SANS 3301-2004 Statistical interpretation of data - Comparison of two means in the case of paired observations《数据的统计学解释 在两个成对观测情形下的手段比较》.pdf

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. ICS 03.120.30 ISBN 0-626-16008-1 SANS 3301:2004 Edition 1 ISO 3301:1975 Edition 1SOUTH AFRICAN NATIONAL STANDARD Statistical interpretation of data Compa

4、rison of two means in the case of paired observations This national standard is the identical implementation of ISO 3301:1975 and is adopted with the permission of the International Organization for Standardization. Published by Standards South Africa 1 dr lategan road groenkloof private bag x191 pr

5、etoria 0001 tel: 012 428 7911 fax: 012 344 1568 international code +27 12 www.stansa.co.za Standards South Africa This standard may only be used and printed by approved subscription and freemailing clients of the SABS.SANS 3301:2004 Edition 1 ISO 3301:1975 Edition 1 Table of changes Change No. Date

6、Scope Abstract Specifies a method for comparing the mean of a population of differences between paired observations with zero or any other preassigned value. It may be applied to establish a difference between two treatments, where “treatment” should be understood in a wide sense, e.g. as a test met

7、hod, measuring instrument or laboratory involved, and is applicable under the conditions that the series of differences can be considered as a series of independent random items and the distribution of the differences is normal or approximately normal. If the distribution deviates from the normal, i

8、t remains valid provided the sample size is sufficiently large (100 or more). Keywords comparison, data, data estimation, data processing, estimation, mean, statistical analysis, statistical data, statistical methods of analysis, statistics, tolerances (measurement). National foreword This South Afr

9、ican standard was approved by National Committee StanSA TC 169, Applications of statistical methods, in accordance with procedures of Standards South Africa, in compliance with annex 3 of the WTO/TBT agreement. This standard may only be used and printed by approved subscription and freemailing clien

10、ts of the SABS.INTERNATIONAL STANDARD INTERNATIONAL ORGANIZATION FOR STANDARDIZATION l MEXYHAPOAHAR OPrAHM3AUWl I-Io CTAHAAPTM3ALViM.ORGANISATION INTERNATIONALE DE NORMALISATION c Statistical interpretation of data - Comparison of two means in the case of paired observations lnterpr6tation statistiq

11、ue des donnkes - Comparaison de deux mo yennes dans Ie cas dobservations apparikes First edition - 1975-08-15 UDC 519.28 Ref. No. ISO 33014975 (E) Descriptors : statistical analysis, mean, comparison. Price based on 6 pages This standard may only be used and printed by approved subscription and free

12、mailing clients of the SABS.FOREWORD ISO (the International Organization for Standardization) is a worldwide federation of national Standards institutes (ISO Member Bodies). The work of developing International Standards is carried out through ISO Technical Committees. Every Member Body interested i

13、n a subject for which a Technical Committee has been set up 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. Draft International Standards adopted by the Technical Committees are circul

14、ated to the Member Bodies for approval before their acceptance as International Standards by the ISO Council. International Standard ISO 3301 was drawn up by Technical Committee lSO/TC 69, Applications of statistica/ methods, and circulated to the Member Bodies in March 1974. lt has been approved by

15、 the Member Bodies of the following countries : Austria . India South Africa, Rep. of Belgium Israel Spain L Brazil I taly Switzerland Czechoslovakia Netherlands Turkey France Poland United Kingdom Germany Portugal U.S.S. R. Hungary Romania Yugoslavia The Member Bodies of the following countries exp

16、ressed disapproval of the document on technical grounds : Sweden U.S.A. 0 International Organkation for Standardkation, 1975 l Printed in Switzerland This standard may only be used and printed by approved subscription and freemailing clients of the SABS.INTERNATIONAL STANDARD ISO 33014975 (E) Statis

17、tical interpretation of data - Comparison of two means in the case of paired observations 0 INTRODUCTION The method specified in this International Standard, known as the method of paired observations, is a special case of the method described in table A of ISO 2854, Statistical in terpretation o f

18、da ta - Techniques of estimation and tests relating to means and variances. 1 ) This special case is mentioned in section two of ISO 2854 immediately after the numerical illustration of table A, and a complete example of applications of the method of paired comparisons has been given in annex A of t

19、hat International Standard. The importante and wide applicability of the method justify a separate International Standard being devoted to it. 1 SCOPE This International Standard specifies a method for comparing the mean of a population of differentes between paired observations with zero or any oth

20、er preassigned value. 2 DEFINITION paired observations : TWO observations x; and yi of a certain property or characteristic are said to be paired if they are made : - on the same element i from a population of elements but under different conditions (for example, comparison of results of two methods

21、 of analysis on the Same product); - on two distinct elements considered similar in all respects except for the systematic differente which is the subject of the test (for example, comparison of the yield from adjacent Plots sown with two distinct varieties of seed). However, it should be noted that

22、 in the second case the efficiency of the test depends on the validity of the hypothesis that there is no other systematic differente between the individuals in the same pair other than the systematic differente under test. 3 FIELD OF APPLICATION The method may be applied to establish a differente b

23、etween two treatments. In this case, the observations Xi are carried out after the first treatment and yi after the second treatment. The two series of results of the observations are not independent because each resultx; of the first series (first treatment) is associated with a result yi of the se

24、cond series (second treatment). The term “treatment” should be understood in a wide sense. The two treatments to be compared may, for instance, be two test methods, two measuring instruments or two laboratories, in Order to detect a possible systematic error. Two treatments carried out successively

25、on the same experimental material might interact and the value obtained might depend on the Order. Good experimental design should enable such biases to be eliminated. Alternatively, only one treatment may be applied and its effect may be compared to the absence of treatment; the purpose of this com

26、parison is then to establish the effect of that treatment. 4 CONDITIONS FOR APPLICATION The method tan be applied validly if the following two conditions are satisfied : - the series of differentes d,- = xi - yi tan be considered as a series of independent random items; - the distribution of the dif

27、ferentes d; = x,- Yj between the paired observations is supposed to be normal or approximately normal. If the distribution of these differentes deviates from the normal, the technique described remains valid, provided the Sample size is sufficiently large; the greater the deviation from normality, t

28、he larger the Sample size required. Even in extreme cases, however, a Sample size of 100 will be sufficient for most practical applications. 1) At present at the Stage of draft. This standard may only be used and printed by approved subscription and freemailing clients of the SABS.ISO 3301-1975 (E)

29、5 FORMAL PRESENTATION OF CALCULATIONS Problem studied . . a . . . . . . . . . m . . . . . . . . . . . . . . . . . . . . Experimental conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statistical data Sample size : n= Calculations Sum of the observed values : ZXi = Zyi = Sum of t

30、he differentes : Edi = Sum of the squares of the differentes : Zdi = Given value : dO = Degrees of freedom : v=n-l= Chosen significance level : , Q!= Results Two-sided case : The hypothesis that the population mean of the differente is equal to dO (null hypothesis) is rejected if : -doiA2 One-sided

31、cases : a) The hypothesis that the population mean of the differentes is at least equal to dO (null hypothesis) is rejected if : ddo +A, NOTE tl -,(v) is the fractile of Order 1 - CY of Students variate t with v degrees of freedom. The values of tl _ = the probability of an error of the second kind

32、is therefore fl. For a given Sample n and error of the first kind, these probabilities depend not only on the true mean D of the observed differentes di = Xi - yi for which one tan Postulate different alternative hypotheses but also on the Standard deviation od of these differentes. This Standard de

33、viation is in general unknown and if n is small the Sample will provide only a poor estimator. The resul t is that it is impossi ble to set an u probabil i ty of an error of the secon d kind. Neverthe less, i n the following graphs the relati on is shown between the power of the test, 1 - PI and the

34、 actual PPer Iimit to the population mean divided by the corresponding Standard deviation, D/Od for one-sided tests of the hypothesis Ho : D 0 and CII different from 0 and from 1. 3) With a significance level of 0,05 and a Sample size of 50, a power of at least 0,95 is already obtained when the true mean differente exceeds one-half of the Standard deviation of the differentes. For n = 20, this power is obtained for D/od = 0,78 or more. This standard may only be used and printed by approved subscription and freemailing clients of the SABS.