rREStDENTIAL ADDRESS, 637 



this, that the chances are theeilme for all the itevis of the group to be sampled, and 

 that the way thet/ are taken is absolntely independent of their ma(/nitudc. 



The forty averages of 10, so obtained, should by Professor Edgeworth's 

 theory be grouped in a normal curve of error, and, in fact, they are, with modulus 

 1'096/. The average of the 400 is found to be 4'7435/., with standard deviation 

 •122^. The original items vary ' from nil to 103/. The average, deduced from 

 the samples, is thus known with practical certainty to be between 41. 7s. and 

 51., and the chances are even that it is or is not between 41. 13s. 3d. and 

 41. 16s. 6d.^ Actually, when the whole 3,878 were added together, the average 

 proved to be 41. 16s. Id. 



It is to be noticed that the precision of this and the following measuretnents 

 does not depend in any ivay on the size of the group sampled, but only on its 

 nature and on the number of sainples taken, if the area of choice is co-e.vte7isive 

 xoith the group. Here I have taken 2 in 19 as samples, hut the results would 

 apply equally well if my original list were extended a hundredfold or to any 

 size; but then the task of verification would he prohibitive. If information were 

 required as to the incomes, for example, of 1,000,000 persons, the labour of 

 sampling to obtain results of given precision would be no greater than for 10,000 

 persons, except that precautions would need to be taken that each of the 1 ,000,000 

 had an equal chance of inclusion. 



Having forecasted the average, I proceed to forecast the grouping. 7 per cent, 

 are found with no dividend, 9 per cent, between 3/. and 31. 10s., and so on, as in the 

 table on the next page. The precision of these measurements is found from them- 

 selves, and varies jointly as the square roots of the number in the whole sample 

 and (nearly) of the fraction the class selected is of tbe whole. The precision can 

 be made as great as 7ce please, the probable and possible errors as little, by 

 ina-easing the size of the sample. It is to he noticed that the deviations in the 

 separate classes are not independent, since their sum is zero, and the problem is 

 thus complicated. If an unlucky sample is taken for one group, there must be 

 one or more bad samples for others. Where, in the table on the next page, no 



' They are, in the list used, grouped according to the nature of the securities. 

 Government, Municipal, Railways, Mines, &c., and tbe averages and standard clevia- 

 lions on successive pages differ materially. An artificial method of sampling is 

 therefore necessary. This aggregation is very similar to that found in wages in 

 different occupations and localities, and in many other practical examples. 



- 400 samples taken at random from a list containing 3,878. 



Average of 400, 4-7435L 



Modulus for 40 averages of 10, deduced from these averages, 1096 = r, where 



, _ /'^Sf 



Hence modnlns for the average of 400 h -=1 ^-jyal. ' Probable error' = 082/ 



^^40 

 Standard deviation, -122/. 



Hence average of all is as lihely as not to be hetneen 4S26I. and 4-66ll. 



The modulus for the average of 100 is .S4(;, and, in fact, the deviations of the 

 four sample averages of 100 taken from the average for 400 are +06 -^"0 —-27 

 + 003. 



The 40 averages of 10 each conform fairly with a normal curve of error thus : 



When the curve of frequency is normal with standard deviation <r, the chance 

 that any particular case shall differ from the central value by as much as <r is -317 • 

 2c, 046 ; 3(r, 0027. ' 



