Past Experience on Future Expectation*. 373 



Table I. 



Frequency of Percentages in a Second Sample 



for Special Case. 



(1) 



(2) 



(3) 



(1) 



(2) 



(3) 



Percentage. 



Frequency. 



Sum of | 

 Frequency. 1 



Percentage. 



Frequency. 



Sum of 

 Frequency. 







•4 



•4 



17 



29-5 



9277 



1 



22 



26 i 



18 



220 



949-7 



2 



74 



9-7 



19 



160 



965-7 



3 



15-9 



25-6 



20 



11-4 



977-1 



4 



28-9 



54-5 



21 



7-9 



985-0 



5 



44-8 



99-3 



22 



5-4 



990-4 



6 



61-3 



160-6 



23 



3-6 



994-0 



7 



761 



236 7 



24 



23 



996-3 



8 



87-0 



323-7 



25 



1-5 



997-8 



9 



92-8 



4165 



26 



•9 



998-8 



10 



933 



509-8 



27 



•6 



999-3 



11 



89-1 



598-9 



28 



•4 



999-6 



12 



812 



680-1 



29 



•2 



999 8 



13 



710 



751-1 



30 



•1 



999-9 



14 



59-9 



811-0 



31 



•o 



1000 



15 

 16 



48 7 

 345 



859-7 



898-2 









Total ... 



1000 



1000 



side of the mean (i. e. 7*85 to 13*1) is clearly too high. If 

 we calculate the median by the rale of the third * and place 

 the probable error either side of this theoretical median, we 

 shall approximate closely to the actual range. The distance 

 from mode to mean =10' 78 — 10 = * 78, and one-third of this 

 is *26. Hence 10*78 — '26 = 10*52 is the theoretical median. 

 This gives 10*52 + 2*93, or a range of 7*59 to 13*45 as compared 

 with the actual 7*65 to 13*48 within which half the number 

 of second samples will fall. This is a far better result than 

 the 7*98 to 12*02 of the usual method. The one amounts to 

 saying that half f the second samples will fall between 7*5 

 .and 13*5, which is sensibly in accord with actuality, while the 

 other crowds them into the range between 8 and 12, which 

 result is far in excess, only about 36 p. c. of the samples 

 falling within this range. 



This first illustration is, I think, sufficient to demonstrate 

 that it is not possible in judging expectancy from past expe- 

 rience (i.) to neglect the relative sizes of the first and second 

 samples, or (ii.) to neglect, even in characteristics which 

 appear in 10 p. c. of the sample, the sensible deviation from 

 the Gaussian distribution. 



The above method of dealing with the problem is merely 

 intended as a ready approximate method. It involves only 



* Phi!. Trans, vol. clxxxvi. A. p. 375. 

 t More accuratel}'- 51*5 per cent. 



