n n
0,50 0,75 0,90 0,95 0,99 0,999 0,50 0,75 0,90 0,95 0,99 0,999
2 0,000 0,675 1,282 1,645 2,327 3,091 2 0,477 1,152 1,759 2,122 2,804 3,568
3 0,000 0,675 1,282 1,645 2,327 3,091 3 0,390 1,064 1,671 2,035 2,716 3,480
4 0,000 0,675 1,282 1,645 2,327 3,091 4 0,338 1,012 1,619 1,983 2,664 3,428
5 0,000 0,675 1,282 1,645 2,327 3,091 5 0,302 0,977 1,584 1,947 2,628 3,392
6 0,000 0,675 1,282 1,645 2,327 3,091 6 0,276 0,950 1,557 1,921 2,602 3,366
7 0,000 0,675 1,282 1,645 2,327 3,091 7 0,255 0,930 1,537 1,900 2,582 3,346
8 0,000 0,675 1,282 1,645 2,327 3,091 8 0,239 0,913 1,521 1,884 2,565 3,329
9 0,000 0,675 1,282 1,645 2,327 3,091 9 0,225 0,900 1,507 1,870 2,552 3,316
10 0,000 0,675 1,282 1,645 2,327 3,091 10 0,214 0,888 1,495 1,859 2,540 3,304
11 0,000 0,675 1,282 1,645 2,327 3,091 11 0,204 0,878 1,485 1,849 2,530 3,294
12 0,000 0,675 1,282 1,645 2,327 3,091 12 0,195 0,870 1,477 1,840 2,522 3,285
13 0,000 0,675 1,282 1,645 2,327 3,091 13 0,188 0,862 1,469 1,832 2,514 3,278
14 0,000 0,675 1,282 1,645 2,327 3,091 14 0,181 0,855 1,462 1,826 2,507 3,271
15 0,000 0,675 1,282 1,645 2,327 3,091 15 0,175 0,849 1,456 1,820 2,501 3,265
16 0,000 0,675 1,282 1,645 2,327 3,091 16 0,169 0,844 1,451 1,814 2,495 3,259
17 0,000 0,675 1,282 1,645 2,327 3,091 17 0,164 0,839 1,446 1,809 2,490 3,254
18 0,000 0,675 1,282 1,645 2,327 3,091 18 0,159 0,834 1,441 1,804 2,486 3,250
19 0,000 0,675 1,282 1,645 2,327 3,091 19 0,155 0,830 1,437 1,800 2,482 3,245

20 0,000 0,675 1,282 1,645 2,327 3,091 20 0,151 0,826 1,433 1,796 2,478 3,242
22 0,000 0,675 1,282 1,645 2,327 3,091 22 0,144 0,819 1,426 1,789 2,471 3,235
24 0,000 0,675 1,282 1,645 2,327 3,091 24 0,138 0,813 1,420 1,783 2,465 3,228
26 0,000 0,675 1,282 1,645 2,327 3,091 26 0,133 0,807 1,414 1,778 2,459 3,223
28 0,000 0,675 1,282 1,645 2,327 3,091 28 0,128 0,802 1,410 1,773 2,454 3,218
30 0,000 0,675 1,282 1,645 2,327 3,091 30 0,124 0,798 1,405 1,768 2,450 3,214
35 0,000 0,675 1,282 1,645 2,327 3,091 35 0,115 0,789 1,396 1,759 2,441 3,205
40 0,000 0,675 1,282 1,645 2,327 3,091 40 0,107 0,782 1,389 1,752 2,433 3,197
45 0,000 0,675 1,282 1,645 2,327 3,091 45 0,101 0,776 1,383 1,746 2,427 3,191
50 0,000 0,675 1,282 1,645 2,327 3,091 50 0,096 0,770 1,377 1,741 2,422 3,186
60 0,000 0,675 1,282 1,645 2,327 3,091 60 0,088 0,762 1,369 1,732 2,414 3,178
70 0,000 0,675 1,282 1,645 2,327 3,091 70 0,081 0,756 1,363 1,726 2,407 3,171
80 0,000 0,675 1,282 1,645 2,327 3,091 80 0,076 0,750 1,357 1,721 2,402 3,166
90 0,000 0,675 1,282 1,645 2,327 3,091 90 0,072 0,746 1,353 1,716 2,398 3,162
100 0,000 0,675 1,282 1,645 2,327 3,091 100 0,068 0,742 1,350 1,713 2,394 3,158
150 0,000 0,675 1,282 1,645 2,327 3,091 150 0,056 0,730 1,337 1,700 2,382 3,146
200 0,000 0,675 1,282 1,645 2,327 3,091 200 0,048 0,723 1,330 1,693 2,375 3,138
250 0,000 0,675 1,282 1,645 2,327 3,091 250 0,043 0,718 1,325 1,688 2,370 3,133
300 0,000 0,675 1,282 1,645 2,327 3,091 300 0,039 0,714 1,321 1,684 2,366 3,130
400 0,000 0,675 1,282 1,645 2,327 3,091 400 0,034 0,709 1,316 1,679 2,361 3,124
500 0,000 0,675 1,282 1,645 2,327 3,091 500 0,031 0,705 1,312 1,676 2,357 3,121
1 000 0,000 0,675 1,282 1,645 2,327 3,091 1 000 0,022 0,696 1,303 1,667 2,348 3,112
∞ 0,000 0,675 1,282 1,645 2,327 3,091 ∞ 0,000 0,675 1,282 1,645 2,327 3,091
14 © ISO 2005 – Tous droits réservés

Tableau B.3 — Niveau de confiance Tableau B.4 — Niveau de confiance
de 90,0 % (1 − α = 0,90) de 95,0 % (1 − α = 0,95)
p p
n n
0,50 0,75 0,90 0,95 0,99 0,999 0,50 0,75 0,90 0,95 0,99 0,999
2 0,907 1,581 2,188 2,552 3,233 3,997 2 1,164 1,838 2,445 2,808 3,490 4,254
3 0,740 1,415 2,022 2,385 3,067 3,831 3 0,950 1,625 2,232 2,595 3,277 4,040
4 0,641 1,316 1,923 2,286 2,968 3,732 4 0,823 1,497 2,104 2,468 3,149 3,913
5 0,574 1,248 1,855 2,218 2,900 3,664 5 0,736 1,411 2,018 2,381 3,062 3,826
6 0,524 1,198 1,805 2,169 2,850 3,614 6 0,672 1,346 1,954 2,317 2,998 3,762
7 0,485 1,159 1,766 2,130 2,811 3,575 7 0,622 1,297 1,904 2,267 2,949 3,712
8 0,454 1,128 1,735 2,098 2,780 3,544 8 0,582 1,257 1,864 2,227 2,908 3,672
9 0,428 1,102 1,709 2,073 2,754 3,518 9 0,549 1,223 1,830 2,194 2,875 3,639
10 0,406 1,080 1,687 2,051 2,732 3,496 10 0,521 1,195 1,802 2,166 2,847 3,611
11 0,387 1,061 1,668 2,032 2,713 3,477 11 0,496 1,171 1,778 2,141 2,823 3,587
12 0,370 1,045 1,652 2,015 2,697 3,461 12 0,475 1,150 1,757 2,120 2,802 3,566
13 0,356 1,030 1,637 2,001 2,682 3,446 13 0,457 1,131 1,738 2,102 2,783 3,547
14 0,343 1,017 1,625 1,988 2,669 3,433 14 0,440 1,115 1,722 2,085 2,766 3,530
15 0,331 1,006 1,613 1,976 2,658 3,422 15 0,425 1,100 1,707 2,070 2,752 3,515
16 0,321 0,995 1,602 1,966 2,647 3,411 16 0,412 1,086 1,693 2,057 2,738 3,502
17 0,311 0,986 1,593 1,956 2,638 3,402 17 0,399 1,074 1,681 2,044 2,726 3,490
18 0,303 0,977 1,584 1,947 2,629 3,393 18 0,388 1,063 1,670 2,033 2,715 3,478
19 0,295 0,969 1,576 1,939 2,621 3,385 19 0,378 1,052 1,659 2,023 2,704 3,468

20 0,287 0,962 1,569 1,932 2,613 3,377 20 0,368 1,043 1,650 2,013 2,695 3,459
22 0,274 0,948 1,555 1,919 2,600 3,364 22 0,351 1,026 1,633 1,996 2,678 3,441
24 0,262 0,937 1,544 1,907 2,588 3,352 24 0,336 1,011 1,618 1,981 2,663 3,426
26 0,252 0,926 1,533 1,897 2,578 3,342 26 0,323 0,998 1,605 1,968 2,649 3,413
28 0,243 0,917 1,524 1,888 2,569 3,333 28 0,311 0,986 1,593 1,956 2,638 3,402
30 0,234 0,909 1,516 1,879 2,561 3,325 30 0,301 0,975 1,582 1,946 2,627 3,391
35 0,217 0,892 1,499 1,862 2,543 3,307 35 0,279 0,953 1,560 1,923 2,605 3,369
40 0,203 0,878 1,485 1,848 2,529 3,293 40 0,261 0,935 1,542 1,905 2,587 3,351
45 0,192 0,866 1,473 1,836 2,518 3,282 45 0,246 0,920 1,527 1,891 2,572 3,336
50 0,182 0,856 1,463 1,827 2,508 3,272 50 0,233 0,908 1,515 1,878 2,559 3,323
60 0,166 0,840 1,447 1,811 2,492 3,256 60 0,213 0,887 1,494 1,858 2,539 3,303
70 0,154 0,828 1,435 1,799 2,480 3,244 70 0,197 0,872 1,479 1,842 2,523 3,287
80 0,144 0,818 1,425 1,789 2,470 3,234 80 0,184 0,859 1,466 1,829 2,511
...