432 Prof. F. Y. Edgeworth on the 



independent elements, each element in random fashion assu- 

 ming either the value or +i with equal frequency in the 

 long run. Then, according to the magnitude of m, and the 

 degree of accuracy required, the group formed by the varying 

 values of H may be regarded as conforming to a probability- 

 curve whose modulus is \ / — h U P *° a distance from the 



central value ( -^ i) amounting to the quartile, octile, decile, 

 &c, as the case may be. Thus of a group of values assumed 

 by S about a quarter occurs between the limits -5- i and 

 q- i-~q a / ^ t; where q is the " quartile u for modulus unity, 

 = '476 .... Another quarter occurs between -^ i + qA/ — i. 

 Similarly the octiles, deciles, &c. are 



^•\JV' f'+V?^* -' 



where r, s, &c. are coefficients obtained from the tables ; either 

 below, or not much above unity. And so on, up to the 

 largest percentile up to which the approximation is accurate. 

 Now let this group be deformed by squaring each of the 

 observations. The new median, quartile, octile, &c. will of 

 course be the squares of the respective old ones. The new 



median will be-^-2 2 ; the new quartiles (^ ±q\/ — ) &\ 

 distant respectively from the new median, — ? 2 , by 



Now the last-written two expressions differ from the ex- 

 pression outside the brackets and from each other by small 

 quantities ; so that but a small proportion of the group occurs 

 between the limits 



and in the corresponding interval in the neighbourhood of the 

 quartile beloic the Mean. Thus the new quartiles are approxi- 



-r-±-y~g)i i ' In like manner it may be shown 



