IEC 60493-1:2011

IEC 60493-1 ®
Edition 2.0 2011-12
INTERNATIONAL
STANDARD
NORME
INTERNATIONALE
Guide for the statistical analysis of ageing test data –
Part 1: Methods based on mean values of normally distributed test results

Guide pour l’analyse statistique de données d’essais de vieillissement –
Partie 1: Méthodes basées sur les valeurs moyennes de résultats d’essais
normalement distribués
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IEC 60493-1 ®
Edition 2.0 2011-12
INTERNATIONAL
STANDARD
NORME
INTERNATIONALE
Guide for the statistical analysis of ageing test data –
Part 1: Methods based on mean values of normally distributed test results

Guide pour l’analyse statistique de données d’essais de vieillissement –
Partie 1: Méthodes basées sur les valeurs moyennes de résultats d’essais
normalement distribués
INTERNATIONAL
ELECTROTECHNICAL
COMMISSION
COMMISSION
ELECTROTECHNIQUE
PRICE CODE
INTERNATIONALE
CODE PRIX V
ICS 29.035.01 ISBN 978-2-88912-834-1

– 2 – 60493-1  IEC:2011
CONTENTS
FOREWORD . 3
INTRODUCTION . 5
1 Scope . 6
2 Normative references . 6
3 Terms, definitions and symbols . 6
3.1 Terms and definitions . 6
3.2 Symbols . 8
4 Calculation procedures . 9
4.1 General considerations . 9
4.2 Single sub-group – Difference of mean and specified value . 9
4.2.1 General . 9
4.2.2 Complete data sub-group . 9
4.2.3 Censored data sub-group . 10
4.3 Two subgroups – Difference of means . 10
4.3.1 General . 10
4.3.2 Both sub-groups complete . 10
4.3.3 One or both subgroups censored . 11
4.4 Two or more subgroups – Analysis of variance . 11
4.5 Three or more subgroups – Regression analysis . 13
4.5.1 Regression analysis – General considerations . 13
4.5.2 Calculations. 14
4.5.3 Test equality of subgroup variances . 15
4.5.4 Test significance of deviations from linearity . 16
4.5.5 Estimate and confidence limit of y . 16
4.5.6 Estimate and confidence limit of x . 16
Annex A (informative) Statistical background . 18
Annex B (informative) Statistical tables . 22
Bibliography . 35

Table B.1 – Coefficients for censored data calculations . 23
Table B.2 – Fractiles of the F-distribution, F . 30
0,95
Table B.3 – Fractiles of the F-distribution, F . 32
0,995
Table B.4 – Fractiles of the t-distribution, t . 34
0,95
Table B.5 – Fractiles of the χ -distribution . 34

60493-1  IEC:2011 – 3 –
INTERNATIONAL ELECTROTECHNICAL COMMISSION
____________
GUIDE FOR THE STATISTICAL ANALYSIS
OF AGEING TEST DATA –
Part 1: Methods based on mean values
of normally distributed test results

FOREWORD
1) The International Electrotechnical Commission (IEC) is a worldwide organization for standardization comprising
all national electrotechnical committees (IEC National Committees). The object of IEC is to promote
international co-operation on all questions concerning standardization in the electrical and electronic fields. To
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Publications.
8) Attention is drawn to the Normative references cited in this publication. Use of the referenced publications is
indispensable for the correct application of this publication.
9) Attention is drawn to the possibility that some of the elements of this IEC Publication may be the subject of
patent rights. IEC shall not be held responsible for identifying any or all such patent rights.
International Standard IEC 60493-1 has been prepared by IEC technical committee 112:
Evaluation and qualification of electrical insulating materials and systems.
This second edition cancels and replaces the first edition, published in 1974, and constitutes
a technical revision.
The main changes with respect to the first edition are that, besides a complete editorial
revision, censored data sub-group are considered.

– 4 – 60493-1  IEC:2011
The text of this standard is based on the following documents:
CDV Report on voting
112/172/CDV 112/192/RVC
Full information on the voting for the approval of this standard can be found in the report on
voting indicated in the above table.
This publication has been drafted in accordance with the ISO/IEC Directives, Part 2.
A list of all the parts in the IEC 60493 series, published under the general title Guide for the
statistical analysis of ageing test data, can be found on the IEC website.
The committee has decided that the contents of this publication will remain unchanged until
the stability date indicated on the IEC web site under "http://webstore.iec.ch" in the data
related to the specific publication. At this date, the publication will be
• reconfirmed,
• withdrawn,
• replaced by a revised edition, or
• amended.
60493-1  IEC:2011 – 5 –
INTRODUCTION
Procedures for estimating ageing properties are described in specific test procedures, or are
covered by the general documents on test procedures for ageing tests with a specific
environmental stress (e.g. temperature, radiation, partial discharges).
In many cases, a certain property is determined as a function of time at different ageing
stresses, and a time to failure based on a chosen end-point criterion is found at each ageing
stress. A plot of time to failure versus ageing stress may be used to obtain an estimate of the
time to failure for similar specimens exposed to a specified stress, or to obtain an estimate of
the value of stress which will cause failure in a specified time.
The physical and chemical laws governing the ageing phenomena may often lead to the
assumption that a linear relationship exists between the property examined and the ageing
time at fixed ageing stresses, or between certain mathematical functions of property and
ageing time, e.g. square root or logarithm. Also, there may be a linear relationship between
time to failure and ageing stress, or mathematical functions of these variables.
The methods described in this part of IEC 60493 apply to such cases of linear relationship.
The methods are illustrated by the example of thermal ageing wherein the case of a simple
chemical process it may be assumed that the degradation obeys the Arrhenius law, i.e. the
logarithm of time to failure is a linear function of the reciprocal thermodynamic temperature.
Numerical examples demonstrating the use of the methods in this case are given in
IEC 60216-3 [1] .
The calculation processes specified in this standard are based on the assumption that the
data under examination are normally distributed. No test for normality of the data is specified,
since the available tests are unreliable for small sample groups of data. However, the
methods have been used for a considerable time without undesirable results and with no
check on the normality of the data distributions.

___________
Figures in square brackets refer to the bibliography.

– 6 – 60493-1  IEC:2011
GUIDE FOR THE STATISTICAL ANALYSIS
OF AGEING TEST DATA –
Part 1: Methods based on mean values
of normally distributed test results

1 Scope
This part of IEC 60493 gives statistical methods which may be applied to the analysis and
evaluation of the results of ageing tests.
It covers numerical methods based on mean values of normally distributed test results.
These methods are only valid under specific assumptions regarding the mathematical and
physical laws obeyed by the test data. Statistical tests for the validity of some of these
assumptions are also given.
This standard deals with data from both complete test sets and censored test sets.
This standard provides data treatment based on the concept of "data sub-group" as defined in
Clause 3. The validity of the coefficients used in the calculation processes to derive statistical
parameters of the data groups are described in [1].
2 Normative references
None.
3 Terms, definitions and symbols
3.1 Terms and definitions
For the purposes of this document, the following terms, definitions and symbols apply.
3.1.1
ordered data
set of data arranged in sequence so that in the appropriate direction through the sequence
each member is greater than or equal to its predecessor
Note 1 to entry: "Ascending order" in this standard implies that the data is ordered in this way, the first being the
smallest.
3.1.2
order-statistic
each individual value in a set of ordered data is referred to as an "order-statistic" identified by
its numerical position in the sequence
3.1.3
incomplete data
ordered data, where the values above and/or below defined points are not known
3.1.4
censored data
incomplete data, where the number of unknown values is known

60493-1  IEC:2011 – 7 –
Note 1 to entry: If the censoring is begun above/below a specified numerical value, the censoring is Type I. If it is
begun above/below a specified order-statistic, it is Type II. This standard is concerned only with Type II.
3.1.5
truncated data
incomplete data where the number of unknown values is not known
Note 1 to entry: This report is not concerned with truncated data.
3.1.6
Saw coefficient
one of the coefficients developed by J.G. Saw for calculating the primary statistical functions
of a single sub-group
Note 1 to entry: There are four coefficients used in this standard. Saw originally defined five, the fifth being
intended for estimating the variance of the variance estimate (see [7]).
3.1.7
degrees of freedom
number of data values minus the number of parameter values
3.1.8
variance of a data group
sum of the squares of the deviations of the data from a reference level
Note 1 to entry: The reference level may be defined by one or more parameters, for example a mean value (one
parameter) or a line (two parameters, slope and intercept), divided by the number of degrees of freedom.
3.1.9
central second moment of a data group
sum of the squares of the differences between the data values and the value of the group
mean, divided by the number of data in the group
3.1.10
covariance of data groups
for two groups of data with equal numbers of elements where each element in one group
corresponds to one in the other, the sum of the products of the deviations of the
corresponding members from their group means, divided by the number of degrees of
freedom
3.1.11
regression analysis
process of deducing the best-fit line expressing the relation of corresponding members of two
data groups by minimizing the sum of squares of deviations of members of one of the groups
from the line
Note 1 to entry: The parameters are referred to as the regression coefficients.
3.1.12
correlation coefficient
number expressing the completeness of the relation between members of two data groups,
equal to the covariance divided by the square root of the product of the variances of the
groups
Note 1 to entry: The value of its square is between 0 (no correlation) and 1 (complete correlation.
3.1.13
data sub-group
single set of data which may be used with other sub-groups to form a compound group

– 8 – 60493-1  IEC:2011
3.2 Symbols
Symbol Definition
a, b
Regression coefficient
e Sample (point) estimate of e
e Lower confidence limit of e
e Upper confidence limit of e
f
Number of degrees of freedom
f (x) Probability density
f (t), f t) Arbitrary function of time
1 3
f (θ) Arbitrary function of stress
f (p) Arbitrary function of property
F Fisher-distributed stochastic variable
F (x) Cumulative probability distribution
i Order No. of partial sample
j Order No. of specimen in partial sample No. i
k
Number of partial samples in total sample
m Order No. of specimen
n Number of observations in sample
n Number of specimens in partial sample No. i
i
N Total number of specimens
p Arbitrary property of test specimens
P (X ≤ x) Probability that X ≤ x
s Sample variance
s Variance within sets
Variance about regression line
s
Partial sample variance
s
t Student-distributed stochastic variable
u Standardized normal (Gaussian) distributed stochastic variable
x Independent variable (for example 1/θ)
x Partial sample value of x
i
x Sample mean
Weighted mean of x
x
~
Sample median
x
X Stochastic variable, specified value of x
y Dependent stochastic variable (for example log v)
y Individual specimen value of y
ij
y
Partial sample mean of y
i
y Total sample mean of y
Y Specified value of y
α Significance level
ε Arbitrary population parameter
Thermodynamic temperature/Kelvin
Θ
θ Ageing stress (for example temperature)

60493-1  IEC:2011 – 9 –
Symbol Definition
Mean value of X
ξ
ξ Median value of X
Standard deviation of X
σ
σ Variance of X
χ Chi-square-distributed test variable
1 – α Confidence level
4 Calculation procedures
4.1 General considerations
For these calculations:
– n is the number of values known in subgroup;
– m is the total number in subgroup (= n for complete data group);
– α, β, µ and ε are the “Saw” coefficients for these values of m and n.
For an uncensored subgroup, the values of the “Saw” coefficients are as follows:
α =1/(n −1)
(1)
β = −1/(n(n −1)) (2)
µ =1− 1/ n (3)
ε =1 (4)
If convenient, these coefficients may be used to calculate the primary statistical functions
(mean and standard deviation) of complete data groups, using the formulae of 4.2.3 (in place
of the usual formulae as in 4.2.2). “Saw” coefficients are given in Table B.1.
4.2 Single sub-group – Difference of mean and specified value
4.2.1 General
The purpose of the procedure is to test the significance of the difference between the sub-
group mean and a specified numerical value.
4.2.2 Complete data sub-group
n
Calculate sub-group mean y = y / n (5)
∑ i
i=1
n
 
2 2
y −n y
 
∑ i
2  i=1 
Calculate sub-group variance σ = (6)
(n −1)
– 10 – 60493-1  IEC:2011
(7)
Calculate t t = y / σ / n
Compare the value of t with the tabulated t values.
4.2.3 Censored data sub-group
n−1
y
j
Calculate sub-group mean y = (1− µ) y + µ (8)
n ∑
(n −1)
j=1
n−1 n−1
 
Calculate sub-group variance σ =α (y − y ) + β (y − y ) (9)
∑ n j ∑ n j 
 
j=1 j=1
 
(1− n / m)
Calculate adjustment for t a = (10)
(6,2 + n / 6,4 − (m − n)/10,7)
Calculate t t = y / ε σ / n (11)
Calculate t 1/ t =1/ t + a (12)
a a
Compare the value of t with the tabulated t values.
a
4.3 Two subgroups – Difference of means
4.3.1 General
The purpose of this procedure is to test the significance of the difference between the sub-
group means.
For these calculations:
– n is the number of values known in subgroup i;
i
– m is the total number of values in subgroup i;
i
– α β μ and ε are the “Saw” coefficients for these values of m and n.
i i i i
For a complete sub-group, ε =1.
i
4.3.2 Both sub-groups complete
n
i
Calculate sub-group means y = y / n (13)
i ij i
∑
j=1
n
 i 
 2 2 
y − n y
ij i
∑
 
 
j=1
2  
Calculate sub-group variances σ = (14)
i
(n −1)
i
60493-1  IEC:2011 – 11 –
 
ε ε
1 2
  (15)
Calculate the group value of ε e = +
 
n n
 1 2 
2 2
((n − 1)σ + (n −1)σ )
2 1 1 2 2
Calculate the merged variance σ σ = (16)
(n + n − 2)
1 2
(y − y )
1 2
Calculate t t = (17)
eσ
Determine probability by reference to tabulated values of t.
4.3.3 One or both subgroups censored
n −1 n −1
i  i 
 
Calculate sub-group variances ( ) ( ) (18)
σ =α y − y + β y − y
i i in ij i in ij
∑ ∑
i i
 
j=1 j=1
 
n −1
i
y
ij
Calculate sub-group means y = (1− µ ) y + µ (19)
i i i i
n ∑
i
(n −1)
i
j=1
 ε ε 
1 2
 
Calculate the group value of ε e = + (20)
 
n n
 1 2 
2 2
((n − 1)σ + (n −1)σ )
2 1 1 2 2
(21)
Calculate the merged variance σ σ =
(n + n − 2)
1 2
(y − y )
1 2
Calculate t t = (22)
eσ
 
 
p n n  (n + n ) 
1 2 1 2
Calculate adjustment, a a = −  + 2 (23)
 
 
m m 20
  
(n + n )  1 2 
1 2
 
where p is the smaller of n and n .
1 2
Apply adjustment t = (24)
a
 
+ a
 
t
 
Determine probability by reference to tabulated values of t.
4.4 Two or more subgroups – Analysis of variance
Individual sub-groups may be complete or censored.

– 12 – 60493-1  IEC:2011
For these calculations:
n is the number of values known in subgroup i;
i
m is the total number in subgroup i;

i
α , β , μ and ε are the “Saw” coefficients for these values of m and n;
i i i i
k is the number of subgroups;
c is the intermediate value for χ calculation;
A is the adjustment factor for χ calculation.
k
Calculate the total number of values M = m (25)
∑ i
i=1
k
Calculate the total number of values known N = n (26)
i
∑
i=1
Calculate subgroup means:
n −1
i
y
ij
y = (1− µ ) y + µ (Censored data)
i i i i
n ∑
i
(n − 1)
i
j=1
n
i (27)
y
ij
∑
j=1
y = (Complete data subgroup)
i
n
i
k
n y
∑ i i
i=1
Calculate group general mean y = (28)
N
Calculate sub-group variances:
n −1 n −1
i  i 
 
s =α (y − y ) + β (y − y )
i i in ij i in ij
∑ ∑
i i
 
j=1 j=1
 
(29)
n
i
2 2
(y − n y )
ij i i
∑
j=1
s = (Complete data subgroup)
i
(n −1)
i
k
ε
∑
i
i=1
ε =
k
Calculate mean variance factor (30)

60493-1  IEC:2011 – 13 –
k
 
2 2
n y − N y
∑ i i
 
2  i=1 
Calculate variance of means         s = (31)
N
(k −1)
k
 
ε s (n −1)
∑ i i
 
2  i=1

Calculate residual variance s = (32)
D
(N − k)
Test equality of subgroup variances:
k
 
1 1
−
∑ 
(n −1) (N − k)
i=1
 i 
Calculate c         c =1+ (33)
3(k −1)
N 12
   
1− × 1−
   
M M
   
Calculate adjustment factor A= 1 + (34)
k
 
 
A s
2 2
D
Calculate χ χ = (N − k) ln   − (n −1) ln(s ) (35)
 
∑ i i
 
c ε
i=1
 
 
Test equality of residual variance and variance of subgroup means.
s
N
Calculate F F = (36)
s
D
Degrees of freedom for F N - k (denominator), k -1 (numerator)
Calculate significance of general mean:
N
Calculate t t = y (37)
s
T
Adjust t for censoring 1/ t =1/ t + a (38)
a
Determine probability by reference to tabulated values of t with N-1 degrees of freedom.
4.5 Three or more subgroups – Regression analysis
4.5.1 Regression analysis – General considerations
These data differ from those of (4.4) in that the y-values in each subgroup are associated with
a value of another variable, referred to in this section as x . The objective of the analysis is to
i
determine whether there is a linear relationship between x and y and, if so, its parameters and
properties.
– 14 – 60493-1  IEC:2011
The parameters and properties in question are as follows:
– slope(b) and intercept (a) of regression line;
– equality of variance of subgroups (χ );
– linearity of regression (F);
– confidence intervals of regression estimates.
For these calculations:
is the number of values known in subgroup i;
n
i
m is the total number in subgroup i;
i
β μ and ε are the “Saw” coefficients for these values of m and n;
α
, ,
i i i i
k is the number of subgroups;
c is the intermediate value for χ calculation;
A is the adjustment factor for χ calculation;
b and a are the slope and intercept of the regression line;
t is the tabulated value of t for probability p and n-1 degrees of freedom.
p,n-1
Sub-groups may be either complete or censored. Values of y are the actual values of
ij
variable j in subgroup i.
4.5.2 Calculations
k
Calculate the total number of values M = m (39)
∑
i
i=1
k
Calculate the total number of values known N = n (40)
∑ i
i=1
Calculate subgroup means:
n −1
i
y
ij
y = (1− µ ) y + µ (Censored data)
i i i i ∑
n
i
(n −1)
i
j=1
n
(41)
i
y
ij
∑
j=1
y = (Complete data subgroup)
i
n
i
Calculate sub-group variances:
n −1 n −1
i  i 
 
s =α (y − y ) + β (y − y ) (Censored data)
i i ∑ in ij i ∑ in ij
i i
 
j=1 j=1
 
(42)
n
i
2 2
(y − n y )
∑ ij i i
j=1
s = (Complete data subgroup)
i
(n −1)
i
60493-1  IEC:2011 – 15 –
k
n x
∑
i i
i=1
Calculate x-mean x = (43)
N
k
n y
∑ i i
i=1
Calculate group general mean y = (44)
N
k
ε
∑ i
i=1
Calculate mean variance factor ε = (45)
k
k
 
ε s (n −1)
∑ i i
 
2  i=1 
Calculate residual variance s = (46)
D
(N − k)
k
2 2
Calculate y-sum of squares SSy = n y − N y (47)
∑ i i
i=1
k
2 2
Calculate x-sum of squares         SSx= n x − N x (48)
∑ i i
i=1
k
Calculate xy-sum of products SPxy = n x y − N x y (49)
∑ i i i
i=1
SPxy
Calculate slope b b= (50)
SSx
(51)
Calculate intercept a a = y −b x
SPxy  SPxy 
2  
Calculate r (52)
r =  
 
 
SSx SSy
 
 
(r is the correlation coefficient)
(1− r )SSy
Calculate non-linearity variance s = (53)
N
(k − 2)
4.5.3 Test equality of subgroup variances
k
 
1 1
−
∑ 
(n −1) (N − k)
i=1
 i 
Calculate c c = 1+ (54)
3(k −1)
– 16 – 60493-1  IEC:2011
 N   12 
1− × 1−
   
M M
   
Calculate adjustment factor A= 1 + (55)
k
 
A  s 
2 2
D
Calculate χ   (56)
χ = (N − k) ln − (n −1) ln(s )
 ∑ 
i i
 
c ε
i=1
 
 
4.5.4 Test significance of deviations from linearity
s
N
(57)
Calculate F F =
s
D
Degrees of freedom for F N-k (denominator), k-2 (numerator)
4.5.5 Estimate and confidence limit of y
SSx
Central second moment of x µ = (58)
2(x)
N
2 2
((N − k)s +(k − 2)s )
D N
Non regression variance s = (59)
T
(N − 2)
 
s (X − x)
T
 
Adjust for extrapolation            s = 1+ (60)
C
 
N µ
2(x)
 
 N 
1−
 
M
 
Correction to t for censoring          a = ` (61)
 N (M − N ) 
6,2 + −
 
6,4 10,7
 
 
 
Correction applied t = −a (62)
c
 
t
p,N −2
 
Estimate of y         yˆ = a + b X (63)
Confidence limit of estimate of y       yˆ = yˆ +t s (64)
c c c
4.5.6 Estimate and confidence limit of x
(y − a)
Estimate xˆ = (65)
b
For simplicity, calculate several temporary variables:

60493-1  IEC:2011 – 17 –
2 2
t s
c T
b =b− (66)
r
N b µ
2,( x)
 
s b (xˆ − x)
T r
 
s = + (67)
r
 
N b µ
2,(x)
 
t s
(y − y)
c r
ˆ
Confidence limit of estimate of x     x = x + + (68)
c
b b
r r
– 18 – 60493-1  IEC:2011
Annex A
(informative)
Statistical background
A.1 Statistical distributions and parameters
The distribution of a stochastic variable X is described by the distribution function:
F(x)= P (X ≤ x) (A.1)
where P(X ≤ x) is the probability that the value of X is ≤ x. Here F(x) goes from 0 to 1 and is a
never-decreasing function of x. If F(x) is a continuous function of x, then the probability
density is determined as:
dF(x)
f (x)= (A.2)
dx
The distribution may be characterized by parameters, of which the most important are:
– the mean value:
∞
ξ = x f (x)dx (A.3)
∫
−∞
– the variance:
∞
( ) ( ) (A.4)
σ = x − ξ f x dx
∫
−∞
The square root of the variance is termed the standard deviation σ .
A.2 Estimates of parameters
From a sample of n stochastic independent specimens from a population, estimates of the
parameters of the population (see Clause A.1) may be derived.
An estimate of the mean value of the population (Formula (A.3)) is calculated as the average
of the individual sample values:
n
x
∑ i
i=1
x = (A.5)
n
where
x represents the individual sample values (i = 1, 2, . n).
i
An estimate of the variance of the population (Formula (A.4)) is the sample variance:

60493-1  IEC:2011 – 19 –
n
( x )
∑ i
(x − x) 2
x − 2
∑ i
∑ i
n x −( x )
∑ i ∑ i
i=1 n
s = = = (A.6)
(n −1) n −1 n(n −1)
where n – 1 = f is called the number of degrees of freedom of s .
A.3 Distribution types
A.3.1 General
The following distribution types are relevant to this application, the t, F, and χ distributions
being the distributions of secondary functions derived from the mean and variance parameter
estimates of normally distributed data.
A.3.2 The normal distribution
The calculation processes specified in this standard are based on the assumption that the
data under examination are normally distributed. No test for normality of the data is specified,
since the available tests are unreliable for small sample groups of data. However, the
methods have been used for a considerable time without undesirable results and with no
check on the normality of the data distributions.
The normal (Gaussian) distribution is defined by:
2 2
exp{− (x − ξ ) / 2σ }
f (x)= (A.7)
2πσ
ξ and variance σ .
and is completely characterized by its mean value
The standardized normal distribution:
u
exp−
f (u)= (A.8)
2π
where
x −ξ
u = (A.9)
σ
and the corresponding distribution function F(u) have been tabulated and computer routines
for their calculation are available (see [1]).
The above use of F should not be confused with the F distribution below.
of a sample of n specimens from a normal distribution is itself a normally
The mean value x
distributed stochastic variable with mean value ξ = ξ and variance σ = σ n and the
x x
corresponding standardized variable is:

– 20 – 60493-1  IEC:2011
x −ξ
u = (A.10)
σ
n
A.3.3 The t distribution
2 2
If the true variance of the normal distribution σ is not known, the sample estimate s from
Formula (A.6) may be substituted and the standardized sample mean value becomes:
x −ξ
u = (A.11)
s
n
The distribution of this variable is called the t distribution (or Student's t) and depends on the
parameter f = n – 1 (the number of degrees of freedom for s ). The t distribution has been
tabulated for different values of f. It is derived from the “Incomplete Beta function.
A.3.4 The F distribution
To test if two sample variances, determined from two different samples, may reasonably be
considered to be estimates of the same theoretical variance (population parameter), the
following test variable is calculated:
s
(A.12)
F =
s
The distribution of this variable is called the F distribution (or Fisher) and depends on the
2 2
parameters f = n – 1 and f = n – 1 (the number of degrees of freedom for s and s ). The
1 1 2 2 1 2
F distribution has been tabulated for different values of f and f . It is derived from the
1 2
“Incomplete Beta function”
A.3.5 The χ distribution
To test if several sample variances, each determined from a different sample, may reasonably
be considered to be estimates of the same theoretical variance, the following test variable is
calculated (Bartlett’s χ ).
k
 
2 2
2,3 f lg s − f lg s 
∑ i i
 i=1 
χ = (A.13)
c
k
 
1 1
−
∑ 
f f
i=1
 i 
where c =1+ (A.14)
3(k −1)
60493-1  IEC:2011 – 21 –
the individual sample variance (i = 1, 2, . k) with f degrees
k is the number of variances, s
i 1
f s
i i
∑
of freedom, and s = is a pooled variance with f = f degrees of freedom. The test
∑
i
f
i
∑
hypothesis is that all k variances s are estimates of the same theoretical variance σ .
i
2 2
The calculated value χ is compared with the tabulated value χ (1 – α, k – 1) which is a
function of k – 1, the number of degrees of freedom for χ and of α, the significance level. If
2 2
χ > χ (1 – α, k – 1), the hypothesis is rejected on significance level α.
The distribution of this variable is called the Bartlett’s χ distribution and depends on the
parameter f = k – 1. The χ distribution has been tabulated for different values of f . It is
derived from the “Incomplete Gamma function”.
Bartlett's test is an approximate test, but a good approximation if the number of degrees of
freedom f of all the individual sample variances is greater than 2.
s
i i
is taken as a pooled estimate of the common variance with f
If the hypothesis is accepted, s
degrees of freedom.
– 22 – 60493-1  IEC:2011
Annex B
(informative)
Statistical tables
B.1 Use of the tables
Statistical tables of cumulative distribution functions F (x) of a stochastic variable X are
usually arranged in such a way that they give that value of x which, for a specified probability,
P, satisfies the condition:
F (x, δ) = P (X ≤ x)
where δ represents possible parameters, which cannot be taken care of by standardization of
the variable For instance, in the case of χ distribution, Table B.5 gives for P = 0,95 and f = 6
2 2
a value of χ = 12,6. This means that when f =6, the probability of getting a value of χ equal
2 2
to or less than χ (P, f) = χ (0,95, 6) = 12.6, is 95 %, or:
P (χ ≤ 12,6) = 0,95 f =6
Expressed in another way, P = 95 % of the χ distribution lies below 12,6, and α = 1 – P = 5 %

α may be considered as a significance level, for example, if by
above this value when f = 6.
hypothesis testing we use the interval 12,6 < χ < + ∞ as reject interval, the risk of making a
false decision by rejecting the hypothesis although true is 5 %. In some cases, α is used as
entrance to the tables instead of P, for example where in Table B.5 for 6 degrees of freedom
2 2
and a probability of 0,05, a value of χ = 12,6 means that the probability of χ being greater
than12,6 is 5 %:
P(χ >12,6) = 0,05 f = 6
60493-1  IEC:2011 – 23 –
Table B.1 – Coefficients for censored data calculations
m n
α β µ ε
5 3 614,4705061728 –100,3801985597 0,0000000000 860,4482888889
5 4 369,3153100012 –70,6712934899 472,4937150842 874,0745894447

6 4 395,4142139605 –58,2701183523 222,6915218468 835,7650306465
6 5 272,5287238052 –44,0988850936 573,5126123815 887,1066681426

7 4 415,5880351563 –46,5401552734 0,0000000000 841,7746734375
7 5 289,1914470089 –38,0060438107 364,2642153815 837,3681267819
7 6 215,5146796875 –30,1363662109 642,2345606152 898,7994404297

8 5 302,2559543304 –32,0455510095 173,7451925589 823,1325022970
8 6 227,1320334900 –26,7149242720 462,3946896558 845,5891673417
8 7 178,0192047851 –21,8909055649 692,0082911498 908,7175231765

9 5 312,9812000000 –26,3842700000 0,0000000000 830,5022000000
9 6 236,3858000000 –23,2986100000 296,0526300000 821,3172600000
9 7 186,6401000000 –19,7898900000 534,4601800000 855,2096700000
9 8 151,5120000000 –16,6140800000 729,7119900000 917,0583200000

10 6 244,1191560890 –20,0047740729 142,3739002847 815,8210886826
10 7 193,6205880047 –17,6663604814 386,9526017618 825,7590437753
10 8 158,2300608320 –15,2437931582 589,6341322307 864,6219294884
10 9 131,8030382363 –13,0347627976 759,2533663842 924,0989192531

11 6 250,6859320988 –16,8530354295 0,0000000000 822,9729127315
11 7 199,4695468487 –15,5836545374 249,2599953079 812,6308986254
11 8 163,6996121337 –13,8371182557 457,2090965743 832,5488161799
11 9 137,2299243827 –12,1001907793 633,2292924678 873,3355410880
11 10 116,5913210464 –10,4969569718 783,0177949444 930,0880372994

12 7 204,5349924229 –13,5767110244 120,5748554921 810,9803051840
12 8 168,3292196600 –12,4439880795 332,5519557674 814,7269021330
12 9 141,6425229674 –11,1219466676 513,1493415383 840,0625045817
12 10 121,0884792448 –9,8359507754 668,5392651269 881,2400322962
12 11 104,5060800375 –8,6333795848 802,5441292356 935,2282230049

13 7 208,9406118284 –11,6456142827 0,0000000000 817,5921863390
13 8 172,3464251400 –11,0865264201 215,2023355151 807,2699422973
13 9 145,4178687827 –10,1472348992 399,3236520338 819,3180095090
13 10 124,7371924225 –9,1300085328 558,7461589055 847,5908596926
13 11 108,3018058633 –8,1510819663 697,7158560873 888,3591181189
13 12 94,6796149706 –7,2252117874 818,8697028778 939,6794196639
–3
NOTE α, β, µ and ε are all in units of 1 × 10 .

– 24 – 60493-1  IEC:2011
Table B.1 (continued)
m n
α β µ ε
14 8 175,9018422090 –9,7746826098 104,5543516980 807,5106793327
14 9 148,7066543210 –9,1891433745 291,5140765844 807,9273940741
14 10 127,8816896780 –8,4224506929 454,0609002065 825,0398828063
14 11 111,3817699729 –7,6266971302 596,6235832604 854,8238304463
14 12 97,9278246914 –6,8636059259 722,2249188477 894,7614153086
14 13 86,5363075231 –6,1355268822 832,7192524487 943,5668941976

15 8 179,0513405762 –8,5071530762 0,0000000000 813,5568182129
15 9 151,6274451540 –8,2566923172 189,3157319524 803,6572346196
15 10 130,6387362674 –7,7228786289 354,3906973785 810,9441335713
15 11 114,0457797966 –7,0973951863 499,7526628800 831,1920110198
15 12 100,5718881836 –6,4648224487 628,5859288205 861,6352648315
15 13 89,3466123861 –5,8578554309 743,0997382709 900,5262051665
15 14 79,6796956870 –5,2751393667 844,6143938637 946,9889014846

16 9 154,2518689085 –7,3527348129 92,2865976624 804,8901545650
16 10 133,0926552303 –7,0374903483 259,4703005026 803,4179489468
16 11 116,3971900144 –6,5718807983 407,1074446942 815,2259119510
16 12 102,8620227960 –6,0590262781 538,4703518878 837,4056164917
16 13 91,6475110414 –5,5485234808 655,9153003723 867,9864133589
16 14 82,1334839298 –5,0573990501 761,0897304685 905,7302132374
16 15 73,8281218530 –4,5839766095 854,9400915790 950,0229759376

17 9 156,6104758421 –6,4764602745 0,0000000000 810,4190113397
17 10 135,3069770991 –6,3698625234 168,9795641122 801,0660748802
17 11 118,4974933487 –6,0543187349 318,5208867246 805,3180627394
17 12 104,8944939376 –5,6546733211 451,9486020413 820,1513691949
17 13 93,6414079430 –5,2310447166 571,6961830632 843,4861778660
17 14 84,1578079201 –4,8133017972 679,5480456810 873,8803351313
17 15 75,9876912684 –4,4100612544 776,7517032846 910,4428918550
17 16 68,7761850391 –4,0203992390 863,9866274899 952,7308021373

18 10 137,3196901001 –5,7208401228 82,5925913725 802,8356541137
18 11 120,3965503416 –5,5477052124 233,7625216775 800,2584198483
18 12 106,7179571420 –5,2548692706 368,9237739923 808,4878348626
18 13 95,4179152353 –4,9135219393 490,5582072725 825,3579958906
18 14 85,9129822797 –4,5606570913 600,5193900565 849,3339891000
18 15 77,7846697341 –4,2145025451 700,1840825530 879,3395044075
18 16 70,6902823246 –3,8792292982 790,5080136386 914,7252389325
18 17 64,3706903919 –3,5548196830 871,9769987244 955,1618993620
–3
NOTE α, β, µ and ε are all in units of 1 × 10 .

60493-1  IEC:2011 – 25 –
Table B.1 (continued)
m n
α β µ ε
19 10 139,1496250000 –5,0900181250 0
...