1、 Collection of SANS standards in electronic format (PDF) 1. Copyright This standard is available to staff members of companies that have subscribed to the complete collection of SANS standards in accordance with a formal copyright agreement. This document may reside on a CENTRAL FILE SERVER or INTRA
2、NET SYSTEM only. Unless specific permission has been granted, this document MAY NOT be sent or given to staff members from other companies or organizations. Doing so would constitute a VIOLATION of SABS copyright rules. 2. Indemnity The South African Bureau of Standards accepts no liability for any
3、damage whatsoever than may result from the use of this material or the information contain therein, irrespective of the cause and quantum thereof. ISBN 968-0-626-23812-4 SANS 16269-7:2010Edition 1ISO 16269-7:2001Edition 1SOUTH AFRICAN NATIONAL STANDARD Statistical interpretation of data Part 7: Medi
4、an Estimation and confidence intervals This national standard is the identical implementation of ISO 16269-7:2001 and is adopted with the permission of the International Organization for Standardization. Published by SABS Standards Division 1 Dr Lategan Road Groenkloof Private Bag X191 Pretoria 0001
5、Tel: +27 12 428 7911 Fax: +27 12 344 1568 www.sabs.co.za SABS SANS 16269-7:2010 Edition 1 ISO 16269-7:2001 Edition 1 Table of changes Change No. Date Scope National foreword This South African standard was approved by National Committee SABS TC 169, Applications of statistical methods, in accordance
6、 with procedures of the SABS Standards Division, in compliance with annex 3 of the WTO/TBT agreement. This SANS document was published in March 2010. Reference numberISO 16269-7:2001(E)ISO 2001INTERNATIONALSTANDARDISO16269-7First edition2001-03-01Statistical interpretation of data Part 7:Median Esti
7、mation and confidenceintervalsInterprtation statistique des donnes Partie 7: Mdiane Estimation et intervalles de confianceSANS 16269-7:2010This s tandard may only be used and printed by approved subscription and freemailing clients of the SABS .ISO 16269-7:2001(E)PDF disclaimerThis PDF file may cont
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10、rswere optimized for printing. Every care has been taken to ensure that the file is suitable for use by ISO member bodies. In the unlikely eventthat a problem relating to it is found, please inform the Central Secretariat at the address given below. ISO 2001All rights reserved. Unless otherwise spec
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12、postale 56 Gb7 CH-1211 Geneva 20Tel. + 41 22 749 01 11Fax + 41 22 749 09 47E-mail copyrightiso.chWeb www.iso.chPrinted in Switzerlandii ISO 2001 All rights reservedSANS 16269-7:2010This s tandard may only be used and printed by approved subscription and freemailing clients of the SABS .ISO 16269-7:2
13、001(E) ISO 2001 All rights reserved iiiContents Page1 Scope 12 Normative references 13 Terms, definitions and symbols.14 Applicability25 Point estimation.26 Confidence interval .3Annex A (informative) Classical method of determining confidence limits for the median7Annex B (informative) Examples .8F
14、ormsForm A Calculation of an estimate of a median9Form B Calculation of a confidence interval for a median 11TableTable 1 Exact values of k for sample sizes varying from 5 to 100: one-sided case4Table 2 Exact values of k for sample sizes varying from 5 to 100: two-sided case 5Table 3 Values of u and
15、 c for the one-sided case6Table 4 Values of u and c for the two-sided case6SANS 16269-7:2010This s tandard may only be used and printed by approved subscription and freemailing clients of the SABS .ISO 16269-7:2001(E)iv ISO 2001 All rights reservedForewordISO (the International Organization for Stan
16、dardization) is a worldwide federation of national standards bodies (ISOmember bodies). The work of preparing International Standards is normally carried out through ISO technicalcommittees. Each member body interested in a subject for which a technical committee has been established hasthe right to
17、 be represented on that committee. International organizations, governmental and non-governmental, inliaison with ISO, also take part in the work. ISO collaborates closely with the International ElectrotechnicalCommission (IEC) on all matters of electrotechnical standardization.International Standar
18、ds are drafted in accordance with the rules given in the ISO/IEC Directives, Part 3.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 least 75 % of the member bodies castin
19、g a vote.Attention is drawn to the possibility that some of the elements of this part of ISO 16269 may be the subject ofpatent 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, App
20、lications of statisticalmethods, Subcommittee SC 3, Application of statistical methods in standaridization.ISO 16269 consists of the following parts, under the general title Statistical interpretation of data:Gbe Part 7: Median Estimation and confidence intervalsThe following will be the subjects of
21、 future parts to ISO 16269:Gbe Part 1: Guide to statistical interpretation of dataGbe Part 2: Presentation of statistical dataGbe Part 3: Tests for departure from normalityGbe Part 4: Detection and treatment of outliersGbe Part 5: Estimation and tests of means and variances for the normal distributi
22、on, with power functions for testsGbe Part 6: Determination of statistical tolerance intervalsAnnexes A and B of this part of ISO 16269 are for information only.SANS 16269-7:2010This s tandard may only be used and printed by approved subscription and freemailing clients of the SABS .INTERNATIONAL ST
23、ANDARD ISO 16269-7:2001(E) ISO 2001 All rights reserved 1Statistical interpretation of data Part 7:Median Estimation and confidence intervals1 ScopeThis part of ISO 16269 specifies the procedures for establishing a point estimate and confidence intervals for themedian of any continuous probability d
24、istribution of a population, based on a random sample size from thepopulation. These procedures are distribution-free, i.e. they do not require knowledge of the family of distributionsto which the population distribution belongs. Similar procedures can be applied to estimate quartiles andpercentiles
25、.NOTE The median is the second quartile and the fiftieth percentile. Similar procedures for other quartiles or percentiles arenot described in this part of ISO 16269.2 Normative referencesThe following normative documents contain provisions which, through reference in this text, constitute provision
26、s ofthis part of ISO 16269. For dated references, subsequent amendments to, or revisions of, any of these publicationsdo not apply. However, parties to agreements based on this part of ISO 16269 are encouraged to investigate thepossibility of applying the most recent editions of the normative docume
27、nts indicated below. For undatedreferences, the latest edition of the normative document referred to applies. Members of ISO and IEC maintainregisters of currently valid International Standards.ISO 2602, Statistical interpretation of test results Estimation of the mean Confidence interval.ISO 3534-1
28、, Statistics Vocabulary and symbols Part 1: Probability and general statistical terms.3 Terms, definitions and symbols3.1 Terms and definitionsFor the purposes of this part of ISO 16269, the terms and definitions given in ISO 2602 and ISO 3534-1 and thefollowing apply.3.1.1kth order statistic of a s
29、amplevalue of the kth element in a sample when the elements are arranged in non-decreasing order of their valuesNOTE For a sample of n elements arranged in non-decreasing order, the kth order statistics is xkwherenxx x1 2 .uuuSANS 16269-7:2010This s tandard may only be used and printed by approved s
30、ubscription and freemailing clients of the SABS .ISO 16269-7:2001(E)2 ISO 2001 All rights reserved3.1.2median of a continuous probability distributionvalue such that the proportions of the distribution lying on either side of it are both equal to one halfNOTE In this part of ISO 16269, the median of
31、 a continuous probability distribution is called the population median and isdenoted by M.3.2 Symbolsa lower bound to the values of the variable in the populationb upper bound to the values of the variable in the populationC confidence levelc constant used for determining the value of k in equation
32、(1)k number of the order statistic used for the lower confidence limitM population mediann sample sizeT1lower confidence limit derived from a sampleT2upper confidence limit derived from a sampleu fractile of the standardized normal distributionxiith smallest element in a sample when the elements are
33、 arranged in a non-decreasing order of their valuesxG25 sample mediany intermediate value calculated to determine k using equation (1)4 ApplicabilityThe method described in this part of ISO 16269 is valid for any continuous population, provided that the sample isdrawn at random.NOTE If the distribut
34、ion of the population can be assumed to be approximately normal, the population median isapproximately equal to the population mean and the confidence limits should be calculated in accordance with ISO 2602.5 Point estimationA point estimate of the population median is given by the sample median, xG
35、25 . The sample median is obtained bynumbering the sample elements in non-decreasing order of their values and taking the value ofGbe the (n G2b 1)/2th order statistic, if n is odd, orGbe the arithmetic mean of the (n/2)th and (n/2) G2b 1th order statistics, if n is even.NOTE This estimator is in ge
36、neral biased for asymmetrical distributions, but an estimator that is unbiased for anypopulation does not exist.SANS 16269-7:2010This s tandard may only be used and printed by approved subscription and freemailing clients of the SABS .ISO 16269-7:2001(E) ISO 2001 All rights reserved 36 Confidence in
37、terval6.1 GeneralA two-sided confidence interval for the population median is a closed interval of the form T1, T2, where T1G3c T2; T1and T2arecalledthelower and upper confidence limits, respectively.If a and b are respectively the lower and upper bounds of the variable in the population, a one-side
38、d confidenceinterval will be of the form T1, b)oroftheform(a, T2.NOTE For practical purposes, a is often taken to be zero for variables that cannot be negative, and b is often taken to beinfinity for variables with no natural upper bound.The practical meaning of a confidence interval is that the exp
39、erimenter claims that the unknown M lies within theinterval, while admitting a small nominal probability that this assertion may be wrong. The probability that intervalscalculated in such a way cover the population median is called the confidence level.6.2 Classical methodThe classical method is des
40、cribed in annex A. It involves solving a pair of inequalities. Alternatives to solving theseinequalities are given below for a range of confidence levels.6.3 Small samples (5 u n u 100)The values of k satisfying the equations in annex A for eight of the most commonly used confidence levels forsample
41、 sizes varying from 5 to 100 sampling units are given in Table 1 for the one-sided case and in Table 2 forthe two-sided case. The values of k are given such that the lower confidence limit iskTx1G3dand the upper confidence limit is21nkTxG2dG2bG3dwhere1 2 , ,.,nxx xare the ordered observed values in
42、the sample.For small values of n, it can happen that confidence limits based on order statistics are unavailable at certainconfidence levels.An example of the calculation of the confidence limits for small samples is given in B.1 and shown in Form A ofannex B.SANS 16269-7:2010This s tandard may only
43、 be used and printed by approved subscription and freemailing clients of the SABS .ISO 16269-7:2001(E)4 ISO 2001 All rights reservedTable 1 Exact values of k for sample sizes varying from 5 to 100: one-sided casekkConfidence level%Confidence level%Samplesizen80 90 95 98 99 99,5 99,8 99,9Samplesizen8
44、0 90 95 98 99 99,5 99,8 99,95211aaa a a55 24 23 21 20 19 18 17 1667891022334122331122211122a1111aa111aaa11aaaa1565758596025252626272324242525222223232420212122221920202121181919202017181819191717171818111213141545556344553344423334222331222311222111226162636465272828292925262627272425252526232323242
45、421222223232121212222192020212119191920201617181920677885667755666445553445533444233342233366676869703030313131282829293026272728282525262626242424252523232324242122222323212121222221222324259991010888997788866777566675556644555444557172737475323233333430313131322929293030272728282926262727272525262
46、626232424252523232324242627282930111112121310101111119910101188999788897778866777566677677787980343535363632333334343131323233293030303128282929302727282829262626272725252526263132333435131414151512121313141111121213101011111199101011899101088899778898182838485373738383935353636373334343435313232333
47、330313131322929303031282828292927272828283637383940151616171714151516161314141415121213131411111212131011111212101010111199101010868788899039404041413738383839353636373734343435353233333434313232323330303131312929303030414243444518181919201617171818151616171714141515161314141415121313141411121213131
48、111121212919293949541424243433940404141383839393936363737383435353636333434353532323333343131323233464748495020212122221919202020171818191916171717181516161617141515161613141415151313131414969798991004444454546424243434440404141423838393940373738383835363637373434353536333334343551525354222323242121
49、2222202021211819191917181819161717181516161715151516aA confidence interval and confidence limit cannot be determined for this sample size at this confidence level.SANS 16269-7:2010This s tandard may only be used and printed by approved subscription and freemailing clients of the SABS .ISO 16269-7:2001(E) ISO 2001 All rights reserved 5Table 2 Exact values of k for sample sizes varying from 5 to 100: two-sided casekkConfidence level%Confidence level%Samplesizen80 90 95 98 99 99,5 99,8 99,9S
