1、BRITISH STANDARD CONFIRMED OCTOBER1985 BS2846-4: 1976 ISO 2854:1976 Guide to Statistical interpretation of data Part4: Techniques of estimation and tests relating to means and variances UDC 31:519.222BS2846-4:1976 This British Standard, having been prepared under the directionof the Advisory Committ
2、ee on Statistical Methodswas published under theauthority of the ExecutiveBoard on 31August1976 BSI 05-1999 The following BSI references relate to the work on this standard: Committee referenceOC/8/2 draft for approvalISO/DIS2854 ISBN 0 580 09176 7 Cooperating organizations The Advisory Committee on
3、 Statistical Methods, under whose supervision this British Standard was prepared, consists of representatives from the following Government departments and scientific and industrial organizations. Biometrika* British National Committee for Surface Active Agents British Paper and Board Industry Feder
4、ation (PIF) Economist Intelligence Unit Electronic Components Board Institute of Petroleum Institute of Quality Assurance* Institute of Statisticians* Institution of Production Engineers Ministry of Defence* National Coal Board* Post Office Royal Statistical Society* Society of Analytical Chemists U
5、niversity of Essex The Government department and scientific and industrial organizations marked with an asterisk in the above list were directly represented on the committee entrusted with the preparation of this British Standard. Amendments issued since publication Amd. No. Date of issue CommentsBS
6、 2846-4:1976 BSI 05-1999 i Contents Page Cooperating organizations Inside front cover Foreword ii Section 1. Presentation of calculations General remarks 1 Section 2. Explanatory notes and examples Introductory remarks 27 Numerical illustration of procedures 36 Annex A Comparison of paired observati
7、ons using Students t-test 41 Annex B Statistical tables 42 Figure 1 Breaking load of yarn in samples 29 Figure 2 Graphical test for normality applied to sample of yarn 2 29 Figure 3 Illustrating the use of normal probability paper, using plots of 12 values of (x i , i/13) for data of yarn 2 30 Figur
8、e 4 Rotating-bend fatigue data. Graphical test for normality 33 Table A Comparison of a mean with a given value (variance known) 2 Table A Comparison of a mean with a given value (variance unknown) 4 Table B Estimation of a mean (variance known) 6 Table B Estimation of a mean (variance unknown) 8 Ta
9、ble C Comparison of two means (variances known) 10 Table C Comparison of two means (variances unknown, but may be assumed equal) 12 Table D Estimation of the difference of two means (variances known) 14 Table D Estimation of the difference of two means (variances unknown, but may be assumed equal) 1
10、6 Table E Comparison of a variance or of a standard deviation with a given value 18 Table F Estimation of a variance or of a standard deviation 20 Table G Comparison of two variances or of two standard deviations 22 Table H Estimation of the ratio of two variances or of two standard deviations 24 Ta
11、ble I Values of the ratiou 1 ! / 43 Table IIa Fractiles of Students distribution 44 Table IIb Values of the ratiot 1 !/2 v/ for v = n 1 44 Table III Fractiles of the chi-squared distribution 45 Table IV Upper percentage points ofF 46 Table V Expected values of normal order statistics,K (i|n) 48 n nB
12、S2846-4:1976 ii BSI 05-1999 Foreword The correct interpretation and presentation of test results have been assuming increasing importance in the analysis of data obtained from manufacturing processes based on sample determinations and prototype evaluations in industry, commerce and educational insti
13、tutions. It was for this reason that Subcommittee2 of Technical Committee69, “Applications of Statistical Methods”, of the International Organization for Standardization (ISO), was charged with the task of preparing a guide to statistical methods for the interpretation of test results. International
14、 agreement has already been reached on the specification of several techniques and it was agreed in the UK to publish these as additional parts to the revision of BS2846:1957 “The reduction and presentation of experimental results” as follows: Statistical interpretation of data Part1: Routine analys
15、is of quantitative data; Part2: Estimation of the meanconfidence interval ISO2602; Part3: Determination of a statistical tolerance interval ISO3207; Part4: Techniques of estimation and tests relating to means and variances ISO2854; Part5: Efficiency of tests relating to means and variances ISO3494 1
16、) ; Part6: Comparison of two means in the case of paired observations ISO3301. Much of the experimentation and testing is concerned with providing test results, which will give some idea of the value of a characteristic under investigation, or lend themselves to testing some theory concerning the ch
17、aracteristic. For example, it may be of interest to estimate the nitrogen content in a batch of some chemical compound, and further to test whether this content differs significantly from a required level. This Part of the standard is concerned with providing techniques which will answer these and s
18、imilar questions. However, some caution is necessary in the application of the techniques and in this respect the General remarks should be read carefully and the validity of the assumptions checked. This Part of this British Standard is identical with ISO2854 “Statistical interpretation of data Tec
19、hniques of estimation and tests relating to means and variances”. The UK made a significant contribution during the later stages in the development of this international standard, Professor E.S. Pearson CBE FRS being primarily responsible for the Explanatory notes and examples in Section 2. For the
20、purposes of this British Standard the text of ISO2854 given in this publication should be modified as follows: Terminology. The words “British Standard” should replace “International Standard” wherever they appear. The decimal point should replace the decimal comma wherever it appears. Cross-referen
21、ce. The references to other International Standards should be replaced by references to British Standards as follows. 1) In course of preparation. Reference to ISO Standard Appropriate British Standard ISO3534 aStatistics Vocabulary BS a ISO2602 Statistical interpretation of test results Estimation
22、of the mean Confidence interval BS2846 Guide to statistical interpretation of data Part2 Estimation of the meanconfidence interval ISO3207 Statistical interpretation of dataDetermination of a statistical tolerance interval BS2846-3 Determination of a statistical tolerance intervalBS 2846-4:1976 BSI
23、05-1999 iii Additional information General remark no.4. In line16 the expression “exp (mean log x)” should be interpreted as the “antilogarithm, to the base e, of the arithmetic average of the logarithms, to the base e, of the observations x”. In the final line “characteristic s 2 ” is the estimated
24、 standard deviation, calculated by dividing the sum of squares by “n 1”, where n is the sample size. General remark no.8. In line6 “value of the maximum value of the error of the first kind” should strictly read “value of the maximum value of the probability of committing the error of the first kind
25、”. General remark no.11. The reference to ISO3207 is incorrect and reference should be made to ISO/DIS3725 2)“StatisticsSymbols”. The following British Standards also provide practical guidance on the application of statistical methods. BS600, Application of statistical methods to industrial standar
26、dization and quality control. BS1313, Fraction-defective charts for quality control. BS2564, Control chart technique when manufacturing to a specification, with special reference to articles machined to dimensional tolerances. BS6000, Guide to the use of BS6001. Sampling procedures and tables for in
27、spection by attributes. BS6001, Sampling procedures and tables for inspection by attributes. BS6002, Sampling procedures and charts for inspection by variables for percent defective 2) . BS, Determination and statistical analysis of precision data for a standard test method 2) . A British Standard d
28、oes 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 Standard does not of itself confer immunity from legal obligations. Summary of pages This document comprises a front cover, an in
29、side front cover, pages i to iv, pages1to49 and a back cover. This standard has been updated (see copyright date) and may have had amendments incorporated. This will be indicated in the amendment table on the inside front cover. Reference to ISO Standard Appropriate British Standard NOTESee “General
30、 remark no.11” under “Additional information”. ISO3301 Statistical interpretation of dataComparison of two means in the case of paired observations BS2846-6 Comparison of two means in the case of paired observations a In course of preparation. 2) In course of preparation.iv blankBS2846-4:1976 BSI 05
31、-1999 1 Section 1. Presentation of calculations General remarks 1) This International Standard specifies the techniques required: a) to estimate the mean or the variance of populations; b) to examine certain hypotheses concerning the value of those parameters, from samples. 2) The techniques used ar
32、e valid only if, in each of the populations under consideration, the sample elements are drawn at random and are independent. In the case of a finite population, elements drawn at random may be considered as independent when the population size is sufficiently large or when the sampling fraction is
33、sufficiently small (for instance smaller than1/10). 3) The distribution of the observed variable is assumed to be normal in each population. However, if the distribution does not deviate very much from the normal, the techniques described remain approximately valid to an extent sufficient for most p
34、ractical applications, provided the sample size is not too small. For Table A, Table B, Table C and Table D, the sample size should be of the order of5 to10 at least; for all the other tables, it should be not less than about20 3) 4) A certain number of techniques exist which permit the verification
35、 of the hypothesis of normality. This subject is dealt with briefly in the examples in section 2 and will also be dealt with in a further document (yet to be prepared). Nevertheless, this hypothesis may be admitted on the basis of information other than that provided by the sample itself. In the cas
36、e where the hypothesis of normality should be rejected, the obvious method to follow is to resort to non-parametric tests or to use suitable transformations for obtaining normally distributed populations, for example1/x, log (x + a), , but the conclusions reached by applying these procedures describ
37、ed in this International Standard are only directly valid for the transformed variate; caution should be used in the translation to the original variate. For example exp (mean log x) is equal to the geometric mean of x not the arithmetic mean. If what is really needed is an estimate of the mean or s
38、tandard deviation of the variate X itself then, whether the population distribution is normal or not, an unbiassed estimation of the mean m and the population variance 2is produced by the sample mean and characteristic s 2 . 5) It is desirable to accompany each statistical operation with all the par
39、ticulars relevant to the source or to the method of obtaining the observations which may clarify this statistical analysis, and in particular to give the unit or the smallest unit of measurement having practical meaning. 6) It is not permissible to discard any observations or to apply any correction
40、s to apparently doubtful observations without a justification based on experimental, technical or other evident grounds which should be clearly given. In any case the discarded or corrected values and the reason for discarding or correcting them must be mentioned. 7) In problems of estimation, the c
41、onfidence level1! is the probability that the confidence interval covers the true value of the estimated parameter. Its most usual values are0,95 and0,99, or !=0,05 and!=0,01. 8) In problems of testing a hypothesis, the significance level is, in the two-sided cases, the probability of rejecting the
42、null hypothesis (or tested hypothesis) if it is true (error of the first kind); in the one-sided cases, the significance level is the maximum value of this probability (maximum value of the error of the first kind). Values of !=0,05(1 in20 chance) or0,01(1 in100 chance) are very commonly employed ac
43、cording to the risk which the user is prepared to take. Since a hypothesis may be rejected using!=0,05, but not when using0,01, it is often appropriate to use the phrase: “the hypothesis is rejected at the5% level” or, if this is the case, “at the1% level”. Attention is drawn to the existence of an
44、error of the second kind. This error is committed if the null hypothesis is accepted when it is false. Terms concerning statistical tests are defined in clause2 of ISO3534, StatisticsVocabulary 4) . 9) The calculations can often be greatly reduced by making a change of origin and/or unit on the data
45、. In the case of data classified into groups, reference may be made to the formulae in ISO2602, “Statistical interpretation of test results Estimation of the meanConfidence interval. NOTE Achange of origin may be essential to obtain sufficient accuracy when calculating a variance using the stated fo
46、rmulae with a low precision calculator or computer. 3) Studies about normal distributions are in progress in TC69/SC2. 4) At present at the stage of draft. xa + xBS2846-4:1976 2 BSI 05-1999 10) The methods shown in Table C and Table C deal with the comparison of two means. They assume that the corre
47、sponding samples are independent. For the study of certain problems, it may be interesting to pair the observations (for instance in the comparison of two methods or the comparison of two instruments). The statistical treatment of paired observations is the subject of ISO3301, Statistical interpreta
48、tion of data Comparison of two means in the case of paired observations, but in Annex A an example of treatment of paired observations is given. It uses formally the data of Table A. 11) The symbols and their definitions used in this International Standard are in conformity with ISO3207, Statistical
49、 interpretation of dataDetermination of a statistical tolerance interval. Table A Comparison of a mean with a given value (variance known) NOTEThe numbers(5),(6) and(8) refer to the corresponding paragraphs of the “General remarks”.BS2846-4:1976 BSI 05-1999 3 Comments 1) The significance level ! (see 8 of the “General remarks”) is the probability of rejecting the null hypothesis when this hypothesis is true. 2) U stands for the standardized normal variate: the value u !is defined by
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