ASTM E2586-10a
(Practice)Standard Practice for Calculating and Using Basic Statistics
Standard Practice for Calculating and Using Basic Statistics
SIGNIFICANCE AND USE
This practice provides approaches for characterizing a sample of n observations that arrive in the form of a data set. Large data sets from organizations, businesses, and governmental agencies exist in the form of records and other empirical observations. Research institutions and laboratories at universities, government agencies, and the private sector also generate considerable amounts of empirical data.
A data set containing a single variable usually consists of a column of numbers. Each row is a separate observation or instance of measurement of the variable. The numbers themselves are the result of applying the measurement process to the variable being studied or observed. We may refer to each observation of a variable as an item in the data set. In many situations, there may be several variables defined for study.
The sample is selected from a larger set called the population. The population can be a finite set of items, a very large or essentially unlimited set of items, or a process. In a process, the items originate over time and the population is dynamic, continuing to emerge and possibly change over time. Sample data serve as representatives of the population from which the sample originates. It is the population that is of primary interest in any particular study.
The data (measurements and observations) may be of the variable type or the simple attribute type. In the case of attributes, the data may be either binary trials or a count of a defined event over some interval (time, space, volume, weight, or area). Binary trials consist of a sequence of 0s and 1s in which a “1” indicates that the inspected item exhibited the attribute being studied and a “0” indicates the item did not exhibit the attribute. Each inspection item is assigned either a “0” or a “1.” Such data are often governed by the binomial distribution. For a count of events over some interval, the number of times the event is observed on the inspection interval is recorded for each of...
SCOPE
1.1 This practice covers methods and equations for computing and presenting basic descriptive statistics using a set of sample data containing a single variable. This practice includes simple descriptive statistics for variable data, tabular and graphical methods for variable data, and methods for summarizing simple attribute data. Some interpretation and guidance for use is also included.
1.2 The system of units for this practice is not specified. Dimensional quantities in the practice are presented only as illustrations of calculation methods. The examples are not binding on products or test methods treated.
1.3 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 and health practices and determine the applicability of regulatory limitations prior to use.
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An American National Standard
Designation:E2586–10a
Standard Practice for
1
Calculating and Using Basic Statistics
This standard is issued under the fixed designation E2586; 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 3.1.1 characteristic, n—a property of items in a sample or
population which, when measured, counted, or otherwise
1.1 This practice covers methods and equations for comput-
observed, helps to distinguish among the items. E2282
ing and presenting basic descriptive statistics using a set of
3.1.2 coeffıcient of variation, CV, n—for a nonnegative
sampledatacontainingasinglevariable.Thispracticeincludes
characteristic, the ratio of the standard deviation to the mean
simple descriptive statistics for variable data, tabular and
for a population or sample
graphical methods for variable data, and methods for summa-
3.1.2.1 Discussion—The coefficient of variation is often
rizing simple attribute data. Some interpretation and guidance
expressed as a percentage.
for use is also included.
3.1.2.2 Discussion—This statistic is also known as the
1.2 The system of units for this practice is not specified.
relative standard deviation, RSD.
Dimensional quantities in the practice are presented only as
3.1.3 estimate, n—sample statistic used to approximate a
illustrations of calculation methods. The examples are not
population parameter.
binding on products or test methods treated.
3.1.4 histogram, n—graphical representation of the fre-
1.3 This standard does not purport to address all of the
quency distribution of a characteristic consisting of a set of
safety concerns, if any, associated with its use. It is the
rectangles with area proportional to the frequency.
responsibility of the user of this standard to establish appro-
ISO 3534-1
priate safety and health practices and determine the applica-
3.1.4.1 Discussion—While not required, equal bar or class
bility of regulatory limitations prior to use.
widths are recommended for histograms.
th
2. Referenced Documents
3.1.5 interquartile range, IQR, n—the 75 percentile (0.75
th
2
quantile) minus the 25 percentile (0.25 quantile), for a data
2.1 ASTM Standards:
set.
E178 Practice for Dealing With Outlying Observations
3.1.6 kurtosis, g,g , n—for a population or a sample, a
E456 Terminology Relating to Quality and Statistics 2 2
measure of the weight of the tails of a distribution relative to
E2282 Guide for Defining the Test Result of a Test Method
3
the center, calculated as the ratio of the fourth central moment
2.2 ISO Standards:
(empiricalifasample,theoreticalifapopulationapplies)tothe
ISO 3534-1 Statistics—Vocabulary and Symbols, part 1:
standard deviation (sample, s, or population, s) raised to the
Probability and General Statistical Terms
fourth power, minus 3 (also referred to as excess kurtosis).
ISO 3534-2 Statistics—Vocabulary and Symbols, part 2:
3.1.7 mean, n—of a population, µ, average or expected
Applied Statistics
value of a characteristic in a population – of a sample, x, sum
3. Terminology
of the observed values in the sample divided by the sample
size.
3.1 Definitions—Unless otherwise noted, terms relating to
th
3.1.8 median, X , n—the 50 percentile in a population or
quality and statistics are as defined in Terminology E456.
sample.
3.1.8.1 Discussion—The sample median is the [(n + 1)/2]
1
This practice is under the jurisdiction ofASTM Committee E11 on Quality and
order statistic if the sample size n is odd and is the average of
Statistics and is the direct responsibility of Subcommittee E11.10 on Sampling /
the [n/2] and [n/2 + 1] order statistics if n is even.
Statistics.
Current edition approved Sept. 1, 2010. Published October 2010. Originally
3.1.9 midrange, n—average of the minimum and maximum
approved in 2007. Last previous edition approved in 2010 as E2586 – 10. DOI:
values in a sample.
10.1520/E2586-10A.
th
2 3.1.10 order statistic, x , n—value of the k observed
(k)
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM value in a sample after sorting by order of magnitude.
Standards volume information, refer to the standard’s Document Summary page on
3.1.10.1 Discussion—For a sample of size n, the first order
the ASTM website.
statistic x is the minimum value, x is the maximum value.
3
(1) (n)
Available fromAmerican National Standards Institute (ANSI), 25 W. 43rd St.,
3.1.11 parameter, n—see population parameter.
4th Floor, New York, NY 10036, http://www.ansi.org.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700
...
This document is not anASTM standard and is intended only to provide the user of anASTM 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:E2586–10 Designation:E2586–10a
Standard Practice for
1
Calculating and Using Basic Statistics
This standard is issued under the fixed designation E2586; 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 covers methods and formulasequations for computing and presenting basic descriptive statistics using a set of
sample data containing a single variable.This practice includes simple descriptive statistics for variable data, tabular and graphical
methods for variable data, and methods for summarizing simple attribute data. Some interpretation and guidance for use is also
included.
1.2 Thesystemofunitsforthispracticeisnotspecified.Dimensionalquantitiesinthepracticearepresentedonlyasillustrations
of calculation methods. The examples are not binding on products or test methods treated.
1.3 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 and health practices and determine the applicability of regulatory
limitations prior to use.
2. Referenced Documents
2
2.1 ASTM Standards:
E178 Practice for Dealing With Outlying Observations
E456 Terminology Relating to Quality and Statistics
E2282 Guide for Defining the Test Result of a Test Method
3
2.2 ISO Standards:
ISO 3534-1 Statistics—Vocabulary and Symbols, part 1: Probability and General Statistical Terms
ISO 3534-2 Statistics—Vocabulary and Symbols, part 2: Applied Statistics
3. Terminology
3.1 Definitions: Unless Definitions—Unless otherwise noted, terms relating to quality and statistics are as defined in
Terminology E456.
3.1.1 characteristic, n—a property of items in a sample or population which, when measured, counted, or otherwise observed,
helps to distinguish among the items. E2282
3.1.2 coeffıcient of variation, CV, n—for a nonnegative characteristic, the ratio of the standard deviation to the mean for a
population or sample
3.1.2.1 Discussion—The coefficient of variation is often expressed as a percentage.
3.1.2.2 Discussion—This statistic is also known as the relative standard deviation, RSD.
3.1.3 estimate, n—sample statistic used to approximate a population parameter.
3.1.4 histogram, n—graphical representation of the frequency distribution of a characteristic consisting of a set of rectangles
with area proportional to the frequency. ISO 3534-1
3.1.4.1 Discussion—While not required, equal bar or class widths are recommended for histograms.
th th
3.1.5 interquartile range, IQR, n—the 75 percentile (0.75 quantile) minus the 25 percentile (0.25 quantile), for a data set.
3.1.6 kurtosis, g ,g , n—forapopulationorasample,ameasureoftheweightofthetailsofadistributionrelativetothecenter,
2 2
calculated as the ratio of the fourth central moment (empirical if a sample, theoretical if a population applies) to the standard
deviation (sample, s, or population, s) raised to the fourth power, minus 3 (also referred to as excess kurtosis).
3.1.7 mean, n—of a population, µ, average or expected value of a characteristic in a population – of a sample, x, sum of the
observed values in the sample divided by the sample size.
th
3.1.8 median, X , n—the 50 percentile in a population or sample.
1
ThispracticeisunderthejurisdictionofASTMCommitteeE11onQualityandStatisticsandisthedirectresponsibilityofSubcommitteeE11.10onSampling/Statistics.
CurrenteditionapprovedMay15,Sept.1,2010.PublishedAugustOctober2010.Originallyapprovedin2007.Lastpreviouseditionapprovedin20072010asE2586 – 107.
DOI: 10.1520/E2586-10A.
2
For referencedASTM standards, visit theASTM website, www.astm.org, or contactASTM Customer Service at service@astm.org. For Annual Book of ASTM Standards
volume information, refer to the standard’s Document Summary page on the ASTM website.
3
Available from American National Standards Institute (ANSI), 25 W. 43rd St., 4th Floor, New York, NY 10036, http://www.ansi.org.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.
1
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E2586–10a
3.1.8.1 Discussion—The sample median is the [(n + 1)/2] order statistic if the sample size n is odd and is the average of the
[n/2] and [n/2 + 1] order statistics if n is even.
3.1.
...
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