ASTM D4686-91(2003)

Standard Guide for Identification and Transformation of Frequency Distributions (Withdrawn 2009)

SIGNIFICANCE AND USE
In measuring, testing, and experimenting, statistical tests are made to determine whether the observed effect of the introduction of a factor is real or simply due to chance. The appropriate statistical test to use depends on the kind of distribution used to model the data. Distribution identification is useful in selecting the most powerful statistical test.
Note 2—There are statistical tests which can be used for data for which a parametric distribution cannot be selected. But these non-parametric tests do not discriminate as well as the distribution-dependent tests.
For certain types of data, a transformation can be made which will make it possible to use the hypothesis that the normal distribution is a suitable model for the transformed data. When this hypothesis can be made, the analysis of the data is made much easier.
SCOPE
1.1 This guide gives the rudiments of identification of some of the most common and useful frequency distributions. It does not give rigorous identification. To achieve exactitude, the procedures similar to those given by Shapiro should be used.
1.2 This guide provides a key to identify frequency distributions.
1.3 This guide gives ways to select the proper transformation to use to transform a particular set of data to one which can be modeled by the normal distribution, if such a transformation can be found at all.
1.4 This guide includes the following topics:Topic TitleSectionScope Referenced Documents Terminology Significance and Use Key to Distributions Binomial Frequency Distribution Poisson Frequency Distribution Normal Frequency Distribution Sample Average Distribution Other Distributions Transformations of Data
WITHDRAWN RATIONALE
This guide gives the rudiments of identification of some of the most common and useful frequency distributions.
Formerly under the jurisdiction of Committee D13 on Textiles, this guide was withdrawn in March 2009 because D13 no longer has the expertise to maintain and statistical standards are being maintained by Committee E11.

General Information

• Status: Withdrawn
• Publication Date: 14-Jul-1991
• Withdrawal Date: 14-Jan-2009
• Technical Committee: D13 - Textiles

Relations

• Replaces: ASTM D4686-91(1998) - Standard Guide for Identification and Transformation of Frequency Distributions
• Effective Date: 15-Jul-1991

Standards Content (Sample)

NOTICE: This standard has either been superseded and replaced by a new version or withdrawn.
Contact ASTM International (www.astm.org) for the latest information
Designation: D 4686 – 91 (Reapproved 2003)
Standard Guide for
Identification and Transformation of Frequency
Distributions
This standard is issued under the fixed designation D4686; 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 (e) indicates an editorial change since the last revision or reapproval.
NOTE 1—Tex-Pac is a group of PC programs on floppy disks, available
1. Scope
throughASTM Headquarters, 100 Barr Harbor Drive, Conshohocken, PA
1.1 This guide gives the rudiments of identification of some
19428, USA. Many of the transformations described in this Guide can be
ofthemostcommonandusefulfrequencydistributions.Itdoes
made using a program in this adjunct.
not give rigorous identification. To achieve exactitude, the
3. Terminology
procedures similar to those given by Shapiro should be used.
1.2 This guide provides a key to identify frequency distri-
3.1 Definitions:
butions.
3.1.1 Bernoulli distribution—see binomial distribution.
1.3 This guide gives ways to select the proper transforma-
3.1.2 binomial distribution, n—the frequency distribution
tiontousetotransformaparticularsetofdatatoonewhichcan
which has the probability function:
bemodeledbythenormaldistribution,ifsuchatransformation
r n2r
P~r! 5 ~n!/@r! ~n 2 r!!#!p q (1)
can be found at all.
1.4 This guide includes the following topics:
where:
Topic Title Section
P(r) = the probability of obtaining exactly r “successes” in
Scope 1
n independent trials,
Referenced Documents 2
p = the probability, constant from trial to trial, of obtain-
Terminology 3
Significance and Use 4
ing a “success” in a single trial, and
Key to Distributions 5
q =1− p.
Binomial Frequency Distribution 6
(Syn. Bernoulli distribution)
Poisson Frequency Distribution 7
Normal Frequency Distribution 8 3.1.3 distribution—see frequency distribution of a sample
Sample Average Distribution 9
and frequency distribution of a population.
Other Distributions 10
3.1.4 frequency distribution, n—of a population, a function
Transformations of Data 11
that, for a specific type of distribution, gives for each value of
2. Referenced Documents
a random discrete variate, or each group of values of a random
2.1 ASTM Standards: continuous variate, the corresponding probability of occur-
D123 Terminology Relating to Textiles rence.
D4392 Terminology for Statistically Related Terms 3.1.5 frequency distribution, n—of a sample, a table giving
E456 Terminology Relating to Quality and Statistics for each value of a discrete variate, or for each group of values
2.2 ASTM Adjuncts: of a continuous variate, the corresponding number of observa-
TEX-PAC tions.
3.1.6 normal distribution, n—the distribution that has the
probability function:
This guide is under the jurisdiction ofASTM Committee D13 on Textiles and
1/2 2 2
f x 5 1/s 2p exp 2 x 2 µ /2s
~ ! ~ !~ ! @ ~ ! #
is the direct responsibility of Subcommittee D13.93 on Statistics.
(2)
Current edition approved July 15, 1991. Published September 1991. Originally
published as D4686–87. Last previous edition D4686–90.
2 where:
Shapiro, Samuel S., How to Test Normality and Other Distribution Assump-
x = a random variate,
tions, American Society for Quality Control, Milwaukee, WI, 1980. Vol. 3 of the
series, The ASQC Basic References in Quality Control: Statistical Techniques,
µ = the mean of the distribution, and
Edward J. Dudewitz, ed.
s = the standard deviation of the distribution.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
(Syn. Gaussian distribution, law of error)
contact ASTM Customer Service at [e-mail]. ForAnnual Book of ASTM
3.1.7 Poisson distribution, n—the distribution which has as
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website.
its probability function:
PC programs on floppy disks are available throughASTM. For a 3 ⁄2 inch disk
2µ r
request PCN:12-429040-18, for a 5 ⁄4 inch disk request PCN:12-429041-18. P~r! 5e µ/r! (3)
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.
D 4686 – 91 (2003)
A,B
TABLE 1 Key to Frequency Distributions
where:
Section
P(r) = probability of obtaining exactly r occurrences of an
1a. Variates are discrete. (2)
event in one unit, such as a unit of time or area,
1b. Variates are continuous. (8)
µ = both mean and variance of distribution, and
2a. Variates are counts of events. (3)
e = base of natural logarithms. 2b. Variates are on an arbitrary scale
.........................UNIDENTIFIED 10, 11
3.1.8 probability function, n—of a continuous variate, the
3a. Counts are of one of a pair of mutually exclusive
mathematical expression whose definite integral gives the
events per unit. (4)
3b. Counts are of other events per unit. (6)
probabilitythatavariatewilltakeavaluewithinthetwolimits
4a. Experiment consists of n identical trials, each
of integration.
conducted independently. (5)
3.1.9 probability function, n—of a discrete variate, the
4b. Experiment otherwise
.........................UNIDENTIFIED 10, 11
mathematical expression which gives the probability that a
5a. Probability of occurrence of counted event
variate will take a particular value.
constant from trial to trial
3.1.10 transformation, n—the change from one set of vari-
............................BINOMIAL 6
5b. Probability otherwise
ables, x, to another set, y, by the use of a function, y= f (x).
.........................UNIDENTIFIED 10, 11
3.1.11 For definitions of textile terms used in this guide,
6a. Probability of event occurrence constant from
refer to Terminology D123. For definitions of other statistical
unit to unit. (7)
6b. Probability otherwise
terms used in this guide, refer to Terminology D4392, Termi-
.........................UNIDENTIFIED 10, 11
nology E456, or appropriate textbooks on statistics.
7a. Number of events independent from unit to unit
............................POISSON 7
4. Significance and Use
7b. Number of events otherwise
.........................UNIDENTIFIED 10, 11
4.1 In measuring, testing, and experimenting, statistical
8a. Variates individual observations. (9)
tests are made to determine whether the observed effect of the
8b. Variates sample averages
introduction of a factor is real or simply due to chance. The
............................. NORMAL 9
9a. Distribution symmetrical. (10)
appropriate statistical test to use depends on the kind of
9b. Distribution otherwise
distribution used to model the data. Distribution identification
.........................UNIDENTIFIED 10, 11
is useful in selecting the most powerful statistical test. 10a. Distribution unimodal. (11)
10b. Distribution multimodal
NOTE 2—Therearestatisticaltestswhichcanbeusedfordataforwhich
.........................UNIDENTIFIED 10, 11
a parametric distribution cannot be selected. But these non-parametric 11a. Distribution bell shaped
............................. NORMAL 8
tests do not discriminate as well as the distribution-dependent tests.
11b. Distribution shape otherwise
4.2 For certain types of data, a transformation can be made
.........................UNIDENTIFIED 10, 11
A
which will make it possible to use the hypothesis that the
See Section 5 for instructions for using this table.
B
See Note 1 concerning unidentified distributions.
normal distribution is a suitable model for the transformed
data. When this hypothesis can be made, the analysis of the
data is made much easier.
ment. Data from a series of such trials will produce a binomial
distribution, provided all of the trials are made on essentially
5. Key to Distributions
the same kind of material and in the same manner.
5.1 Table 1 is a key to the identification of frequency
6.3 Approximation of the Binomial Distribution by a Nor-
distributions. The table consists of a series of pairs of state-
mal Distribution—Binomial data can be transformed to a near
ments. The statement in the pair that is true either (1) directs
normal distribution. (See Section 11.) Under a certain condi-
the user to another pair of statements to solicit additional
tion, the binomial distribution can be approximated by the
information or (2) identifies the distribution.
normal distribution without transformation. This condition is
6. Binomial Frequency Distribution
met when the interval, p 6 3s,or p 6 3 =~p ~1 2p!! does
not contain zero or one.
6.1 Abinomial distribution is the probability distribution of
a binomial experiment. This fact provides a means for its
7. Poisson Frequency Distribution
identification. Such an experiment has the following four
7.1 APoissondistributionistheprobabilitydistributionofa
characteristics:
6.1.1 The experiment consists of n independent, identical Poisson experiment. This fact provides a m
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