1、BRITISH STANDARD BS 2846-7: 1997 ISO 5479:1997 Guide to Statistical interpretation of data Part 7: Tests for departure from normality ICS 03.120.30BS2846-7:1997 This British Standard, having been prepared under the directionof the Management Systems Sector Board, was published under the authority of
2、the Standards Board and comesinto effect on 15September1997 BSI 10-1999 The following BSI references relate to the work on this standard: Committee reference SS/2 Draft for comment 95/402132 DC ISBN 0 580 28343 7 Committees responsible for this British Standard The preparation of this British Standa
3、rd was entrusted to Technical Committee SS/2, Statistical interpretation of data, upon which the following bodies were represented: Clay Pipe Development Association Limited Royal Statistical Society Amendments issued since publication Amd. No. Date CommentsBS2846-7:1997 BSI 10-1999 i Contents Page
4、Committees responsible Inside front cover National foreword ii Foreword iii Text of ISO 5479 1BS2846-7:1997 ii BSI 10-1999 National foreword This Part of BS2846 has been prepared by Technical Committee SS/2. It is identical with ISO5479:1997 Statistical interpretation of data Tests for departure fro
5、m the normal distribution, published by the International Organization for Standardization (ISO). This Part of BS2846 supersedes BS2846-7:1984 which is withdrawn. The importance of correct interpretation and presentation of test results has been increasingly recognized in the analysis of data obtain
6、ed from manufacturing processes based on sample determinations and prototype evaluations in industry, commerce and educational institutions. It was for this reason that a series of guides on statistical interpretation of data was prepared. At present this British Standard consists of the following P
7、arts: Part 1: Routine analysis of quantitative data; Part 2: Estimation of the mean: confidence interval; (Identical to ISO2602) Part 3: Determination of a statistical tolerance interval; (Identical to ISO3207) Part 4: Techniques of estimation and tests relating to means and variances; (Identical to
8、 ISO 2854) Part 5: Power of tests relating to means and variances; (IdenticaltoISO3494) Part 6: Comparison of two means in the case of paired observations; (Identical to ISO 3301) Part 7: Tests for departure from normality; (Identical to ISO 5479) Part 8: Estimation of a median; Part 9 is in prepara
9、tion and will deal with proportions as follows: Section 1: Estimation of a proportion; Section 2: Comparison of a proportion with a given value; Section 3: Comparison of two proportions; Section 4: Fractiles of the F distribution (set of tables). This Part of BS2846 is the national implementation of
10、 ISO5479. However, the latter is itself an adaptation of BS2846-7:1984, and the present version of this Part of BS2846 is therefore effectively a direct revision of the version it replaces. Although much of the text has been rewritten, the present version differs fundamentally only to the extent tha
11、t of the omnibus tests the dAgostino test has been replaced by the Epps-Pulley test. Extensive comparative studies have shown that the latter test is generally superior to the former. Cross-references The British Standards that implement international publications referred to in this document may be
12、 found in the BSI Catalogue under the section entitled “International Standards Correspondence Index”. A British Standard does not purport to include all the necessary provisions of a contract. Users of British Standards are responsible for their correct application. Compliance with a British Standa
13、rd does not of itself confer immunity from legal obligations. Summary of pages This document comprises a front cover, an inside front cover, pages i and ii, theISO title page, pages ii to iv, pages 1 to 31 and a back cover. This standard has been updated (see copyright date) and may have had amendme
14、nts incorporated. This will be indicated in the amendment table on the inside front cover.BS2846-7:1997 ii BSI 10-1999 Contents Page Foreword iii Introduction 1 1 Scope 1 2 Normative references 1 3 Definitions and symbols 2 4 General 3 5 Graphical method 4 6 Directional tests 10 7 Joint test usingan
15、d b 2(multidirectional test) 14 8 Omnibus tests 14 9 Joint test using several independent samples 19 10 Statistical tables 22 Annex A (informative) Blank normal probability graph paper 30 Annex B (informative) Bibliography 31 Figure 1 Annotated normal probability graph paper 5 Figure 2 Graph of a se
16、ries of observations on normal probability graph paper 6 Figure 3 Density function with kurtosis in default 8 Figure 4 Density function with kurtosis in excess 8 Figure 5 Density function with positive skewness 9 Figure 6 Density function with negative skewness 9 Figure 7 Superposition of two differ
17、ent density functions 10 Figure 8 Flow chart for the computation of the test statistic T EPof the Epps-Pulley test 17 Figure 9 Joint test using and b 2(multidirectional test) 24 Table 1 Results, x (k)of a series of 15 rotating-bend fatigue tests and corresponding values of lg (10 x (k) ) 7 Table 2 D
18、epth of sapwood 12 Table 3 Series of 50 observations suspected of being affected by a variation in the dispersion of measurements 13 Table 4 Annual amount of rainfall collected at a weather station 16 Table 5 Breaking strength of rayon yarn 18 Table 6 Breaking strength of rayon yarn Calculation of t
19、he test statistic T EP 19 Table 7 Values of W jand G jfor 22 samples of size n = 20 dawn from the same population 21 Table 8 Test for skewness, (p-quantiles of for p = 1 = 0,95 and 0,99) 22 Table 9 Test for kurtosis, b 2(p-quantiles of b 2for p = = 0.01 and 0,05 and p = 1 = 0,95 and 0,99) 23 Table 1
20、0 Shapiro-Wilk test coefficients a kfor calculating the test statistic W 26 Table 11 Shapiro-Wilk test: p-quantiles of the test statistic W for p = = 0,01 and 0,05 28 Table 12 Epps-Pulley test: p-quantiles of the test statistic T EPfor p = 1 = 0,90; 0,95; 0,975 and 0,99 28 Table 13 Joint test using
21、several independent samples: Coefficients for converting W to a standardized normal variate for n = 8(1)50 29 Table 14 Quantities u pof the standardized normal distribution 29 Descriptors: Statistical analysis, statistical quality control, statistical distribution, tests, estimation, use. b 1 b 1 b
22、1 b 1BS2846-7:1997 BSI 10-1999 iii 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 in
23、terested 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 Electrotechni
24、cal Commission (IEC) on all matters of electrotechnical standardization. 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 least75% of the member bodies casting a vote. In
25、ternational Standard ISO5479 was prepared by Technical Committee ISO/TC69, Applications of statistical methods, Subcommittee SC6, Measurement methods and results. Annex A andAnnex B of this International Standard are for information only.iv blankBS2846-7:1997 BSI 10-1999 1 Introduction Many of the s
26、tatistical methods recommended in International Standards, such as those described in ISO28541, are based on the assumption that the random variable(s) to which these methods apply are independently distributed according to a normal distribution with one or both of its parameters unknown. The follow
27、ing question therefore arises. Is the distribution that is represented by the sample sufficiently close to the normal distribution that the methods provided by these International Standards can be used reliably? There is no simple yes or no answer to this question which is valid in all cases. For th
28、is reason a large number of “tests of normality” have been developed, each of which is more or less sensitive to a particular feature of the distribution under consideration; e.g.asymmetry or kurtosis. Generally the test used is designed to correspond to a predetermined a priori risk that the hypoth
29、esis of normality is rejected even if it is true (error of the first kind). On the other hand, the probability that this hypothesis is not rejected when it is not true (error of the second kind) cannot be determined unless the alternative hypothesis (i.e.that which is opposed to the hypothesis of no
30、rmality) can be precisely defined. This is not possible in general and, furthermore, it requires computational effort. For a distinct test, this risk is particularly large if the sample size is small. 1 Scope 1.1 This International Standard gives guidance on methods and tests for use in deciding whe
31、ther or not the hypothesis of a normal distribution should be rejected, assuming that the observations are independent. 1.2 Whenever there are doubts as to whether the observations are normally distributed, the use of a test for departure from the normal distribution may be useful or even necessary.
32、 In the case of robust methods, however (i.e.where the results are only altered very slightly when the real probability distribution of the observations is not a normal distribution), a test for departure from the normal distribution is not very helpful. This is the case, for example, when the mean
33、of a single random sample of observations is to be checked against a given theoretical value using a t-test. 1.3 It is not strictly necessary to use such a test whenever one refers to statistical methods based on the hypothesis of normality. It is possible that there is no doubt at all as to the nor
34、mal distribution of the observations, whether theoretical (e.g.physical) reasons are present which confirm the hypothesis or because this hypothesis is deemed to be acceptable according to prior information. 1.4 The tests for departure from the normal distribution selected in this International Stan
35、dard are primarily intended for complete data, not grouped data. They are unsuitable for censored data. 1.5 The tests for departure from the normal distribution selected in this International Standard may be applied either to observed values or to functions of them, such as the logarithm or the squa
36、re root. 1.6 Tests for departure from the normal distribution are very ineffective for samples of size less than eight. Accordingly, this International Standard is restricted to samples of eight or more. 2 Normative reference The following standard contains provisions which, through reference in thi
37、s text, constitute provisions of this International Standard. At the time of publication, the edition indicated was valid. All standards are subject to revision, and parties to agreements based on this International Standard are encouraged to investigate the possibility of applying the most recent e
38、dition of the standard indicated below. Members of IEC and ISO maintain registers of currently valid International Standards. ISO 3534-1:1993, Statistics Vocabulary and symbols Part 1: Probability and general statistical terms. BS2846-7:1997 2 BSI 10-1999 3 Definitions and symbols 3.1 Definitions Fo
39、r the purposes of this International Standard, the definitions given in ISO3534-1 apply. 3.2 Symbols a k coefficient of the Shapiro-Wilk test A auxiliary quantity for the Epps-Pulley test b 2 empirical kurtosis empirical skewness B auxiliary quantity for the Epps-Pulley test E expectation G j auxili
40、ary quantity for the joint test using several independent samples h number of consecutive samples H 0 null hypothesis H 1 alternative hypothesis k within the sample, arranged in non-decreasing order, the number of the observed value x m j central moment of order j of the sample n sample size p proba
41、bility associated with the p-quantile of a distribution P probability P k probability associated with X (k) S auxiliary quantity for the Shapiro-Wilk test T test statistic T EP test statistic of the Epps-Pulley test u p p-quantile of the standardized normal distribution v j auxiliary quantity for th
42、e joint test using several independent samples W test statistic of the Shapiro-Wilk test W j auxiliary quantity for the joint test using several independent samples x value of X X random variable x (j) j thvalue in the sample, arranged in non-decreasing order x (k) k thvalue in the sample, arranged
43、in non-decreasing order arithmetic average significance level probability of an error of the second kind 2 kurtosis of the population 2 3 excess of the population skewness of the population auxiliary quantity for the joint test using several independent samples (n) coefficient of the joint test usin
44、g several independent samples b 1 x 1BS2846-7:1997 BSI 10-1999 3 4 General 4.1 There are several categories of tests for departure from normality. In this International Standard, graphical methods, moment tests, regression tests and characteristic function tests are considered. Chi-squared tests are
45、 appropriate for grouped data only but, because grouping results in a loss of information, they are not considered in this International Standard. 4.2 If no additional information about the sample is available, it is recommended first to do a normal probability plot; i.e.to plot the cumulative distr
46、ibution function of the observed values on normal probability graph paper consisting of a system of coordinate axes where the cumulative distribution function of the normal distribution is represented by a straight line. This method, which is described in clause5, allows one to “see” immediately whe
47、ther the distribution observed is close to the normal distribution or not. With this additional information it can be decided whether to carry out a directional test, or to carry out either a regression test or a characteristic function test, or no test at all. In addition, although such a graphical
48、 representation cannot be considered as a rigorous test, the summary information that it provides is an essential supplement to any test for departure from the normal distribution. In the case of rejection of the null hypothesis it is often possible to envisage by this means the type of alternative
49、that might be applicable. 4.3 A test for departure from the normal distribution is a test of the null hypothesis that the sample consists of n independent observations coming from one and the same normal distribution. It consists of the calculation of a function T of the observations, which is called the test statistic. The null hypothesis of a normal distribution is then not rejected or rejected depending on whether or not the value of T lies within a set of values
