1、BRITISH STANDARD BS 2846-8:2001 ISO 16269-7:2001 Incorporating Corrigendum No.1 Statistical interpretation of data Part 8: Median Estimation and confidence intervals ICS 03.120.30 NO COPYING WITHOUT BSI PERMISSION EXCEPT AS PERMITTED BY COPYRIGHT LAWBS 2846-8:2001 This British Standard, having been
2、prepared under the direction of the Management Systems Sector Committee, was published under the authority of the Standards Committee and comes into effect on 15 June 2001 BSI 06-2001 ISBN 0 580 37549 8 National foreword This British Standard reproduces verbatim ISO 16269-7:2001 and implements it as
3、 the UK national standard. It supersedes BS 2846-8:1995 which is withdrawn. The UK participation in its preparation was entrusted to Technical Committee SS/2, Statistical interpretation of data, which has the responsibility to: A list of organizations represented on this committee can be obtained on
4、 request to its secretary. Cross-references The British Standards which implement international publications referred to in this document may be found in the BSI Standards Catalogue under the section entitled “International Standards Correspondence Index”, or by using the “Find” facility of the BSI
5、Standards Electronic Catalogue. 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 Standard does not of itself confer immunity from legal obligations. aid enqui
6、rers to understand the text; present to the responsible international/European committee any enquiries on the interpretation, or proposals for change, and keep the UK interests informed; monitor related international and European developments and promulgate them in the UK. Summary of pages This docu
7、ment comprises a front cover, an inside front cover, the ISO title page, pages ii to iv, pages 1 to 11 and a back cover. The BSI copyright date displayed in this document indicates when the document was last issued. Amendments issued since publication Amd. No. Date Comments 13246 Corr. No.1 June 200
8、1 Missing text on pages 3-8 reinstatedReference number ISO 16269-7:2001(E) INTERNATIONAL STANDARD ISO 16269-7 First edition 2001-03-01 Statistical interpretation of data Part 7: Median Estimation and confidence intervals Interprtation statistique des donnes Partie 7: Mdiane Estimation et intervalles
9、 de confiance BS28468:2001ii BS28468:2001 iii Contents Page 1 Scope 1 2 Normative references 1 3 Terms, definitions and symbols.1 4 Applicability2 5 Point estimation.2 6 Confidence interval .3 Annex A (informative) Classical method of determining confidence limits for the median7 Annex B (informativ
10、e) Examples .8 Forms Form A Calculation of an estimate of a median9 Form B Calculation of a confidence interval for a median 11 Table Table 1 Exact values of k for sample sizes varying from 5 to 100: one-sided case4 Table 2 Exact values of k for sample sizes varying from 5 to 100: two-sided case 5 T
11、able 3 Values of u and c for the one-sided case6 Table 4 Values of u and c for the two-sided case6 BS28468:2001iv Foreword ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO member bodies). The work of preparing International Standard
12、s is normally carried out through ISO technical committees. Each member body interested 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
13、part in the work. ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization. International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 3. Draft International Standards adopted by t
14、he technical committees are circulated to the member bodies for voting. Publication as an International Standard requires approval by at least 75 % of the member bodies casting a vote. Attention is drawn to the possibility that some of the elements of this part of ISO 16269 may be the subject of pat
15、ent rights. ISO shall not be held responsible for identifying any or all such patent rights. International Standard ISO 16269-7 was prepared by Technical Committee ISO/TC 69, Applications of statistical methods, Subcommittee SC 3, Application of statistical methods in standaridization. ISO 16269 con
16、sists of the following parts, under the general title Statistical interpretation of data: Part 7: Median Estimation and confidence intervals The following will be the subjects of future parts to ISO 16269: Part 1: Guide to statistical interpretation of data Part 2: Presentation of statistical data P
17、art 3: Tests for departure from normality Part 4: Detection and treatment of outliers Part 5: Estimation and tests of means and variances for the normal distribution, with power functions for tests Part 6: Determination of statistical tolerance intervals Annexes A and B of this part of ISO 16269 are
18、 for information only. BS28468:2001INTENRATIONAL TSANDADR ISO -962617:(1002E)1 Statistical interpretation of data Part 7: Median Estimation and confidence intervals 1 Scope This part of ISO 16269 specifies the procedures for establishing a point estimate and confidence intervals for the median of an
19、y continuous probability distribution of a population, based on a random sample size from the population. These procedures are distribution-free, i.e. they do not require knowledge of the family of distributions to which the population distribution belongs. Similar procedures can be applied to estim
20、ate quartiles and percentiles. NOTE The median is the second quartile and the fiftieth percentile. Similar procedures for other quartiles or percentiles are not described in this part of ISO 16269. 2 Normative references The following normative documents contain provisions which, through reference i
21、n this text, constitute provisions of this part of ISO 16269. For dated references, subsequent amendments to, or revisions of, any of these publications do not apply. However, parties to agreements based on this part of ISO 16269 are encouraged to investigate the possibility of applying the most rec
22、ent editions of the normative documents indicated below. For undated references, the latest edition of the normative document referred to applies. Members of ISO and IEC maintain registers of currently valid International Standards. ISO 2602, Statistical interpretation of test results Estimation of
23、the mean Confidence interval. ISO 3534-1, Statistics Vocabulary and symbols Part 1: Probability and general statistical terms. 3 Terms, definitions and symbols 3.1 Terms and definitions For the purposes of this part of ISO 16269, the terms and definitions given in ISO 2602 and ISO 3534-1 and the fol
24、lowing apply. 3.1.1 kth order statistic of a sample value of the kth element in a sample when the elements are arranged in non-decreasing order of their values NOTE For a sample of n elements arranged in non-decreasing order, the kth order statistics is x k where n xx x 1 2 . uuu BS28468:20012 3.1.2
25、 median of a continuous probability distribution value such that the proportions of the distribution lying on either side of it are both equal to one half NOTE In this part of ISO 16269, the median of a continuous probability distribution is called the population median and is denoted by M. 3.2 Symb
26、ols a lower bound to the values of the variable in the population b upper bound to the values of the variable in the population C confidence level c constant used for determining the value of k in equation (1) k number of the order statistic used for the lower confidence limit M population median n
27、sample size T 1 lower confidence limit derived from a sample T 2 upper confidence limit derived from a sample u fractile of the standardized normal distribution x i ith smallest element in a sample when the elements are arranged in a non-decreasing order of their values x sample median y intermediat
28、e value calculated to determine k using equation (1) 4 Applicability The method described in this part of ISO 16269 is valid for any continuous population, provided that the sample is drawn at random. NOTE If the distribution of the population can be assumed to be approximately normal, the populatio
29、n median is approximately equal to the population mean and the confidence limits should be calculated in accordance with ISO 2602. 5 Point estimation A point estimate of the population median is given by the sample median, x . The sample median is obtained by numbering the sample elements in non-dec
30、reasing order of their values and taking the value of the (n 1)/2th order statistic, if n is odd, or the arithmetic mean of the (n/2)th and (n/2) 1th order statistics, if n is even. NOTE This estimator is in general biased for asymmetrical distributions, but an estimator that is unbiased for any pop
31、ulation does not exist. BS28468:2001 3 6 Confidence interval 6.1 General A two-sided confidence interval for the population median is a closed interval of the form T 1 , T 2 , where T 1 T 2 ; T 1 and T 2 a recal le dth elower and upper confidence limits, respectively. If a and b are respectively the
32、 lower and upper bounds of the variable in the population, a one-sided confidence interval will be of the form T 1 , b)o ro ft h efo r m( a, T 2 . NOTE For practical purposes, a is often taken to be zero for variables that cannot be negative, and b is often taken to be infinity for variables with no
33、 natural upper bound. The practical meaning of a confidence interval is that the experimenter claims that the unknown M lies within the interval, while admitting a small nominal probability that this assertion may be wrong. The probability that intervals calculated in such a way cover the population
34、 median is called the confidence level. 6.2 Classical method The classical method is described in annex A. It involves solving a pair of inequalities. Alternatives to solving these inequalities are given below for a range of confidence levels. 6.3 Small samples (5u u u u nu u u u 100) The values of
35、k satisfying the equations in annex A for eight of the most commonly used confidence levels for sample sizes varying from 5 to 100 sampling units are given in Table 1 for the one-sided case and in Table 2 for the two-sided case. The values of k are given such that the lower confidence limit is k Tx
36、1 and the upper confidence limit is 21 nk Tx where 1 2 , ,., n xx x are the ordered observed values in the sample. For small values of n, it can happen that confidence limits based on order statistics are unavailable at certain confidence levels. An example of the calculation of the confidence limit
37、s for small samples is given in B.1 and shown in Form A of annex B. BS28468:20014 Table 1 Exact values of k for sample sizes varying from 5 to 100: one-sided case kk Confidence level % Confidence level % Sample size n 80 90 95 98 99 99,5 99,8 99,9 Sample size n 80 90 95 98 99 99,5 99,8 99,9 5211 aaa
38、aa 55 24 23 21 20 19 18 17 16 6 7 8 9 10 2 2 3 3 4 1 2 2 3 3 1 1 2 2 2 1 1 1 2 2 a 1 1 1 1 a a 1 1 1 a a a 1 1 a a a a 1 56 57 58 59 60 25 25 26 26 27 23 24 24 25 25 22 22 23 23 24 20 21 21 22 22 19 20 20 21 21 18 19 19 20 20 17 18 18 19 19 17 17 17 18 18 11 12 13 14 15 4 5 5 5 6 3 4 4 5 5 3 3 4 4 4
39、 2 3 3 3 4 2 2 2 3 3 1 2 2 2 3 1 1 2 2 2 1 1 1 2 2 61 62 63 64 65 27 28 28 29 29 25 26 26 27 27 24 25 25 25 26 23 23 23 24 24 21 22 22 23 23 21 21 21 22 22 19 20 20 21 21 19 19 19 20 20 16 17 18 19 20 6 7 7 8 8 5 6 6 7 7 5 5 6 6 6 4 4 5 5 5 3 4 4 5 5 3 3 4 4 4 2 3 3 3 4 2 2 3 3 3 66 67 68 69 70 30 3
40、0 31 31 31 28 28 29 29 30 26 27 27 28 28 25 25 26 26 26 24 24 24 25 25 23 23 23 24 24 21 22 22 23 23 21 21 21 22 22 21 22 23 24 25 9 9 9 10 10 8 8 8 9 9 7 7 8 8 8 6 6 7 7 7 5 6 6 6 7 5 5 5 6 6 4 4 5 5 5 4 4 4 5 5 71 72 73 74 75 32 32 33 33 34 30 31 31 31 32 29 29 29 30 30 27 27 28 28 29 26 26 27 27
41、27 25 25 26 26 26 23 24 24 25 25 23 23 23 24 24 26 27 28 29 30 11 11 12 12 13 10 10 11 11 11 9 9 10 10 11 8 8 9 9 9 7 8 8 8 9 7 7 7 8 8 6 6 7 7 7 5 6 6 6 7 76 77 78 79 80 34 35 35 36 36 32 33 33 34 34 31 31 32 32 33 29 30 30 30 31 28 28 29 29 30 27 27 28 28 29 26 26 26 27 27 25 25 25 26 26 31 32 33
42、34 35 13 14 14 15 15 12 12 13 13 14 11 11 12 12 13 10 10 11 11 11 9 9 10 10 11 8 9 9 10 10 8 8 8 9 9 7 7 8 8 9 81 82 83 84 85 37 37 38 38 39 35 35 36 36 37 33 34 34 34 35 31 32 32 33 33 30 31 31 31 32 29 29 30 30 31 28 28 28 29 29 27 27 28 28 28 36 37 38 39 40 15 16 16 17 17 14 15 15 16 16 13 14 14
43、14 15 12 12 13 13 14 11 11 12 12 13 10 11 11 12 12 10 10 10 11 11 9 9 10 10 10 86 87 88 89 90 39 40 40 41 41 37 38 38 38 39 35 36 36 37 37 34 34 34 35 35 32 33 33 34 34 31 32 32 32 33 30 30 31 31 31 29 29 30 30 30 41 42 43 44 45 18 18 19 19 20 16 17 17 18 18 15 16 16 17 17 14 14 15 15 16 13 14 14 14
44、 15 12 13 13 14 14 11 12 12 13 13 11 11 12 12 12 91 92 93 94 95 41 42 42 43 43 39 40 40 41 41 38 38 39 39 39 36 36 37 37 38 34 35 35 36 36 33 34 34 35 35 32 32 33 33 34 31 31 32 32 33 46 47 48 49 50 20 21 21 22 22 19 19 20 20 20 17 18 18 19 19 16 17 17 17 18 15 16 16 16 17 14 15 15 16 16 13 14 14 15
45、 15 13 13 13 14 14 96 97 98 99 100 44 44 45 45 46 42 42 43 43 44 40 40 41 41 42 38 38 39 39 40 37 37 38 38 38 35 36 36 37 37 34 34 35 35 36 33 33 34 34 35 51 52 53 54 22 23 23 24 21 21 22 22 20 20 21 21 18 19 19 19 17 18 18 19 16 17 17 18 15 16 16 17 15 15 15 16 a A confidence interval and confidenc
46、e limit cannot be determined for this sample size at this confidence level. BS28468:2001 5 Table 2 Exact values of k for sample sizes varying from 5 to 100: two-sided case kk Confidence level % Confidence level % Sample size n 80 90 95 98 99 99,5 99,8 99,9 Sample size n 80 90 95 98 99 99,5 99,8 99,9
47、 511 aaaaaa 55 23 21 20 19 18 17 16 15 6 7 8 9 10 1 2 2 3 3 1 1 2 2 2 1 1 1 2 2 a 1 1 1 1 a a 1 1 1 a a a 1 1 a a a a 1 a a a a a 56 57 58 59 60 23 24 24 25 25 22 22 23 23 24 21 21 22 22 22 19 20 20 21 21 18 19 19 20 20 18 18 18 19 19 17 17 17 18 18 16 16 17 17 17 11 12 13 14 15 3 4 4 5 5 3 3 4 4 4
48、2 3 3 3 4 2 2 2 3 3 1 2 2 2 3 1 1 2 2 2 1 1 1 2 2 1 1 1 1 2 61 62 63 64 65 25 26 26 27 27 24 25 25 25 26 23 23 24 24 25 21 22 22 23 23 21 21 21 22 22 20 20 20 21 21 19 19 19 20 20 18 18 19 19 19 16 17 18 19 20 5 6 6 7 7 5 5 6 6 6 4 5 5 5 6 3 4 4 5 5 3 3 4 4 4 3 3 3 4 4 2 2 3 3 3 2 2 2 3 3 66 67 68 6
49、9 70 28 28 29 29 30 26 27 27 28 28 25 26 26 26 27 24 24 24 25 25 23 23 23 24 24 22 22 23 23 23 21 21 21 22 22 20 20 21 21 21 21 22 23 24 25 8 8 8 9 9 7 7 8 8 8 6 6 7 7 8 5 6 6 6 7 5 5 5 6 6 4 5 5 5 6 4 4 4 5 5 3 4 4 4 5 71 72 73 74 75 30 31 31 31 32 29 29 29 30 30 27 28 28 29 29 26 26 27 27 27 25 25 26 26 26 24 24 25 25 25 23 23 23 24 24 22 22 23 23 23 26 27 28 29 30 10 10 11 11 11 9 9 10 10 11 8 8 9 9 10 7 8 8 8 9 7 7 7 8 8 6 6 7 7 7 5 6 6 6 7 5 5 6 6 6 76 77
