1、BRITISH STANDARD CONFIRMED NOVEMBER 1983 BS 5350-E1: 1976 Incorporating Amendment No. 1 Methods of test for Adhesives Part E1: Guide to statistical analysis UDC 665.93:620.1:519.23BS5350-E1:1976 This British Standard, having been prepared under the authorityof the Adhesives Standards Committee, wasp
2、ublished under the authorityofthe Executive Boardon 30 November1976 BSI 07-1999 The following BSI references relate to work on this standard: Committee reference ADC/9/2 Draft for comment 74/50318 ISBN 0 580 09335 2 Foreword Physical testing invariably yields scattered results and simple attempts to
3、 obtain improved reliability in the estimate of the quantity under study involve replicate experiments. Tests on adhesives are particularly prone to scatter and it is therefore desirable to apply a rigorous statistical treatment to such test data so that an objective interpretation can be made. A Br
4、itish 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. Summary of pages This document comprises a fr
5、ont cover, an inside front cover, pages i and ii, pages1 to 4 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. Amendments issued since publication Amd. No. Date of i
6、ssue Comments 6285 April 1990 Indicated by a sideline in the marginBS5350-E1:1976 BSI 07-1999 i Contents Page Foreword Inside front cover 1 Scope 1 2 Reference 1 3 Definitions 1 4 Use of arithmetic mean and standard deviation 1 5 Calculations 1 6 Confidence limits for the mean 1 7 Significance of di
7、fference between means 1 8 Estimate of number of observations required 2 Appendix A Specimen calculation 3 Appendix B Estimation of standard deviation from small numbers of test results 3 Appendix C Values of t for 95 % confidence limits 4 Appendix D Approximate number of observations related to obs
8、erved coefficient of variation and required range about the mean 4 Appendix E Effect of number of observations on the confidence interval 4ii blankBS5350-E1:1976 BSI 07-1999 1 1 Scope This Part of BS5350 describes statistical methods which are valid even if a small number of test results (fewer than
9、10) is available. More detailed information is given in BS2846. 2 Reference This Part makes reference to the following standards publication: BS2846, Guide to statistical interpretation of data. 3 Definitions For the purposes of this Part of this British Standard the following definitions apply. 3.1
10、 arithmetic mean ( ) of a series of n observations of x 1 , x 1 , x 2 , x 3x n , is the sum of all the values divided by their number, i.e. this quantity is also commonly known as the average 3.2 standard deviation and variance the standard deviation (s) is the positive square root of the variance (
11、s 2 ) which is a measure of the dispersion of the test results. For a series of n observations x 1 , x 2 , x 3 , . x nand an arithmetic mean, the variance is given by the following expression: 3.3 coefficient of variation (v) the standard deviation expressed as a percentage of the mean 4 Use of arit
12、hmetic mean and standard deviation The arithmetic mean in most cases gives the statistically best estimate of the true value of the quantity observed, i.e.the value it would have were all random errors eliminated. The scatter of the results is described by the standard deviation. It is assumed that
13、the errors follow Gaussian or normal distribution. 5 Calculations The calculation of means and standard deviations is straightforward but laborious and the use of a desk calculator is recommended. The equation for estimating the standard deviation can be written as: This form is particularly useful
14、when a desk calculator is available. A specimen calculation showing a recommended tabular layout is given inAppendix A. Alternatively, when the number of test results is small (between2 and10) a simplified formula is available which gives an approximate but reasonable estimate of the standard deviat
15、ion. Details of this method are given inAppendix B. 6 Confidence limits for the mean In general, the mean of a set of observations approaches the true value more closely as the number of observations is increased provided that there are no systematic errors. From any finite number of observations, h
16、owever, the true value cannot be determined exactly; thus, not only the mean of a particular set of observations but also the limits which have have a given probability of including the true value should be calculated. For many purposes the95% confidence limits are found convenient. These define a r
17、ange within which the true value will lie on95% of occasions when the calculation is used. 7 Significance of difference between means It is not always immediately obvious whether the difference between the means of two series of observations is significant (i.e.a real one). One test is that if eithe
18、r95% confidence limits of the difference between the means include zero then the means are not significantly different at a5% level of significance. x x v 100s x - =BS5350-E1:1976 2 BSI 07-1999 The confidence limits, L, of the difference between two means are given by the formula: where and are the
19、two means; n 1and n 2are the number of observations in each set; t is read from the table inAppendix C for (n 1+ n 2 1) observations; and s is defined by where S 1and S 2are the two standard deviations. Alternatively a statistic t can be calculated, defined by: and compared with the value of t as de
20、fined above. The difference between the means is significant ift 1 t, and is not significant if t 1u t, at the5% significance level. The method described is only valid if it can be assumed that the two series of observations come from populations with similar standard deviations, i.e.s 1and s 2are e
21、stimates of the same quantity. 8 Estimate of number of observations required If it is desired that the95% confidence interval of a mean shall lie within a predetermined range around the mean, then it is necessary first to determine the variance and then to calculate the number of observations that n
22、eed to be made. The table inAppendix D relates the approximate number of observations required in an isolated series of tests to the observed value of the coefficient of variation and to the required limits expressed as a percentage of the mean. Thus, if it is desired that the95% confidence limits s
23、hall lie within10% of the mean and if the coefficient of variation is20%, it will be necessary to take the mean of about18 observations. The figures inAppendix D, however, constitute a guide only, as variations in and s will occur in successive sets of observations. Hence the values of s and L for a
24、 set of observations based onAppendix D should always be determined to confirm that they do in fact conform to the requirements. Appendix E shows an example of the effect on the statistically derived quantities of increasing the number of observations. For an estimate, , of the mean, the limits, L,
25、are estimated by the formula: where t is a particular value of Students t distribution tabulated for a range of probability levels in standard text books. The values of t for the estimation of95% confidence limits are given in Appendix C. A specimen calculation of L is included in Appendix A. x 1 x
26、2 x xBS5350-E1:1976 BSI 07-1999 3 Appendix A Specimen calculation NOTEWhen using a hand calculator for calculating the standard deviation, writing down intermediate calculations should be minimized, because this practice gives the greatest source of error. With modern calculators this practice shoul
27、d not be necessary and pressing a single key will give s immediately. Appendix B Estimation of standard deviation from small numbers of test results B.1 Modified linear estimate of s The following method allows rapid estimation of the standard deviation from small numbers of test results (10or less)
28、 ranked in order of decreasing magnitude (i.e.x 1largest, x nsmallest). B.2 Criteria for testing for outliers by Dixons test If a test for the presence of an outlier is required, the following procedure may be used. This test is designed to detect results due to gross blunders (of calculation, opera
29、tion or assembly). The critical value, r nis based on the intention to operate at the95% level of confidence. Number of observation Observation a x x 2 1 2 3 4 5 6 7 8 9 10 11 12 13 4.42 4.47 4.70 4.72 4.53 4.55 4.60 4.64 4.29 4.52 4.57 4.58 4.66 19.536 19.981 22.090 22.278 20.521 20.703 21.160 21.5
30、30 18.404 20.430 20.885 20.976 21.716 Totals Cx = 59.25 Cx 2= 270.210 a e.g. a set of bond strengths Sample size (n) Estimator for s Efficiency a 2 3 4 5 6 7 8 9 10 0.886 (x 1 x 2 ) 0.591 (x 1 x 3 ) 0.486 (x 1 x 4 ) 0.430 (x 1 x 5 ) 0.262 (x 1+ x 2 x 5 x 6 ) 0.237 (x 1+ x 2 x 6 x 7 ) 0.220 (x 1+ x 2
31、 x 7 x 8 ) 0.207 (x 1+ x 2 x 8 x 9 ) 0.197 (x 1+ x 2 x 9 x 10 ) 1.000 0.992 0.975 0.955 0.975 0.967 0.970 0.968 0.964 a Efficiency is the ratio of the variance of the distribution to the variance of the “modified linear estimator” given in the table. It is thus a measure of the relative precision of
32、 the estimator. Sample size (n) Statistics Critical value (r n ) 3 4 5 6 7 whichever is the greater 0.941 0.765 0.642 0.560 0.507 8 0.554 9 0.512 10 whichever is the greater 0.477BS5350-E1:1976 4 BSI 07-1999 The above criteria may be used to test for extremes at the upper or lower end of the sample
33、range. If the ratio r 10or r 11exceeds the appropriate critical value r, there are grounds grounds for believing the outer value to be a statistical outlier and it may with justification be excluded from the calculation of mean and standard deviation. However, before rejecting a test result, technic
34、al (other than statistical) explanations as to why the result is unacceptable should always be sought. The above test should be used only as a last resort. Appendix C Values of t for95% confidence limits Appendix D Approximate number of observations related to observed coefficient of variation and r
35、equired range about the mean Appendix E Effect of number of observations on the confidence interval Number of observations t Number of observations t 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 12.7 4.30 3.18 2.78 2.57 2.45 2.36 2.31 2.26 2.23 2.20 2.18 2.16 2.14 2.13 2.12 2.11 19 20 21 22 23 24 25 2
36、6 27 28 29 30 40 60 120 Z 2.10 2.09 2.09 2.08 2.07 2.07 2.06 2.06 2.06 2.05 2.05 2.05 2.02 2.00 1.98 1.96 Coefficient of variation (%) Required range about mean 5 % 10 % 15 % 20 % 2 4 6 8 10 12 14 16 18 20 3 5 8 13 18 20 3 3 4 5 7 8 10 13 15 18 3 3 3 4 5 5 6 7 8 10 3 3 3 3 4 4 5 5 6 7 Observation nu
37、mber Observed quantity Observation number Observed quantity 1 2 3 4 5 6 7 8 9 10 672 447 720 613 604 562 644 728 789 550 11 12 13 14 15 16 17 18 19 20 483 604 577 586 698 789 674 582 689 711 First 5 observations First 10 observations All 20 observations Mean 611 633 636 Standard deviation 103 100 92
38、 Coefficient of variance, % 16.9 15.8 14.5 95 % confidence limits 128 ( 21 %) 71.6 ( 11.3 %) 43.0 ( 6.8 %)5 blankBS 5350-E1: 1976 BSI 389 Chiswick High Road London W4 4AL BSIBritishStandardsInstitution BSI is the independent national body responsible for preparing BritishStandards. It presents the U
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