cvi Tables for Statisticians and Bio metricians [XVIII XX 



discussed by Irwin, and the probability of differences PI (X) and P 2 (X), as great or 

 greater than Xo- computed by him*. These are given in Tables XIX and XX for 

 samples ranging from 2 and 3 respectively, to 1000. Table XVIII provides the 

 constants of the curves used by Irwin to describe the frequency distributions of the 

 first and second intervals. The frequency distribution of the differences between 

 the th and (p + l)th individuals (p = 1 and p = 2) is given closely by: 



from #=0 to #= oo . The table gives the values of h and S in terms of cr, the standard 

 deviation of the data. 



Illustrations, (i) Chauvenet gives the following fifteen observations of the 

 vertical diameter of Venus as deviations from their mean made by Lieut. Herndonf : 

 -1"-40, -0"-44, -0"-30, -0"-24, - 0"-22, - 0"-13, -0"'05, 

 + 1"-01, + 0"'63, + 0"'48, + 0"-39, + 0"'20, + 0"18, + 0"'10, + 0""06. 

 Are the outliers 1"'40 and 1"'01 to be rejected or not? 



The standard deviation J of the 15 observations is o- = 0" p 5326, with a probable 

 error calculated by the ordinary formula of %2" =0"'0656. Thus the actual 

 standard deviation of the parent population would roughly be as likely to lie out- 

 side as inside the limits 0"'47 and 0"'60. 



(a) We ask first, what is the chance that in 15 observations one individual 

 would lie outside the limits + 1"'01 ? 



1-01/-5326 = 1-8964, and f (1 + a) = nearly -97 1|. Thus the chance would be "03 

 or outside both limits '06. This is the chance at a single draw, and in fifteen trials, 

 we should expect 15 x '06 individuals or '90 individuals beyond 1*01, actually there 

 is one. This certainly would not justify its rejection. Let us consider the matter 

 further. How many individuals ought we to expect beyond 1"'39? 



Here l'39/'5326 = 2'61 nearly; hence | (1 + a) = '9955, or the chance ='0045, 

 and for the double limits '009. We should therefore be prepared for 0'135 individuals 

 outside 1"'39 in 15 observations, and we find one. We therefore decide that the 

 1"'40 should probably be rejected. A like result follows if we increase the 

 standard deviation to 0"'60. If we reject the 1"'40 observation, we shall alter 

 our mean and standard deviation. We find the mean = +0"'1193 and the standard 

 deviation = 0"'3869f . After + 1"'01 the next highest observation is 0"'63, and we 

 may ask how many individuals we ought to expect over 0"'70, say 



* Biometrika, Vol. xvn. pp. 238 250. t Loc. cit. Vol. n. p. 562. 



J The actual mean of Herndon's observations as they stand is not zero, as it should be, but +0"-018. 

 He probably found his mean to more decimals than his observations, and then cut down the deviations 

 to two decimals. I have treated his mean as zero for present purposes. 



Part i, Table V. || Part i, Table II. 



IT The value given in Biometrika, Vol. xvn. p. 245, appears to have been obtained by neglecting the 

 change in mean. 



