1、BRITISH STANDARD CONFIRMED BS 2846-2: 1981 ISO 2602-1980 Guide to Statistical interpretation of data Part 2: Estimation of the mean: confidence interval ISO title: Statistical interpretation of test resultsEstimation of the meanConfidence interval UDC 31:519.222:312BS2846-2:1981 This British Standar
2、d, having been prepared under the direction of the Quality Management and Statistics Standards Committee, was published under the authority of the Executive Board and comes into effect on 30 January 1981 BSI 02-1999 First published June 1975 First revision January 1981 The following BSI references r
3、elate to the work on this standard: Committee reference QMS/1 Draft for comment 80/63802 DC ISBN 0 580 11810 X Cooperating organizations The Quality Management and Statistics Standards Committee, under whose direction this British Standard was prepared, consists of representatives from the following
4、: Confederation of British Industry Consumers Association Department of the Environment (Building Research Establishment) Institute of Cost and Management Accountants Institute of Quality Assurance* Institute of Statisticians* Institution of Electrical Engineers Institution of Production Engineers*
5、Ministry of Defence* National Council for Quality and Reliability National Terotechnology Centre Retail Consortium The organizations marked with an asterisk in the above list, together with the following, were directly represented on the Technical Committee entrusted with the preparation of this Bri
6、tish Standard: Association of Public Analysts Cement and Concrete Association Chemical Industries Association Department of Industry (Laboratory of the Government Chemist) Department of Industry (National Engineering Laboratory) Economist Intelligence Unit Limited Institute of Petroleum Ministry of
7、Agriculture, Fisheries and Food National Coal Board Post Office Royal Statistical Society Amendments issued since publication Amd. No. Date of issue CommentsBS2846-2:1981 BSI 02-1999 i Contents Page Cooperating organizations Inside front cover National foreword ii 0 Introduction 1 1 Scope 1 2 Field
8、of application 1 3 References 2 4 Definitions and symbols 2 5 Estimation of the mean 2 6 Confidence interval for the mean 2 7 Presention of the results 3 Annex Confidence interval for the mean from the range 5 Table 1 Values of t 1 aand of the ratio 4 Table 2 Untitled 4 Table 3 Untitled 5 Publicatio
9、ns referred to Inside back cover t 1 a n BS2846-2:1981 ii BSI 02-1999 National foreword This Part of this British Standard is identical with ISO2602 “Statistical interpretation of test resultsEstimation of the meanConfidence interval” prepared by Subcommittee 2 of Technical Committee 69, Application
10、s of statistical methods, and published in 1980 by the International Organization for Standardization (ISO). The correct interpretation and presentation of test results has been assuming increasing importance in the analysis of data obtained from manufacturing processes based on sample determination
11、s and prototype evaluations in industry, commerce and educational institutions. It was for this reason that a series of guides was prepared. The standard consists of the following Parts: Part 1: Routine analysis of quantitative data; Part 2: Estimation of the mean: confidence interval (Identical wit
12、h ISO2602); Part 3: Determination of a statistical tolerance interval (Identical with ISO3207); Part 4: Problems of estimation and tests relating to means and variances (Identical with ISO2854); Part 5: Efficiency of tests relating to means and variances (Identical with ISO3494); Part 6: Comparison
13、of two means in the case of paired observations (Identical with ISO3301). Of all the ways of representing a group of observations usually the most satisfactory single measure is the mean; it indicates their central position and as such has extensive applications in the assessment of statistical data
14、. This Part of this British Standard specifies techniques for the estimation of the mean of a normal population on the basis of a series of tests applied to a random sample of individuals drawn from this population, but deals only with the case where the variance of the population is unknown. This s
15、tandard supersedes BS2846-2:1975, which is now withdrawn. Terminology and conventions. The text of the international standard has been approved as suitable for publication, without deviation, as a British Standard. Some terminology and certain conventions are not identical with those used in British
16、 Standards; attention is especially drawn to the following. The comma has been used throughout as a decimal marker. In British Standards it is current practice to use a full point on the baseline as the decimal marker. Wherever the words “International Standard” appear, referring to this standard, t
17、hey should be read as “British Standard”. NOTETextual errors. When adopting the text of the international standard the printing errors listed below were noticed. They have been marked in this British Standard and have been reported to ISO in a proposal to amend the text of the international standard
18、: a) in6.2.2, item b), line 8, “n = n + 1” should be read as “n = n 1”; b) inTable 1, column 10, line 14 “0,668” should be read as “0,678”; c) inTable 1, column 10, line 28 “0,658” should be read as “0,458”. Cross-references International standard Corresponding British Standard ISO 2854-1976 BS 2846
19、 Guide to statistical interpretation of data Part4:1976 Techniques of estimation and tests relating to means and variances (Identical) ISO 3534-1977 BS 5532:1978 StatisticsVocabulary and symbols (Identical)BS2846-2:1981 BSI 02-1999 iii In order to clarify the meaning of the standard the following ch
20、anges to the existing text of ISO2602-1980 have been recommended for amendment by the Quality Management and Statistics Standards Committee: a) in clause2, paragraph 3 to be read as: “The normality assumption is very widely satisfied: the distribution of the results obtained under test conditions is
21、 frequently a normal or near normal distribution.”; b) in6.1.2, the final paragraph to be read as: “In the case of grouped results, the calculated value of s should be corrected (“Shephards correction”). As this correction is of secondary importance, it has not been given here.”. A British Standard
22、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 Standard does not of itself confer immunity from legal obligations. Summary of pages This document comprises a front cover, an i
23、nside front cover, pages i to iv, pages1to 6, an inside back cover 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 theinside front cover.iv blankBS2846-2:1981 BSI 02-1999 1 0 Introduction
24、 The scope of this International Standard is limited to a special question. It concerns only the estimation of the mean of a normal population on the basis of a series of tests applied to a random sample of individuals drawn from this population, and deals only with the case where the variance of th
25、e population is unknown. It is not concerned with the calculation of an interval containing, with a fixed probability, at least a given fraction of the population (statistical tolerance limits). It is recalled that ISO2854 relates to the following collection of problems (including the problem treate
26、d in this International Standard): estimation of a mean and of the difference between two means (the variances being either known or unknown); comparison of a mean with a given value and of two means with one another (the variances being either known or unknown, but equal); estimation of a variance
27、and of the ratio of two variances; comparison of a variance with a given value and of two variances with one another. The test methods generally provide for several determinations which are carried out: on the same item (where the test is not destructive); on distinct portions of a very homogeneous
28、product (a liquid, for example); on distinct items sampled from an aggregate with a certain amount of variability. In the first two cases, the deviations between the results obtained depend only upon the repeatability of the method. In the third case, they depend also on the variability of the produ
29、ct itself. The statistical treatment of the results allows the calculation of an interval which contains, with a given probability, the mean of the population of results that would be obtained from a very large number of determinations, carried out under the same conditions. In the case of items wit
30、h a variability, this International Standard assumes that the individuals on which the determinations are carried out constitute a random sample from the original population and may be considered as independent. The interval so calculated is called the confidence interval for the mean. Associated wi
31、th it is a confidence level (sometimes termed a confidence coefficient), which is the probability, usually expressed as a percentage, that the interval does contain the mean of the population. Only the 95% and 99% levels are provided for in this International Standard. 1 Scope This International Sta
32、ndard specifies the statistical treatment of test results needed to calculate a confidence interval for the mean of a population. 2 Field of application The test results are expressed by measurements of a continuous character. This International Standard does not cover tests of a qualitative charact
33、er (for example presence or absence of a property, number of defectives, etc.). The probability distribution taken as a mathematical model for the total population is a normal distribution for which parameters, mean m and standard deviation s, are unknown. The normality assumption is very widely sat
34、isfied: the distribution of the results obtained under test conditions is generally a normal or nearly normal distribution. It may, however, be useful to check the validity of the assumption of normality by means of appropriate methods 1) . The calculations may be simplified by a change of the origi
35、n or the unit of the test results but it is dangerous to round off these results. It is not permissible to discard any observations or to apply any corrections to apparently doubtful observations without a justification based on experimental, technical or other evident grounds which should be clearl
36、y stated. The test method may be subject to systematic errors, the determination of which is not taken into consideration here. It should be noted, however, that the existence of such errors may invalidate the methods which follow. In particular, if there is an unsuspected bias the increase of the s
37、ample size n has no influence on bias. The methods that are treated in ISO2854 may be useful in certain cases for identifying systematic errors. 1) This subject is in preparation.BS2846-2:1981 2 BSI 02-1999 3 References ISO 2854, Statistical treatment of data Problems of estimation and tests of mean
38、s and variances. ISO 3534, Statistics Vocabulary and symbols. 4 Definitions and symbols The vocabulary and symbols used in this International Standard are in conformity with ISO3534. 5 Estimation of the mean 5.1 Case of ungrouped results After the discarding of any doubtful results, the series compr
39、ises n measurements x i(where i = 1, 2, 3, ., n), some of which may have the same value. The mean m of the underlying normal distribution is estimated by the arithmetic mean of the n results: 5.2 Case of results grouped in classes When the number of results is sufficiently high (above 50 for example
40、), it may be advantageous to group them into classes of the same width. In certain cases, the results may also have been directly obtained grouped into classes. The frequency of the ith class, i.e. the number of results in class i, is denoted by n i . The number of classes being denoted by k, we hav
41、e: The midpoint of class i is designated by y i . The mean m is then estimated by the weighted mean of all midpoints of classes: 6 Confidence interval for the mean The confidence interval for the population mean is calculated from the estimates of the mean and of the standard deviation. The alternat
42、ive method of calculating the confidence interval by use of the range is given in the annex. 6.1 Estimation of the standard deviation 6.1.1 Case of ungrouped results The estimate of the standard deviation s, calculated from the squares of the deviations from the arithmetic mean, is given by the form
43、ula: where For ease of calculation, the use of the following formula is recommended: 6.1.2 Case of grouped results In the case of grouping by classes, the formula for the estimate of the standard deviation is written: For ease of calculation, the use of the following formula is recommended: where In
44、 the case of grouped results, the calculated value of s should be corrected (“Sheppards correction”). As this correction is of secondary importance, it has not been mentioned here. x x i is the value of the ith measurement (i = 1, 2, 3, ., n); n is the total number of measurements; is the arithmetic
45、 mean of the n measurements, calculated as in clause5.1. y i is the mid-point of the ith class (i = 1, 2, 3, ., k); k is the number of classes; n is the total number of measurements; is the weighted mean of all mid-points of classes calculated as in sub-clause5.2. x yBS2846-2:1981 BSI 02-1999 3 6.2
46、Confidence interval for the mean For a chosen confidence level (95% or 99%), according to the specific case, a two-sided or a one-sided confidence interval has to be determined. 6.2.1 Two-sided confidence interval The two-sided confidence interval for the population mean is defined by the following
47、double inequality: a) at the confidence level 95 %: b) at the confidence level 99% 6.2.2 One-sided confidence interval The one-sided confidence interval for the population mean is defined by one or other of the following inequalities: a) at the confidence level 95%: or b) at the confidence level 99%
48、: or with , if necessary, replaced by , in the case of results grouped in classes. The values t 0,975 , t 0,995 , t 0,95 , t 0,99are those of Students t distribution with n = n + 1 degrees of freedom. 2) These values are given inTable 1. This table gives also the values of ratios When values of n ar
49、e greater than 60, it is preferable to calculate the value of t by linear interpolation from using Table 2. Example: 7 Presentation of the results 7.1 Give the expression of the mean according to5.1 or5.2. 7.2 Express the confidence interval in the form of the double inequality of6.2.1 or one of the inequalities of6.2.2, stating the confidence level (95% or 99%). Indicate the number of results discarded as being doubtful and the reasons for discarding. x y 2) See national foreword for details of printing errors in
