ASTM D4686-91(1998)

Standard Guide for Identification and Transformation of Frequency Distributions

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 Title Section Scope 1 Referenced Documents 2 Terminology 3 Significance and Use 4 Key to Distributions 5 Binomial Frequency Distribution 6 Poisson Frequency Distribution 7 Normal Frequency Distribution 8 Sample Average Distribution 9 Other Distributions 10 Transformations of Data 11

General Information

• Status: Historical
• Publication Date: 09-Mar-1998
• ICS: 59.080.30 - Textile fabrics
• Technical Committee: D13 - Textiles

Relations

• Replaced By: ASTM D4686-91(2003) - Standard Guide for Identification and Transformation of Frequency Distributions (Withdrawn 2009)
• Effective Date: 15-Jul-1991

Standards Content (Sample)

NOTICE: This standard has either been superceded and replaced by a new version or discontinued.
Contact ASTM International (www.astm.org) for the latest information.
Designation: D 4686 – 91 (Reapproved 1998)
Standard Guide for
Identification and Transformation of Frequency
Distributions
This standard is issued under the fixed designation D 4686; 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 (e) indicates an editorial change since the last revision or reapproval.
19428, USA. Many of the transformations described in this Guide can be
1. Scope
made using a program in this adjunct.
1.1 This guide gives the rudiments of identification of some
of the most common and useful frequency distributions. It does
3. Terminology
not give rigorous identification. To achieve exactitude, the
3.1 Definitions:
procedures similar to those given by Shapiro should be used.
3.1.1 Bernoulli distribution—see binomial distribution.
1.2 This guide provides a key to identify frequency distri-
3.1.2 binomial distribution, n—the frequency distribution
butions.
which has the probability function:
1.3 This guide gives ways to select the proper transforma-
r n2r
P~r! 5 ~n!/@r! ~n 2 r!!#!p q (1)
tion to use to transform a particular set of data to one which can
be modeled by the normal distribution, if such a transformation
where:
can be found at all.
P(r) 5 the probability of obtaining exactly r “successes” in
1.4 This guide includes the following topics:
n independent trials,
Topic Title Section
p 5 the probability, constant from trial to trial, of obtain-
Scope 1
ing a “success” in a single trial, and
Referenced Documents 2
Terminology 3 q 5 1− p.
Significance and Use 4
(Syn. Bernoulli distribution)
Key to Distributions 5
3.1.3 distribution—see frequency distribution of a sample
Binomial Frequency Distribution 6
Poisson Frequency Distribution 7 and frequency distribution of a population.
Normal Frequency Distribution 8
3.1.4 frequency distribution, n—of a population, a function
Sample Average Distribution 9
that, for a specific type of distribution, gives for each value of
Other Distributions 10
Transformations of Data 11 a random discrete variate, or each group of values of a random
continuous variate, the corresponding probability of occur-
2. Referenced Documents
rence.
2.1 ASTM Standards:
3.1.5 frequency distribution, n—of a sample, a table giving
D 123 Terminology Relating to Textiles
for each value of a discrete variate, or for each group of values
D 4392 Terminology for Statistically Related Terms
of a continuous variate, the corresponding number of observa-
E 456 Terminology Relating to Quality and Statistics
tions.
2.2 ASTM Adjuncts:
3.1.6 normal distribution, n—the distribution that has the
TEX-PAC
probability function:
1/2 2 2
NOTE 1—Tex-Pac is a group of PC programs on floppy disks, available
f~x! 5 ~1/s!~2p! exp@2~x 2 μ! /2s #
through ASTM Headquarters, 100 Barr Harbor Drive, Conshohocken, PA
(2)
where:
1 x 5 a random variate,
This guide is under the jurisdiction of ASTM Committee D-13 on Textiles and
μ 5 the mean of the distribution, and
is the direct responsibility of Subcommittee D13.93 on Statistics.
Current edition approved July 15, 1991. Published September 1991. Originally
s5 the standard deviation of the distribution.
published as D 4686 – 87. Last previous edition D 4686 – 90.
(Syn. Gaussian distribution, law of error)
Shapiro, Samuel S., How to Test Normality and Other Distribution Assump-
3.1.7 Poisson distribution, n—the distribution which has as
tions, American Society for Quality Control, Milwaukee, WI, 1980. Vol. 3 of the
series, The ASQC Basic References in Quality Control: Statistical Techniques, its probability function:
Edward J. Dudewitz, ed.
2μ r
P~r! 5 e μ /r! (3)
Annual Book of ASTM Standards, Vol 07.01.
Annual Book of ASTM Standards, Vol 07.02.
Annual Book of ASTM Standards, Vol 14.02. where:
PC programs on floppy disks are available through ASTM. For a 3 ⁄2 inch disk
request PCN:12-429040-18, for a 5 ⁄4 inch disk request PCN:12-429041-18.
Copyright © ASTM, 100 Barr Harbor Drive, West Conshohocken, PA 19428-2959, United States.
NOTICE: This standard has either been superceded and replaced by a new version or discontinued.
Contact ASTM International (www.astm.org) for the latest information.
D 4686
A,B
TABLE 1 Key to Frequency Distributions
P(r) 5 probability of obtaining exactly r occurrences of an
Section
event in one unit, such as a unit of time or area,
1a. Variates are discrete. (2)
μ 5 both mean and variance of distribution, and
1b. Variates are continuous. (8)
e 5 base of natural logarithms.
2a. Variates are counts of events. (3)
2b. Variates are on an arbitrary scale
3.1.8 probability function, n—of a continuous variate, the
........................ .UNIDENTIFIED 10, 11
mathematical expression whose definite integral gives the
3a. Counts are of one of a pair of mutually exclusive
probability that a variate will take a value within the two limits
events per unit. (4)
3b. Counts are of other events per unit. (6)
of integration.
4a. Experiment consists of n identical trials, each
3.1.9 probability function, n—of a discrete variate, the
conducted independently. (5)
mathematical expression which gives the probability that a
4b. Experiment otherwise
........................ .UNIDENTIFIED 10, 11
variate will take a particular value.
5a. Probability of occurrence of counted event
3.1.10 transformation, n—the change from one set of vari-
constant from trial to trial
ables, x, to another set, y, by the use of a function, y 5 f (x).
............................ BINOMIAL 6
5b. Probability otherwise
3.1.11 For definitions of textile terms used in this guide,
........................ .UNIDENTIFIED 10, 11
refer to Terminology D 123. For definitions of other statistical
6a. Probability of event occurrence constant from
terms used in this guide, refer to Terminology D 4392, Termi-
unit to unit. (7)
6b. Probability otherwise
nology E 456, or appropriate textbooks on statistics.
........................ .UNIDENTIFIED 10, 11
7a. Number of events independent from unit to unit
4. Significance and Use
............................ POISSON 7
7b. Number of events otherwise
4.1 In measuring, testing, and experimenting, statistical
........................ .UNIDENTIFIED 10, 11
tests are made to determine whether the observed effect of the
8a. Variates individual observations. (9)
introduction of a factor is real or simply due to chance. The 8b. Variates sample averages
............................. NORMAL 9
appropriate statistical test to use depends on the kind of
9a. Distribution symmetrical. (10)
distribution used to model the data. Distribution identification
9b. Distribution otherwise
........................ .UNIDENTIFIED 10, 11
is useful in selecting the most powerful statistical test.
10a. Distribution unimodal. (11)
NOTE 2—There are statistical tests which can be used for data for which 10b. Distribution multimodal
........................ .UNIDENTIFIED 10, 11
a parametric distribution cannot be selected. But these non-parametric
11a. Distribution bell shaped
tests do not discriminate as well as the distribution-dependent tests.
............................. NORMAL 8
11b. Distribution shape otherwise
4.2 For certain types of data, a transformation can be made
........................ .UNIDENTIFIED 10, 11
which will make it possible to use the hypothesis that the
A
See Section 5 for instructions for using this table.
normal distribution is a suitable model for the transformed
B
See Note 1 concerning unidentified distributions.
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
5. Key to Distributions
distribution, provided all of the trials are made on essentially
the same kind of material and in the same manner.
5.1 Table 1 is a key to the identification of frequency
distributions. The table consists of a series of pairs of state- 6.3 Approximation of the Binomial Distribution by a Nor-
mal Distribution—Binomial data can be transformed to a near
ments. The statement in the pair that is true either (1) directs
the user to another pair of statements to solicit additional normal distribution. (See Section 11.) Under a certain condi-
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 2 p!! does
=
not contain zero or one.
6.1 A binomial 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
characteristics: 7.1 A Poisson distribution is the probability distribution of a
6.1.1 The experiment consists of n independent, identical Poisson experiment. This fact provides a means for its identi-
trials. Each trial is conducted independently. fication. Such an experiment has the following four character-
6.1.2 There are only two possible outcomes on each trial. istics:
...