ASTM D7915-22

Standard Practice for Application of Generalized Extreme Studentized Deviate (GESD) Technique to Simultaneously Identify Multiple Outliers in a Data Set

SIGNIFICANCE AND USE
3.1 The GESD procedure can be used to simultaneously identify up to a pre-determined number of outliers (r) in a data set, without having to pre-examine the data set and make a priori decisions as to the location and number of potential outliers.  
3.2 The GESD procedure is robust to masking. Masking describes the phenomenon where the existence of multiple outliers can prevent an outlier identification procedure from declaring any of the observations in a data set to be outliers.  
3.3 The GESD procedure is automation-friendly, and hence can easily be programmed as automated computer algorithms.
SCOPE
1.1 This practice provides a step by step procedure for the application of the Generalized Extreme Studentized Deviate (GESD) Many-Outlier Procedure to simultaneously identify multiple outliers in a data set. (See Bibliography.)  
1.2 This practice is applicable to a data set comprising observations that is represented on a continuous numerical scale.  
1.3 This practice is applicable to a data set comprising a minimum of six observations.  
1.4 This practice is applicable to a data set where the normal (Gaussian) model is reasonably adequate for the distributional representation of the observations in the data set.  
1.5 The probability of false identification of outliers associated with the decision criteria set by this practice is 0.01.  
1.6 It is recommended that the execution of this practice be conducted under the guidance of personnel familiar with the statistical principles and assumptions associated with the GESD technique.  
1.7 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appropriate safety, health, and environmental practices and determine the applicability of regulatory limitations prior to use.  
1.8 This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.

General Information

• Status: Published
• Publication Date: 30-Apr-2022
• ICS: 03.120.30 - Application of statistical methods
• Technical Committee: D02 - Petroleum Products, Liquid Fuels, and Lubricants
• Drafting Committee: D02.94 - Coordinating Subcommittee on Quality Assurance and Statistics

Overview

ASTM D7915-22: Standard Practice for Application of Generalized Extreme Studentized Deviate (GESD) Technique to Simultaneously Identify Multiple Outliers in a Data Set provides a systematic methodology for detecting multiple outliers in data. This internationally recognized standard from ASTM is used to enhance the reliability of statistical analyses, particularly for data sets where the normal (Gaussian) distribution is assumed. The GESD approach is robust and allows simultaneous identification of up to a specified number of outliers without prior assumptions about their number or position.

Key Topics

• Multivariate Outlier Detection: Employs the GESD procedure to test and remove outliers within a continuous, normally distributed data set containing at least six observations.
• Robustness to Masking: GESD addresses "masking," a situation where multiple outliers make standard detection methods ineffective.
• Automation-Friendly Method: Easily programmable, the GESD procedure supports automated quality control and statistical analysis systems.
• Probability of False Identification: The test criteria maintain a strict 1% chance of falsely identifying an observation as an outlier, supporting high data integrity.
• Guidance Requirement: Best practices recommend that statistical experts oversee the application of the technique, as proper use depends on understanding its underlying assumptions.

Applications

The ASTM D7915-22 standard finds widespread use across industries and laboratory settings where data quality and integrity are critical. Key applications include:

• Quality Assurance in Manufacturing: Detects anomalous readings that could indicate process deviations or faulty products.
• Petroleum and Fuel Testing: Commonly used in laboratories analyzing petroleum products, ensuring that outliers do not skew results for compliance and safety.
• Environmental Data Analysis: Identifies data points that may result from measurement errors, instrument malfunctions, or rare events.
• Research and Development: Ensures data sets used in R&D are accurate, enabling reliable conclusions in scientific studies.
• Automated Statistical Software: The GESD procedure can be embedded within automated data analysis tools for real-time monitoring.

By effectively identifying multiple outliers in a set of continuous numerical observations, ASTM D7915-22 benefits any application requiring rigorous data validation under the normal distribution assumption.

Related Standards

Understanding the context and interoperability of ASTM D7915-22 is enhanced by familiarity with related standards and references:

• ASTM Research Report D2-1481: Tutorial for Generalized Extreme Studentized Deviate (GESD) Many-Outlier Procedure.
• ASQC Basic References in Quality Control: Statistical Techniques, Volume 16: “How to Detect and Handle Outliers” by Boris Iglewicz and David Hoaglin.
• Rosner, Bernard: “Percentage Points for a Generalized ESD Many-Outlier Procedure,” Technometrics 25: 165-172.
• General ASTM Statistical Standards: Provides additional guidelines for statistical analysis and quality assurance across industries.

Practical Value

Implementing ASTM D7915-22 enables organizations to:

• Increase accuracy of statistical reports by systematically identifying and treating multiple outliers.
• Streamline quality control processes using robust, automation-friendly algorithms.
• Apply internationally recognized quality standards to cross-border and multi-site data analysis.
• Meet the requirements for compliance and data integrity in regulated industries.

Leveraging this standard supports reliable decision-making, enhances product and data quality, and enables consistent application of best statistical practices in industry and research.

Standards Content (Sample)

This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
Designation: D7915 − 22 An American National Standard
Standard Practice for
Application of Generalized Extreme Studentized Deviate
(GESD) Technique to Simultaneously Identify Multiple
Outliers in a Data Set
This standard is issued under the fixed designation D7915; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope* 2.1.1 outlier, n—anobservation(orasubsetofobservations)
whichappearstobeinconsistentwiththeremainderofthedata
1.1 This practice provides a step by step procedure for the
set.
application of the Generalized Extreme Studentized Deviate
(GESD) Many-Outlier Procedure to simultaneously identify
3. Significance and Use
multiple outliers in a data set. (See Bibliography.)
3.1 The GESD procedure can be used to simultaneously
1.2 This practice is applicable to a data set comprising
identify up to a pre-determined number of outliers (r) in a data
observations that is represented on a continuous numerical
set, without having to pre-examine the data set and make a
scale.
priori decisions as to the location and number of potential
1.3 This practice is applicable to a data set comprising a
outliers.
minimum of six observations.
3.2 The GESD procedure is robust to masking. Masking
1.4 Thispracticeisapplicabletoadatasetwherethenormal
describes the phenomenon where the existence of multiple
(Gaussian) model is reasonably adequate for the distributional
outliers can prevent an outlier identification procedure from
representation of the observations in the data set.
declaring any of the observations in a data set to be outliers.
1.5 The probability of false identification of outliers asso-
3.3 The GESD procedure is automation-friendly, and hence
ciated with the decision criteria set by this practice is 0.01.
can easily be programmed as automated computer algorithms.
1.6 It is recommended that the execution of this practice be
conducted under the guidance of personnel familiar with the 4. Procedure
statistical principles and assumptions associated with the
4.1 Specifythemaximumnumberofoutliers(r)inadataset
GESD technique.
to be identified. This is the number of cycles required to be
1.7 This standard does not purport to address all of the
executed (see 4.2) for the identification of up to r outliers.
safety concerns, if any, associated with its use. It is the
4.1.1 The recommended maximum number of outliers (r)
responsibility of the user of this standard to establish appro-
by this practice is two (2) for data sets with six to twelve
priate safety, health, and environmental practices and deter-
observations.
mine the applicability of regulatory limitations prior to use.
4.1.2 For data sets with more than twelve observations, the
1.8 This international standard was developed in accor-
recommended maximum number of outliers (r) is the lesser of
dance with internationally recognized principles on standard-
ten (10) or 20%.
ization established in the Decision on Principles for the
4.1.3 The recommended values for r in 4.1.1 and 4.1.2 are
Development of International Standards, Guides and Recom-
not intended to be mandatory. Users can specify other values
mendations issued by the World Trade Organization Technical
based on their specific needs.
Barriers to Trade (TBT) Committee.
4.2 Set the current cycle number cto1(c = 1).
4.2.1 Assign the original data set to be assessed (in 4.1)as
2. Terminology
the data set for the current cycle 1 and label it as DTS .
2.1 Definitions of Terms Specific to This Standard:
4.3 Compute test statistic T for each observation in the data
set assigned to the current cycle (DTS ) as follows:
c
This practice is under the jurisdiction ofASTM Committee D02 on Petroleum
Products, Liquid Fuels, and Lubricants and is the direct responsibility of Subcom-
T 5 |x 2 x¯|⁄s (1)
mittee D02.94 on Coordinating Subcommittee on QualityAssurance and Statistics.
Current edition approved May 1, 2022. Published May 2022. Originally where:
approved in 1988. Last previous edition approved in 2018 as D7915–18. DOI:
x = an observation in the data set,
10.1520/D7915-22.
*A Summary of Changes section appears at the end of this standard
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
D7915 − 22
(DTS ) and all observations associated with maximum T from
x¯ = average calculated using all observations in the data set,
c
DTS to DTS are declared as outliers.
and c−1 1
s = sample standard deviation calculated using all observa-
4.8 Theoutlieridentificationprocedureisdeclaredcomplete
tions in the data set.
at the first occurrence of maximum T exceeding λ for
critical
cycle c in 4.7, or completion of comparison for c=1.
4.4 Identify the observation associated with the largest
absolute magnitude of the test statistic T in the data set of the
5. Worked Example
current cycle.
5.1 Listed below is a data set comprising 30 observations:
4.5 If current cycle c is less than r, execute 4.5.1 to 4.5.4;
35.0 36.6 34.7 36.2 37.0 25.3 37.2 41.3 26.0 24.6
otherwise go to 4.6.
33.5 35.5 35.4 39.9 39.2 36.6 37.2 33.2 34.0 35.7
39.2 42.1 35.7 40.2 36.6 41.1 41.1 39.1 40.6 41.3
4.5.1 Removetheobservationidentifiedin4.4fromthedata
5.1.1 The total number of observations (N) = 30.
set of the current cycle.
5.1.2 From 4.1.2, the maximum number of outliers r to be
4.5.2 Increment the current cycle number by 1:
identified is six (20% of 30), since six is less than ten.
c = c +1.
current
4.5.3 Assign the reduced data set in 4.5.1 (that is, data set
5.2 Refer to Table 1 for the following discussions:
with the observation identified in 4.4 removed) as the data set
5.2.1 DatasetlabeledDTS istheoriginaldataset,whichis
for the new cycle number and label it as DTS .
assigned to cycle (c)=1.
c
4.5.4 Repeat steps 4.3 to 4.5. 5.2.2 The observation 24.6, corresponding to the maximum
T value 2.60 in DTS , is identified and removed to form a
4.6 Beginning with c = r, compare the maximum T com-
reduced data set DTS .
puted in the dataset DTS , to a critical value λ associated
c critical
5.2.3 The above is repeated up to DTS , where the obser-
with the data set for cycle c, where λ is chosen based on
critical
vation 33.5 is identified as the having the maximum T value
a false identification probability of 0.01. See Table A1.1 in
1.65 but not removed since this is the last cycle for identifying
Annex A1 for λ values applicable to different data set
critical
up to r = 6 outliers.
sizes and cycle numbers (c).
5.2.4 In accordance with 4.7, working backwards from
4.7 If maximum T for data set DTS exceeds λ for
c critical c=6, the cycle number for which the first occurrence of
cycle c = r, the observation associated with maximum T in the
maximum T value of the data set DTS exceeds λ is cycle
c critical
dataset DTS and all observations associated with maximum T
c
number three (see data set column labeled DTS ).
from DTS to DTS are declared as outliers.
c–1 1 5.2.5 In accordance with 4.7, observations 24.6 from DTS ,
4.7.1 If maximum T for data set DTS does not exceed
c 25.3 from DTS , and 26.0 from DTS are all declared as
2 3
λ for cycle c, reduce c by 1 (that is, c = c − 1) and repeat
outliers by this practice.
critical
the comparison of maximum T to λ until the first
critical
6. Keywords
occurrence of maximum T exceeding λ is encountered.
critical
The observation associated with maximum T for this cycle 6.1 GESD; outliers
D7915 − 22
TABLE 1 Example Execution of the GESD Procedure for Worked Example in 5.1
NOTE 1—Explanation of Table 1:
The cell marked by a border for each DTS column is the observation with the most extreme T values (T ) in the data set i. For the convenience
i max
ofreaders T isre-showninthethirdlastrowfromthebottomofthistable.Forinstance,inDTS ,thevalue24.6hasacorresponding T valueof2.60,
max 1 1
which is the largest T value (T ) for DTS . Marking of these cells with the border is only to help the readers. It does not mean these cells are outliers.
max 1
What it means is that the marked cell is to be removed for the next required cycle. For example, in next required cycle DTS , the value 24.6 identified
as the most extreme from the previous cycle DTS is removed, and the removed cell is shown as a blank entry in DTS .
1 2
Thedecisiononwhichofthesehighlightedcellsareoutliersismadeonlyaftercompletionoftherequiredcycles(inthiscase,uptoDTS ,since r=6).
To make the outlier decision, start from DTS . Compare the T value to the critical value (λ ), both are listed at the bottom of this table for
6 max critical
readers’convenience. If T does not exceed the critical value below it, move to the previous DTS (DTS ), and if it does not exceed the critical value
max 5
below it, move to DTS . and so forth. Stop at the first DTS where the T exceeds the critical value, which is DTS in the example, where T is
4 i max 3 max
3.27,versusthecriticalvalueof3.20.Theoutliersarethendeclaredasthevalueassociatedwith T atDTS (whichis26.0),andalltheex
...

This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
Designation: D7915 − 18 D7915 − 22 An American National Standard
Standard Practice for
Application of Generalized Extreme Studentized Deviate
(GESD) Technique to Simultaneously Identify Multiple
Outliers in a Data Set
This standard is issued under the fixed designation D7915; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope*
1.1 This practice provides a step by step procedure for the application of the Generalized Extreme Studentized Deviate (GESD)
Many-Outlier Procedure to simultaneously identify multiple outliers in a data set. (See Bibliography.)
1.2 This practice is applicable to a data set comprising observations that is represented on a continuous numerical scale.
1.3 This practice is applicable to a data set comprising a minimum of six observations.
1.4 This practice is applicable to a data set where the normal (Gaussian) model is reasonably adequate for the distributional
representation of the observations in the data set.
1.5 The probability of false identification of outliers associated with the decision criteria set by this practice is 0.01.
1.6 It is recommended that the execution of this practice be conducted under the guidance of personnel familiar with the statistical
principles and assumptions associated with the GESD technique.
1.7 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility
of the user of this standard to establish appropriate safety, health, and environmental practices and determine the applicability of
regulatory limitations prior to use.
1.8 This international standard was developed in accordance with internationally recognized principles on standardization
established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued
by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
2. Terminology
2.1 Definitions of Terms Specific to This Standard:
2.1.1 outlier, n—an observation (or a subset of observations) which appears to be inconsistent with the remainder of the data set.
This practice is under the jurisdiction of ASTM Committee D02 on Petroleum Products, Liquid Fuels, and Lubricants and is the direct responsibility of Subcommittee
D02.94 on Coordinating Subcommittee on Quality Assurance and Statistics.
Current edition approved July 1, 2018May 1, 2022. Published August 2018May 2022. Originally approved in 1988. Last previous edition approved in 20142018 as
D7915 – 14.D7915 – 18. DOI: 10.1520/D7915-18.10.1520/D7915-22.
*A Summary of Changes section appears at the end of this standard
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
D7915 − 22
3. Significance and Use
3.1 The GESD procedure can be used to simultaneously identify up to a pre-determined number of outliers (r) in a data set,
without having to pre-examine the data set and make a priori decisions as to the location and number of potential outliers.
3.2 The GESD procedure is robust to masking. Masking describes the phenomenon where the existence of multiple outliers can
prevent an outlier identification procedure from declaring any of the observations in a data set to be outliers.
3.3 The GESD procedure is automation-friendly, and hence can easily be programmed as automated computer algorithms.
4. Procedure
4.1 Specify the maximum number of outliers (r) in a data set to be identified. This is the number of cycles required to be executed
(see 4.2) for the identification of up to r outliers.
4.1.1 The recommended maximum number of outliers (r) by this practice is two (2) for data sets with six to twelve observations.
4.1.2 For data sets with more than twelve observations, the recommended maximum number of outliers (r) is the lesser of ten (10)
or 20 %.
4.1.3 The recommended values for r in 4.1.1 and 4.1.2 are not intended to be mandatory. Users can specify other values based
on their specific needs.
4.2 Set the current cycle number c to 1 (c = 1).
4.2.1 Assign the original data set to be assessed (in 4.1) as the data set for the current cycle 1 and label it as DTS .
4.3 Compute test statistic T for each observation in the data set assigned to the current cycle (DTS ) as follows:
c
T 5 |x 2 x¯ | ⁄s (1)
where:
x = an observation in the data set,
x¯ = average calculated using all observations in the data set, and
s = sample standard deviation calculated using all observations in the data set.
4.4 Identify the observation associated with the largest absolute magnitude of the test statistic T in the data set of the current cycle.
4.5 If current cycle c is less than r, execute 4.5.1 to 4.5.4; otherwise go to 4.6.
4.5.1 Remove the observation identified in 4.4 from the data set of the current cycle.
4.5.2 Increment the current cycle number by 1:
c = c + 1.
current
4.5.3 Assign the reduced data set in 4.5.1 (that is, data set with the observation identified in 4.4 removed) as the data set for the
new cycle number and label it as DTS .
c
4.5.4 Repeat steps 4.3 to 4.5.
4.6 Beginning with c = r, compare the maximum T computed in the dataset DTS , to a critical value λ associated with the
c critical
data set for cycle c, where λ is chosen based on a false identification probability of 0.01. See Table A1.1 in Annex A1 for
critical
λ values applicable to different data set sizes and cycle numbers (c).
critical
4.7 If maximum T for data set DTS does not exceed exceeds λ for cycle c, reduce c by 1 (that is, c= cr, − 1) and repeat the
c critical
D7915 − 22
comparison of maximum the T to λ until the first occurrence of maximum T exceeding λ is encountered. The observation
critical critical
associated with maximum T for this cycle (DTSin the dataset DTS ) and all observations associated with maximum T from DTS
c c
−– 1 to DTS are declared as outliers.
4.7.1 If maximum T for data set DTS does not exceed λ for cycle c, reduce c by 1 (that is, c = c − 1) and repeat the
c critical
comparison of maximum T to λ until the first occurrence of maximum T exceeding λ is encountered. The observation
critical critical
associated with maximum T for this cycle (DTS ) and all observations associated with maximum T from DTS to DTS are
c c − 1 1
declared as outliers.
4.8 The outlier identification procedure is declared complete at the first occurrence of maximum T exceeding λ for cycle c
critical
in 4.7, or completion of comparison for c = 1.
5. Worked Example
5.1 Listed below is a data set comprising 30 observations:
35.0 36.6 34.7 36.2 37.0 25.3 37.2 41.3 26.0 24.6
33.5 35.5 35.4 39.9 39.2 36.6 37.2 33.2 34.0 35.7
39.2 42.1 35.7 40.2 36.6 41.1 41.1 39.1 40.6 41.3
5.1.1 The total number of observations (N) = 30.
5.1.2 From 4.1.2, the maximum number of outliers r to be identified is six (20 % of 30), since six is less than ten.
5.2 Refer to Table 1 for the following discussions:
5.2.1 Data set labeled DTS is the original data set, which is assigned to cycle (c) = 1.
5.2.2 The observation 24.6, corresponding to the maximum T value 2.60 in DTS , is identified and removed to form a reduced
data set DTS .
5.2.3 The above is repeated up to DTS , where the observation 33.5 is identified as the having the maximum T value 1.65 but not
removed since this is the last cycle for identifying up to r = 6 outliers.
5.2.4 In accordance with 4.7, working backwards from c = 6, the cycle number for which the first occurrence of maximum T value
of the data set DTS exceeds λ is cycle number three (see data set column labeled DTS ).
c critical 3
5.2.5 In accordance with 4.7, observations 24.6 from DTS , 25.3 from DTS , and 26.0 from DTS are all declared as outliers by
1 2 3
this practice.
6. Keywords
6.1 GESD; outliers
D7915 − 22
TABLE 1 Example Execution of the GESD Procedure for Worked Example in 5.1
NOTE 1—Explanation of Table 1:
The cell marked by a border for each DTS column is the observation with the most extreme T values (T ) in the data set i. For the convenience
i max
of readers T is re-shown in the third last row from the bottom of this table. For instance, in DTS , the value 24.6 has a corresponding T value of 2.60,
max 1 1
which is the largest T value (T ) for DTS . Marking of these cells with the border is only to help the readers. It does not mean these cells are outliers.
max 1
What it means is that the marked cell is to be removed for the next required cycle. For example, in next required cycle DTS , the value 24.6 identified
as the most extreme from the previous cycle DTS is removed, and the removed cell is shown as a blank entry in DTS .
1 2
The decision on which of these highlighted cells are outliers is made only after completion of the required cycles (in this case, up to DTS , since r = 6).
To make the outlier decision, start from DTS . Compare the T value to the critical value (λ ), both are listed at the bottom of this table for
6 max critical
readers’ convenience. If T does not exceed the critical value below it, move to the previous DTS ( DTS(DTS ), and if it does not exceed the critical
max 5
value below it, move to DTS . and so forth. Stop at the first DTS where the T exceeds the
...