ISO 16269-4:2010
(Main)Statistical interpretation of data — Part 4: Detection and treatment of outliers
Statistical interpretation of data — Part 4: Detection and treatment of outliers
ISO 16269-4:2010 provides detailed descriptions of sound statistical testing procedures and graphical data analysis methods for detecting outliers in data obtained from measurement processes. It recommends sound robust estimation and testing procedures to accommodate the presence of outliers. ISO 16269-4:2010 is primarily designed for the detection and accommodation of outlier(s) from univariate data. Some guidance is provided for multivariate and regression data.
Interprétation statistique des données — Partie 4: Détection et traitement des valeurs aberrantes
Statistično tolmačenje podatkov - 4. del: Zaznavanje in obravnava osamelcev
Ta del standarda ISO 16269 zagotavlja podrobne opise učinkovitih postopkov za statistično preskušanje in metod za analizo grafičnih podatkov za zaznavanje osamelcev v podatkih, pridobljenih na podlagi merilnih procesov. Priporoča učinkovite in zanesljive ocenjevalne in preskusne postopke za prilagajanje prisotnosti osamelcev. Ta del standarda ISO 16269 je namenjen zlasti zaznavanju in prilagajanju prisotnosti osamelcev iz univariatnih podatkov. Zagotovljeni so delni napotki za multivariatne in regresijske podatke.
General Information
Standards Content (Sample)
INTERNATIONAL ISO
STANDARD 16269-4
First edition
2010-10-15
Statistical interpretation of data —
Part 4:
Detection and treatment of outliers
Interprétation statistique des données —
Partie 4: Détection et traitement des valeurs aberrantes
Reference number
ISO 16269-4:2010(E)
©
ISO 2010
---------------------- Page: 1 ----------------------
ISO 16269-4:2010(E)
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electronic or mechanical, including photocopying and microfilm, without permission in writing from either ISO at the address below or
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Published in Switzerland
ii © ISO 2010 – All rights reserved
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ISO 16269-4:2010(E)
Contents Page
Foreword .iv
Introduction.v
1 Scope.1
2 Terms and definitions .1
3 Symbols.10
4 Outliers in univariate data .11
4.1 General .11
4.1.1 What is an outlier? .11
4.1.2 What are the causes of outliers? .11
4.1.3 Why should outliers be detected?.11
4.2 Data screening.12
4.3 Tests for outliers .14
4.3.1 General .14
4.3.2 Sample from a normal distribution.14
4.3.3 Sample from an exponential distribution.16
4.3.4 Samples taken from some known non-normal distributions.18
4.3.5 Sample taken from unknown distributions.19
4.3.6 Cochran's test for outlying variance .21
4.4 Graphical test of outliers .22
5 Accommodating outliers in univariate data.23
5.1 Robust data analysis.23
5.2 Robust estimation of location.24
5.2.1 General .24
5.2.2 Trimmed mean .24
5.2.3 Biweight location estimate .25
5.3 Robust estimation of dispersion .25
5.3.1 General .25
5.3.2 Median-median absolute pair-wise deviation.25
5.3.3 Biweight scale estimate.26
6 Outliers in multivariate and regression data .26
6.1 General .26
6.2 Outliers in multivariate data .26
6.3 Outliers in linear regression.28
6.3.1 General .28
6.3.2 Linear regression models.29
6.3.3 Detecting outlying Y observations.31
6.3.4 Identifying outlying X observations.31
6.3.5 Detecting influential observations.32
6.3.6 A robust regression procedure.35
Annex A (informative) Algorithm for the GESD outliers detection procedure .36
Annex B (normative) Critical values of outliers test statistics for exponential samples .37
Annex C (normative) Factor values of the modified box plot .44
Annex D (normative) Values of the correction factors for the robust estimators of the scale
parameter .47
Annex E (normative) Critical values of Cochran's test statistic .48
Annex F (informative) A structured guide to detection of outliers in univariate data .51
Bibliography.54
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ISO 16269-4:2010(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
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 16269-4 was prepared by Technical Committee ISO/TC 69, Applications of statistical methods.
ISO 16269 consists of the following parts, under the general title Statistical interpretation of data:
⎯ Part 4: Detection and treatment of outliers
⎯ Part 6: Determination of statistical tolerance intervals
⎯ Part 7: Median — Estimation and confidence intervals
⎯ Part 8: Determination of prediction intervals
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ISO 16269-4:2010(E)
Introduction
Identification of outliers is one of the oldest problems in interpreting data. Causes of outliers include
measurement error, sampling error, intentional under- or over-reporting of sampling results, incorrect
recording, incorrect distributional or model assumptions of the data set, and rare observations, etc.
Outliers can distort and reduce the information contained in the data source or generating mechanism. In the
manufacturing industry, the existence of outliers will undermine the effectiveness of any process/product
design and quality control procedures. Possible outliers are not necessarily bad or erroneous. In some
situations, an outlier may carry essential information and thus it should be identified for further study.
The study and detection of outliers from measurement processes leads to better understanding of the
processes and proper data analysis that subsequently results in improved inferences.
In view of the enormous volume of literature on the topic of outliers, it is of great importance for the
international community to identify and standardize a sound subset of methods used in the identification and
treatment of outliers. The implementation of this part of ISO 16269 enables business and industry to recognize
the data analyses conducted across member countries or organizations.
Six annexes are provided. Annex A provides an algorithm for computing the test statistic and critical values of
a procedure in detecting outliers in a data set taken from a normal distribution. Annexes B, D and E provide
the tables needed to implement the recommended procedures. Annex C provides the tables and statistical
theory that underlie the construction of modified box plots in outlier detection. Annex F provides a structured
guide and flow chart to the procedures recommended in this part of ISO 16269.
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INTERNATIONAL STANDARD ISO 16269-4:2010(E)
Statistical interpretation of data —
Part 4:
Detection and treatment of outliers
1 Scope
This part of ISO 16269 provides detailed descriptions of sound statistical testing procedures and graphical
data analysis methods for detecting outliers in data obtained from measurement processes. It recommends
sound robust estimation and testing procedures to accommodate the presence of outliers.
This part of ISO 16269 is primarily designed for the detection and accommodation of outlier(s) from univariate
data. Some guidance is provided for multivariate and regression data.
2 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
2.1
sample
data set
subset of a population made up of one or more sampling units
NOTE 1 The sampling units could be items, numerical values or even abstract entities depending on the population of
interest.
NOTE 2 A sample from a normal (2.22), a gamma (2.23), an exponential (2.24), a Weibull (2.25), a
lognormal (2.26) or a type I extreme value (2.27) population will often be referred to as a normal, a gamma, an
exponential, a Weibull, a lognormal or a type I extreme value sample, respectively.
2.2
outlier
member of a small subset of observations that appears to be inconsistent with the remainder of a given
sample (2.1)
NOTE 1 The classification of an observation or a subset of observations as outlier(s) is relative to the chosen model for
the population from which the data set originates. This or these observations are not to be considered as genuine
members of the main population.
NOTE 2 An outlier may originate from a different underlying population, or be the result of incorrect recording or gross
measurement error.
NOTE 3 The subset may contain one or more observations.
2.3
masking
presence of more than one outlier (2.2), making each outlier difficult to detect
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ISO 16269-4:2010(E)
2.4
some-outside rate
probability that one or more observations in an uncontaminated sample will be wrongly classified as
outliers (2.2)
2.5
outlier accommodation method
method that is insensitive to the presence of outliers (2.2) when providing inferences about the population
2.6
resistant estimation
estimation method that provides results that change only slightly when a small portion of the data values in a
data set (2.1) is replaced, possibly with very different data values from the original ones
2.7
robust estimation
estimation method that is insensitive to small departures from assumptions about the underlying probability
model of the data
NOTE An example is an estimation method that works well for, say, a normal distribution (2.22), and remains
reasonably good if the actual distribution is skew or heavy-tailed. Classes of such methods include the L-estimation
[weighted average of order statistics (2.10)] and M-estimation methods (see Reference [9]).
2.8
rank
position of an observed value in an ordered set of observed values
NOTE 1 The observed values are arranged in ascending order (counting from below) or descending order (counting
from above).
NOTE 2 For the purposes of this part of ISO 16269, identical observed values are ranked as if they were slightly
different from one another.
2.9
depth
〈box plot〉 smaller of the two ranks (2.8) determined by counting up from the smallest value of the
sample (2.1), or counting down from the largest value
NOTE 1 The depth may not be an integer value (see Annex C).
NOTE 2 For all summary values other than the median (2.11), a given depth identifies two (data) values, one below
the median and the other above the median. For example, the two data values with depth 1 are the smallest value
(minimum) and largest value (maximum) in the given sample (2.1).
2.10
order statistic
statistic determined by its ranking in a non-decreasing arrangement of random variables
[ISO 3534-1:2006, definition 1.9]
NOTE 1 Let the observed values of a random sample be {x , x , …, x }. Reorder the observed values in non-
1 2 n
decreasing order designated as x u x u … u x u … u x ; then x is the observed value of the kth order statistic in
(1) (2) (k) (n) (k)
a sample of size n.
NOTE 2 In practical terms, obtaining the order statistics for a sample (2.1) amounts to sorting the data as formally
described in Note 1.
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ISO 16269-4:2010(E)
2.11
median
sample median
median of a set of numbers
Q
2
[(n + 1)/2]th order statistic (2.10), if the sample size n is odd; sum of the [n/2]th and the [(n/2) + 1]th order
statistics divided by 2, if the sample size n is even
[ISO 3534-1:2006, definition 1.13]
NOTE The sample median is the second quartile (Q ).
2
2.12
first quartile
sample lower quartile
Q
1
for an odd number of observations, median (2.11) of the smallest (n − 1)/2 observed values; for an even
number of observations, median of the smallest n/2 observed values
NOTE 1 There are many definitions in the literature of a sample quartile, which produce slightly different results. This
definition has been chosen both for its ease of application and because it is widely used.
NOTE 2 Concepts such as hinges or fourths (2.19 and 2.20) are popular variants of quartiles. In some cases
(see Note 3 to 2.19), the first quartile and the lower fourth (2.19) are identical.
2.13
third quartile
sample upper quartile
Q
3
for an odd number of observations, median of the largest (n − 1)/2 observed values; for an even number of
observations, median of the largest n/2 observed values
NOTE 1 There are many definitions in the literature of a sample quartile, which produce slightly different results. This
definition has been chosen both for its ease of application and because it is widely used.
NOTE 2 Concepts such as hinges or fourths (2.19 and 2.20) are popular variants of quartiles. In some cases
(see Note 3 to 2.20), the third quartile and the upper fourth (2.20) are identical.
2.14
interquartile range
IQR
difference between the third quartile (2.13) and the first quartile (2.12)
NOTE 1 This is one of the widely used statistics to describe the spread of a data set.
NOTE 2 The difference between the upper fourth (2.20) and the lower fourth (2.19) is called the fourth-spread and is
sometimes used instead of the interquartile range.
2.15
five-number summary
the minimum, first quartile (2.12), median (2.11), third quartile (2.13), and maximum
NOTE The five-number summary provides numerical information about the location, spread and range.
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ISO 16269-4:2010(E)
2.16
box plot
horizontal or vertical graphical representation of the five-number summary (2.15).
NOTE 1 For the horizontal version, the first quartile (2.12) and the third quartile (2.13) are plotted as the left and
right sides, respectively, of a box, the median (2.11) is plotted as a vertical line across the box, the whiskers stretching
downwards from the first quartile to the smallest value at or above the lower fence (2.17) and upwards from the third
quartile to the largest value at or below the upper fence (2.18), and value(s) beyond the lower and upper fences are
marked separately as outlier(s) (2.2). For the vertical version, the first and third quartiles are plotted as the bottom and the
top, respectively, of a box, the median is plotted as a horizontal line across the box, the whiskers stretching downwards
from the first quartile to the smallest value at or above the lower fence and upwards from the third quartile to the largest
value at or below the upper fence and value(s) beyond the lower and upper fences are marked separately as outlier(s).
NOTE 2 The box width and whisker length of a box plot provide graphical information about the location, spread,
skewness, tail lengths, and outlier(s) of a sample. Comparisons between box plots and the density function of a) uniform,
b) bell-shaped, c) right-skewed, and d) left-skewed distributions are given in the diagrams in Figure 1. In each distribution,
a histogram is shown above the boxplot.
NOTE 3 A box plot constructed with its lower fence (2.17) and upper fence (2.18) evaluated by taking k to be a value
based on the sample size n and the knowledge of the underlying distribution of the sample data is called a modified box
plot (see example, Figure 2). The construction of a modified box plot is given in 4.4.
a) Uniform distribution b) Bell-shaped distribution
Figure 1 (continued)
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ISO 16269-4:2010(E)
c) Right-skewed distribution d) Left-skewed distribution
Key
X data values
Y frequency
In each distribution, a histogram is shown above the box plot.
Figure 1 — Box plots and histograms for a) uniform, b) bell-shaped, c) right-skewed,
and d) left-skewed distributions
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ISO 16269-4:2010(E)
Figure 2 — Modified box plot with lower and upper fences
2.17
lower fence
lower outlier cut-off
lower adjacent value
value in a box plot (2.16) situated k times the interquartile range (2.14) below the first quartile (2.12), with
a predetermined value of k
NOTE In proprietary statistical packages, the lower fence is usually taken to be Q − k (Q − Q ) with k taken to be
1 3 1
either 1,5 or 3,0. Classically, this fence is called the “inner lower fence” when k is 1,5, and “outer lower fence” when k is
3,0.
2.18
upper fence
upper outlier cut-off
upper adjacent value
value in a box plot situated k times the interquartile range (2.14) above the third quartile (2.13), with a
predetermined value of k
NOTE In proprietary statistical packages, the upper fence is usually taken to be Q + k (Q − Q ), with k taken to be
3 3 1
either 1,5 or 3,0. Classically, this fence is called the “inner upper fence” when k is 1,5, and the “outer upper fence” when k
is 3,0.
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ISO 16269-4:2010(E)
2.19
lower fourth
x
L:n
for a set x u x u … u x of observed values, the quantity 0,5 [x + x ] when f = 0 or x when f > 0,
(1) (2) (n) (i) (i + 1) (i + 1)
where i is the integral part of n/4 and f is the fractional part of n/4
NOTE 1 This definition of a lower fourth is used to determine the recommended values of k and k given in Annex C
L U
and is the default or optional setting in some widely used statistical packages.
NOTE 2 The lower fourth and the upper fourth (2.20) as a pair are sometimes called hinges.
NOTE 3 The lower fourth is sometimes referred to as the first quartile (2.12).
NOTE 4 When f = 0, 0,5 or 0,75, the lower fourth is identical to the first quartile. For example:
Sample size i = integral f = fractional First quartile Lower fourth
n part of n/4 part of n/4
9 2 0,25 [x + x ]/2 x
(2) (3) (3)
10 2 0,50 x x
(3) (3)
11 2 0,75 x x
(3) (3)
12 3 0 [x + x]/2 [x + x ]/2
(3) (4) (3) (4)
2.20
upper fourth
x
U:n
for a set x u x u … u x of observed values, the quantity 0,5 [x + x ] when f = 0 or x
(1) (2) (n) (n − i) (n − i + 1) (n − i)
when f > 0, where i is the integral part of n/4 and f is the fractional part of n/4
NOTE 1 This definition of an upper fourth is used to determine the recommended values of k and k given in Annex C
L U
and is the default or optional setting in some widely used statistical packages.
NOTE 2 The lower fourth (2.19) and the upper fourth as a pair are sometimes called hinges.
NOTE 3 The upper fourth is sometimes referred to as the third quartile (2.13).
NOTE 4 When f = 0, 0,5 or 0,75, the upper fourth is identical to the third quartile. For example:
Sample size i = integral f = fractional Third quartile Upper fourth
n part of n/4 part of n/4
9 2 0,25 [x + x ]/2 x
(7) (8) (7)
10 2 0,50 x x
(8) (8)
11 2 0,75 x x
(9) (9)
12 3 0 [x + x ]/2 [x + x ]/2
(9) (10) (9) (10)
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ISO 16269-4:2010(E)
2.21
Type I error
rejection of the null hypothesis when in fact it is true
[ISO 3534-1:2006, definition 1.46]
NOTE 1 A Type I error is an incorrect decision. Hence, it is desired to keep the probability of making such an incorrect
decision as small as possible.
NOTE 2 It is possible in some situations (for example, testing the binomial parameter p) that a pre-specified
significance level such as 0,05 is not attainable due to discreteness in outcomes.
2.22
normal distribution
Gaussian distribution
continuous distribution having the probability density function
2
⎧⎫
x − µ
1 ()
⎪⎪
fx()=−exp
⎨⎬
2
σπ2
2σ
⎪⎪
⎩⎭
where −∞ < x < ∞ and with parameters −∞ < µ < ∞ and σ > 0
[ISO 3534-1:2006, definition 2.50]
NOTE 1 The location parameter µ is the mean and the scale parameter σ is the standard deviation of the normal
distribution.
NOTE 2 A normal sample is a random sample (2.1) taken from a population that follows a normal distribution.
2.23
gamma distribution
continuous distribution having the probability density function
α −1
xxexp − / β
()
fx() =
α
βαΓ()
where x > 0 and parameters α > 0, β > 0
[ISO 3534-1:2006, definition 2.56]
NOTE 1 The gamma distribution is used in reliability applications for modelling time to failure. It includes the
exponential distribution (2.24) as a special case as well as other cases with failure rates that increase with age.
2
NOTE 2 The mean of the gamma distribution is αβ. The variance of the gamma distribution is αβ .
NOTE 3 A gamma sample is a random sample (2.1) taken from a population that follows a gamma distribution.
2.24
exponential distribution
continuous distribution having the probability density function
−1
fx()=−ββexp x/
()
where x > 0 and with parameter β > 0
[ISO 3534-1:2006, definition 2.58]
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ISO 16269-4:2010(E)
NOTE 1 The exponential distribution provides a baseline in reliability applications, corresponding to the case of “lack of
ageing” or memory-less property.
2
NOTE 2 The mean of the exponential distribution is β. The variance of the exponential distribution is β .
NOTE 3 An exponential sample is a random sample (2.1) taken from a population that follows an exponential
distribution.
2.25
Weibull distribution
type III extreme-value distribution
continuous distribution having the distribution function
κ
⎧⎫
⎛⎞x −θ
⎪⎪
Fx()=−1 exp −
⎨⎬
⎜⎟
β
⎝⎠
⎪⎪
⎩⎭
where x > θ with parameters −∞ < θ < ∞, β > 0, κ > 0
[ISO 3534-1:2006, definition 2.63]
NOTE 1 In addition to serving as one of the three possible limiting distributions of extreme order statistics, the Weibull
distribution occupies a prominent place in diverse applications, particularly reliability and engineering. The Weibull
distribution has been demonstrated to provide usable fits to a variety of data sets.
NOTE 2 The parameter θ is a location or threshold parameter in the sense that it is the minimum value that a Weibull
variate can achieve. The parameter β is a scale parameter (related to the standard deviation of a Weibull variate). The
parameter κ is a shape parameter.
NOTE 3 A Weibull sample is a random sample (2.1) taken from a population that follows a Weibull distribution.
2.26
lognormal distribution
continuous distribution having the probability density function
2
⎧⎫
1(⎪⎪ln x − µ)
fx()=−exp
⎨⎬
2
xσ 2π
2σ
⎪⎪
⎩⎭
where x > 0 and with parameters −∞ < µ < ∞ and σ > 0
[ISO 3534-1:2006, definition 2.52]
2.27
type I extreme-value distribution
Gumbel distribution
continuous distribution having the distribution function
−−()x µ/σ
Fx()=−exp e
{ }
where −∞ < x < ∞ and with parameters −∞ < µ < ∞ and σ > 0
NOTE Extreme-value distributions provide appropriate reference distributions for the extreme order statistics (2.10)
x and x .
(1) (n)
[ISO 3534-1:2006, definition 2.61]
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ISO 16269-4:2010(E)
3 Symbols
The symbols and abbreviated terms used in this part of ISO 16269 are as follows:
GESD generalized extreme studentized deviate
G Greenwood's statistic
E
g critical value of the Greenwood's test statistic for sample size n
E;n
(0)
I reduced sample of size n − l after removing the most extreme observation x in the original sample
l
(1)
I of size n, removing the most extreme observation x in the reduced sample I of size n − 1,….,
0 1
(l − 1)
and removing the most extreme observation x in the reduced sample I of size n − l + 1
l − 1
F pth percentile of a F-distribution with ν and ν degrees of freedom
p;,ν ν 1 2
12
(l)
λ critical value of the GESD test in testing whether the value x is an outlier
l
L lower fence of a modified box plot
F
U upper fence of a modified box plot
F
M or Q sample median
2
M median absolute deviation about the median
ad
Q first quartile
1
Q third quartile
3
(l)
R GESD test statistic for
...
SLOVENSKI STANDARD
SIST ISO 16269-4:2014
01-januar-2014
6WDWLVWLþQRWROPDþHQMHSRGDWNRYGHO =D]QDYDQMHLQREUDYQDYDRVDPHOFHY
Statistical interpretation of data - Part 4: Detection and treatment of outliers
Interprétation statistique des données - Partie 4: Détection et traitement des valeurs
aberrantes
Ta slovenski standard je istoveten z: ISO 16269-4:2010
ICS:
03.120.30 8SRUDEDVWDWLVWLþQLKPHWRG Application of statistical
methods
SIST ISO 16269-4:2014 en
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.
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SIST ISO 16269-4:2014
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SIST ISO 16269-4:2014
INTERNATIONAL ISO
STANDARD 16269-4
First edition
2010-10-15
Statistical interpretation of data —
Part 4:
Detection and treatment of outliers
Interprétation statistique des données —
Partie 4: Détection et traitement des valeurs aberrantes
Reference number
ISO 16269-4:2010(E)
©
ISO 2010
---------------------- Page: 3 ----------------------
SIST ISO 16269-4:2014
ISO 16269-4:2010(E)
PDF disclaimer
This PDF file may contain embedded typefaces. In accordance with Adobe's licensing policy, this file may be printed or viewed but
shall not be edited unless the typefaces which are embedded are licensed to and installed on the computer performing the editing. In
downloading this file, parties accept therein the responsibility of not infringing Adobe's licensing policy. The ISO Central Secretariat
accepts no liability in this area.
Adobe is a trademark of Adobe Systems Incorporated.
Details of the software products used to create this PDF file can be found in the General Info relative to the file; the PDF-creation
parameters were optimized for printing. Every care has been taken to ensure that the file is suitable for use by ISO member bodies. In
the unlikely event that a problem relating to it is found, please inform the Central Secretariat at the address given below.
COPYRIGHT PROTECTED DOCUMENT
© ISO 2010
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means,
electronic or mechanical, including photocopying and microfilm, without permission in writing from either ISO at the address below or
ISO's member body in the country of the requester.
ISO copyright office
Case postale 56 • CH-1211 Geneva 20
Tel. + 41 22 749 01 11
Fax + 41 22 749 09 47
E-mail copyright@iso.org
Web www.iso.org
Published in Switzerland
ii © ISO 2010 – All rights reserved
---------------------- Page: 4 ----------------------
SIST ISO 16269-4:2014
ISO 16269-4:2010(E)
Contents Page
Foreword .iv
Introduction.v
1 Scope.1
2 Terms and definitions .1
3 Symbols.10
4 Outliers in univariate data .11
4.1 General .11
4.1.1 What is an outlier? .11
4.1.2 What are the causes of outliers? .11
4.1.3 Why should outliers be detected?.11
4.2 Data screening.12
4.3 Tests for outliers .14
4.3.1 General .14
4.3.2 Sample from a normal distribution.14
4.3.3 Sample from an exponential distribution.16
4.3.4 Samples taken from some known non-normal distributions.18
4.3.5 Sample taken from unknown distributions.19
4.3.6 Cochran's test for outlying variance .21
4.4 Graphical test of outliers .22
5 Accommodating outliers in univariate data.23
5.1 Robust data analysis.23
5.2 Robust estimation of location.24
5.2.1 General .24
5.2.2 Trimmed mean .24
5.2.3 Biweight location estimate .25
5.3 Robust estimation of dispersion .25
5.3.1 General .25
5.3.2 Median-median absolute pair-wise deviation.25
5.3.3 Biweight scale estimate.26
6 Outliers in multivariate and regression data .26
6.1 General .26
6.2 Outliers in multivariate data .26
6.3 Outliers in linear regression.28
6.3.1 General .28
6.3.2 Linear regression models.29
6.3.3 Detecting outlying Y observations.31
6.3.4 Identifying outlying X observations.31
6.3.5 Detecting influential observations.32
6.3.6 A robust regression procedure.35
Annex A (informative) Algorithm for the GESD outliers detection procedure .36
Annex B (normative) Critical values of outliers test statistics for exponential samples .37
Annex C (normative) Factor values of the modified box plot .44
Annex D (normative) Values of the correction factors for the robust estimators of the scale
parameter .47
Annex E (normative) Critical values of Cochran's test statistic .48
Annex F (informative) A structured guide to detection of outliers in univariate data .51
Bibliography.54
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Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
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 16269-4 was prepared by Technical Committee ISO/TC 69, Applications of statistical methods.
ISO 16269 consists of the following parts, under the general title Statistical interpretation of data:
⎯ Part 4: Detection and treatment of outliers
⎯ Part 6: Determination of statistical tolerance intervals
⎯ Part 7: Median — Estimation and confidence intervals
⎯ Part 8: Determination of prediction intervals
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Introduction
Identification of outliers is one of the oldest problems in interpreting data. Causes of outliers include
measurement error, sampling error, intentional under- or over-reporting of sampling results, incorrect
recording, incorrect distributional or model assumptions of the data set, and rare observations, etc.
Outliers can distort and reduce the information contained in the data source or generating mechanism. In the
manufacturing industry, the existence of outliers will undermine the effectiveness of any process/product
design and quality control procedures. Possible outliers are not necessarily bad or erroneous. In some
situations, an outlier may carry essential information and thus it should be identified for further study.
The study and detection of outliers from measurement processes leads to better understanding of the
processes and proper data analysis that subsequently results in improved inferences.
In view of the enormous volume of literature on the topic of outliers, it is of great importance for the
international community to identify and standardize a sound subset of methods used in the identification and
treatment of outliers. The implementation of this part of ISO 16269 enables business and industry to recognize
the data analyses conducted across member countries or organizations.
Six annexes are provided. Annex A provides an algorithm for computing the test statistic and critical values of
a procedure in detecting outliers in a data set taken from a normal distribution. Annexes B, D and E provide
the tables needed to implement the recommended procedures. Annex C provides the tables and statistical
theory that underlie the construction of modified box plots in outlier detection. Annex F provides a structured
guide and flow chart to the procedures recommended in this part of ISO 16269.
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Statistical interpretation of data —
Part 4:
Detection and treatment of outliers
1 Scope
This part of ISO 16269 provides detailed descriptions of sound statistical testing procedures and graphical
data analysis methods for detecting outliers in data obtained from measurement processes. It recommends
sound robust estimation and testing procedures to accommodate the presence of outliers.
This part of ISO 16269 is primarily designed for the detection and accommodation of outlier(s) from univariate
data. Some guidance is provided for multivariate and regression data.
2 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
2.1
sample
data set
subset of a population made up of one or more sampling units
NOTE 1 The sampling units could be items, numerical values or even abstract entities depending on the population of
interest.
NOTE 2 A sample from a normal (2.22), a gamma (2.23), an exponential (2.24), a Weibull (2.25), a
lognormal (2.26) or a type I extreme value (2.27) population will often be referred to as a normal, a gamma, an
exponential, a Weibull, a lognormal or a type I extreme value sample, respectively.
2.2
outlier
member of a small subset of observations that appears to be inconsistent with the remainder of a given
sample (2.1)
NOTE 1 The classification of an observation or a subset of observations as outlier(s) is relative to the chosen model for
the population from which the data set originates. This or these observations are not to be considered as genuine
members of the main population.
NOTE 2 An outlier may originate from a different underlying population, or be the result of incorrect recording or gross
measurement error.
NOTE 3 The subset may contain one or more observations.
2.3
masking
presence of more than one outlier (2.2), making each outlier difficult to detect
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2.4
some-outside rate
probability that one or more observations in an uncontaminated sample will be wrongly classified as
outliers (2.2)
2.5
outlier accommodation method
method that is insensitive to the presence of outliers (2.2) when providing inferences about the population
2.6
resistant estimation
estimation method that provides results that change only slightly when a small portion of the data values in a
data set (2.1) is replaced, possibly with very different data values from the original ones
2.7
robust estimation
estimation method that is insensitive to small departures from assumptions about the underlying probability
model of the data
NOTE An example is an estimation method that works well for, say, a normal distribution (2.22), and remains
reasonably good if the actual distribution is skew or heavy-tailed. Classes of such methods include the L-estimation
[weighted average of order statistics (2.10)] and M-estimation methods (see Reference [9]).
2.8
rank
position of an observed value in an ordered set of observed values
NOTE 1 The observed values are arranged in ascending order (counting from below) or descending order (counting
from above).
NOTE 2 For the purposes of this part of ISO 16269, identical observed values are ranked as if they were slightly
different from one another.
2.9
depth
〈box plot〉 smaller of the two ranks (2.8) determined by counting up from the smallest value of the
sample (2.1), or counting down from the largest value
NOTE 1 The depth may not be an integer value (see Annex C).
NOTE 2 For all summary values other than the median (2.11), a given depth identifies two (data) values, one below
the median and the other above the median. For example, the two data values with depth 1 are the smallest value
(minimum) and largest value (maximum) in the given sample (2.1).
2.10
order statistic
statistic determined by its ranking in a non-decreasing arrangement of random variables
[ISO 3534-1:2006, definition 1.9]
NOTE 1 Let the observed values of a random sample be {x , x , …, x }. Reorder the observed values in non-
1 2 n
decreasing order designated as x u x u … u x u … u x ; then x is the observed value of the kth order statistic in
(1) (2) (k) (n) (k)
a sample of size n.
NOTE 2 In practical terms, obtaining the order statistics for a sample (2.1) amounts to sorting the data as formally
described in Note 1.
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2.11
median
sample median
median of a set of numbers
Q
2
[(n + 1)/2]th order statistic (2.10), if the sample size n is odd; sum of the [n/2]th and the [(n/2) + 1]th order
statistics divided by 2, if the sample size n is even
[ISO 3534-1:2006, definition 1.13]
NOTE The sample median is the second quartile (Q ).
2
2.12
first quartile
sample lower quartile
Q
1
for an odd number of observations, median (2.11) of the smallest (n − 1)/2 observed values; for an even
number of observations, median of the smallest n/2 observed values
NOTE 1 There are many definitions in the literature of a sample quartile, which produce slightly different results. This
definition has been chosen both for its ease of application and because it is widely used.
NOTE 2 Concepts such as hinges or fourths (2.19 and 2.20) are popular variants of quartiles. In some cases
(see Note 3 to 2.19), the first quartile and the lower fourth (2.19) are identical.
2.13
third quartile
sample upper quartile
Q
3
for an odd number of observations, median of the largest (n − 1)/2 observed values; for an even number of
observations, median of the largest n/2 observed values
NOTE 1 There are many definitions in the literature of a sample quartile, which produce slightly different results. This
definition has been chosen both for its ease of application and because it is widely used.
NOTE 2 Concepts such as hinges or fourths (2.19 and 2.20) are popular variants of quartiles. In some cases
(see Note 3 to 2.20), the third quartile and the upper fourth (2.20) are identical.
2.14
interquartile range
IQR
difference between the third quartile (2.13) and the first quartile (2.12)
NOTE 1 This is one of the widely used statistics to describe the spread of a data set.
NOTE 2 The difference between the upper fourth (2.20) and the lower fourth (2.19) is called the fourth-spread and is
sometimes used instead of the interquartile range.
2.15
five-number summary
the minimum, first quartile (2.12), median (2.11), third quartile (2.13), and maximum
NOTE The five-number summary provides numerical information about the location, spread and range.
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2.16
box plot
horizontal or vertical graphical representation of the five-number summary (2.15).
NOTE 1 For the horizontal version, the first quartile (2.12) and the third quartile (2.13) are plotted as the left and
right sides, respectively, of a box, the median (2.11) is plotted as a vertical line across the box, the whiskers stretching
downwards from the first quartile to the smallest value at or above the lower fence (2.17) and upwards from the third
quartile to the largest value at or below the upper fence (2.18), and value(s) beyond the lower and upper fences are
marked separately as outlier(s) (2.2). For the vertical version, the first and third quartiles are plotted as the bottom and the
top, respectively, of a box, the median is plotted as a horizontal line across the box, the whiskers stretching downwards
from the first quartile to the smallest value at or above the lower fence and upwards from the third quartile to the largest
value at or below the upper fence and value(s) beyond the lower and upper fences are marked separately as outlier(s).
NOTE 2 The box width and whisker length of a box plot provide graphical information about the location, spread,
skewness, tail lengths, and outlier(s) of a sample. Comparisons between box plots and the density function of a) uniform,
b) bell-shaped, c) right-skewed, and d) left-skewed distributions are given in the diagrams in Figure 1. In each distribution,
a histogram is shown above the boxplot.
NOTE 3 A box plot constructed with its lower fence (2.17) and upper fence (2.18) evaluated by taking k to be a value
based on the sample size n and the knowledge of the underlying distribution of the sample data is called a modified box
plot (see example, Figure 2). The construction of a modified box plot is given in 4.4.
a) Uniform distribution b) Bell-shaped distribution
Figure 1 (continued)
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c) Right-skewed distribution d) Left-skewed distribution
Key
X data values
Y frequency
In each distribution, a histogram is shown above the box plot.
Figure 1 — Box plots and histograms for a) uniform, b) bell-shaped, c) right-skewed,
and d) left-skewed distributions
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Figure 2 — Modified box plot with lower and upper fences
2.17
lower fence
lower outlier cut-off
lower adjacent value
value in a box plot (2.16) situated k times the interquartile range (2.14) below the first quartile (2.12), with
a predetermined value of k
NOTE In proprietary statistical packages, the lower fence is usually taken to be Q − k (Q − Q ) with k taken to be
1 3 1
either 1,5 or 3,0. Classically, this fence is called the “inner lower fence” when k is 1,5, and “outer lower fence” when k is
3,0.
2.18
upper fence
upper outlier cut-off
upper adjacent value
value in a box plot situated k times the interquartile range (2.14) above the third quartile (2.13), with a
predetermined value of k
NOTE In proprietary statistical packages, the upper fence is usually taken to be Q + k (Q − Q ), with k taken to be
3 3 1
either 1,5 or 3,0. Classically, this fence is called the “inner upper fence” when k is 1,5, and the “outer upper fence” when k
is 3,0.
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2.19
lower fourth
x
L:n
for a set x u x u … u x of observed values, the quantity 0,5 [x + x ] when f = 0 or x when f > 0,
(1) (2) (n) (i) (i + 1) (i + 1)
where i is the integral part of n/4 and f is the fractional part of n/4
NOTE 1 This definition of a lower fourth is used to determine the recommended values of k and k given in Annex C
L U
and is the default or optional setting in some widely used statistical packages.
NOTE 2 The lower fourth and the upper fourth (2.20) as a pair are sometimes called hinges.
NOTE 3 The lower fourth is sometimes referred to as the first quartile (2.12).
NOTE 4 When f = 0, 0,5 or 0,75, the lower fourth is identical to the first quartile. For example:
Sample size i = integral f = fractional First quartile Lower fourth
n part of n/4 part of n/4
9 2 0,25 [x + x ]/2 x
(2) (3) (3)
10 2 0,50 x x
(3) (3)
11 2 0,75 x x
(3) (3)
12 3 0 [x + x]/2 [x + x ]/2
(3) (4) (3) (4)
2.20
upper fourth
x
U:n
for a set x u x u … u x of observed values, the quantity 0,5 [x + x ] when f = 0 or x
(1) (2) (n) (n − i) (n − i + 1) (n − i)
when f > 0, where i is the integral part of n/4 and f is the fractional part of n/4
NOTE 1 This definition of an upper fourth is used to determine the recommended values of k and k given in Annex C
L U
and is the default or optional setting in some widely used statistical packages.
NOTE 2 The lower fourth (2.19) and the upper fourth as a pair are sometimes called hinges.
NOTE 3 The upper fourth is sometimes referred to as the third quartile (2.13).
NOTE 4 When f = 0, 0,5 or 0,75, the upper fourth is identical to the third quartile. For example:
Sample size i = integral f = fractional Third quartile Upper fourth
n part of n/4 part of n/4
9 2 0,25 [x + x ]/2 x
(7) (8) (7)
10 2 0,50 x x
(8) (8)
11 2 0,75 x x
(9) (9)
12 3 0 [x + x ]/2 [x + x ]/2
(9) (10) (9) (10)
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2.21
Type I error
rejection of the null hypothesis when in fact it is true
[ISO 3534-1:2006, definition 1.46]
NOTE 1 A Type I error is an incorrect decision. Hence, it is desired to keep the probability of making such an incorrect
decision as small as possible.
NOTE 2 It is possible in some situations (for example, testing the binomial parameter p) that a pre-specified
significance level such as 0,05 is not attainable due to discreteness in outcomes.
2.22
normal distribution
Gaussian distribution
continuous distribution having the probability density function
2
⎧⎫
x − µ
1 ()
⎪⎪
fx()=−exp
⎨⎬
2
σπ2
2σ
⎪⎪
⎩⎭
where −∞ < x < ∞ and with parameters −∞ < µ < ∞ and σ > 0
[ISO 3534-1:2006, definition 2.50]
NOTE 1 The location parameter µ is the mean and the scale parameter σ is the standard deviation of the normal
distribution.
NOTE 2 A normal sample is a random sample (2.1) taken from a population that follows a normal distribution.
2.23
gamma distribution
continuous distribution having the probability density function
α −1
xxexp − / β
()
fx() =
α
βαΓ()
where x > 0 and parameters α > 0, β > 0
[ISO 3534-1:2006, definition 2.56]
NOTE 1 The gamma distribution is used in reliability applications for modelling time to failure. It includes the
exponential distribution (2.24) as a special case as well as other cases with failure rates that increase with age.
2
NOTE 2 The mean of the gamma distribution is αβ. The variance of the gamma distribution is αβ .
NOTE 3 A gamma sample is a random sample (2.1) taken from a population that follows a gamma distribution.
2.24
exponential distribution
continuous distribution having the probability density function
−1
fx()=−ββexp x/
()
where x > 0 and with parameter β > 0
[ISO 3534-1:2006, definition 2.58]
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NOTE 1 The exponential distribution provides a baseline in reliability applications, corresponding to the case of “lack of
ageing” or memory-less property.
2
NOTE 2 The mean of the exponential distribution is β. The variance of the exponential distribution is β .
NOTE 3 An exponential sample is a random sample (2.1) taken from a population that follows an exponential
distribution.
2.25
Weibull distribution
type III extreme-value distribution
continuous distribution having the distribution function
κ
⎧⎫
⎛⎞x −θ
⎪⎪
Fx()=−1 exp −
⎨⎬
⎜⎟
β
⎝⎠
⎪⎪
⎩⎭
where x > θ with parameters −∞ < θ < ∞, β > 0, κ > 0
[ISO 3534-1:2006, definition 2.63]
NOTE 1 In addition to serving as one of the three possible limiting distributions of extreme order statistics, the Weibull
distribution occupies a prominent place in diverse applications, particularly reliability and engineering. The Weibull
distribution has been demonstrated to provide usable fits to a variety of data sets.
NOTE 2 The parameter θ is a location or threshold parameter in the sense that it is the minimum value that a Weibull
variate can achieve. The parameter β is a scale parameter (related to the standard deviation of a Weibull variate). The
parameter κ is a shape parameter.
NOTE 3 A Weibull sample is a random sample (2.1) taken from a population that follows a Weibull distribution.
2.26
lognormal distribution
continuous distribution having the probability density function
2
⎧⎫
1(⎪⎪ln x − µ)
fx()=−exp
⎨⎬
2
xσ 2π
2σ
⎪⎪
⎩⎭
where x > 0 and with parameters −∞ < µ < ∞ and σ > 0
[ISO 3534-1:2006, definition 2.52]
2.27
type I extreme-value distribution
Gumbel distribution
continuous distribution having the distribution function
−−()x µ/σ
Fx()=−exp e
{ }
where −∞ < x < ∞ and with parameters −∞ < µ < ∞ and σ > 0
NOTE Extreme-value distributions provide appropriate reference distributions for the extreme
...
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