276 PROFESSOR K. PEARSON AND MR, L. N. G. FILON 



This remains finite, even if p be indefinitely great. On the other hand, the 

 probable error of a, and even its percentage probable error, becomes indefinitely great 

 with p. It is to be noted that a in this case becomes infinite. 



(y.) Results (Ixv.) to (Ixvii.) give the probable errors of the second, third, and 

 fourth moments. It will be seen that roughly, for a large p, the percentage error of 

 the fourth moment is about double that of the second. It might thus appear, at 

 first sight, safer to work with the second than with the fourth, but this is by no 

 means necessarily the case, for to deduce any quantity from one or the other they 

 must be reduced to the same order. For example, the square root of /u, 2 must be 

 compared with the fourth root of /* 4 . and the probable errors of ^/^ ar >d 

 be sensibly of the same order. 



Remembering that fi 3 = ~~T~ = ~77~ ~~T)> we ma y wr 't- e 



6g // 1 



*M,= 77IT> VU hl + 

 v(* 11 ) \p + j- 



(P + 



18$(p + I 



This tends to a finite limit as p increases indefinitely, and we conclude that the 

 probable error of /i 3 is always finite, and will in general be a small fraction of the 

 cube of the standard deviation. The above remarks are a justification for the use of 

 higher moments in frequency calculations. 



Equations (Ixviii.) to (Ixx.) give the error-correlations between the first three 

 moments. They show that an error in the value of one of these moments will most 

 probably lead to an error in the other two. We see that for p fairly large R^ 4 is a 

 large correlation, while R^ and R MjMt are small. In other words a random selection 

 of an even moment makes a far larger correlated change in another even moment 

 than in an odd moment. If p increase indefinitely we find the ratio R M-1J / R Wi 

 approaches the value 2/3 ; in other words, fj. 3 is more closely correlated to the higher 

 moment /* 4 than to the lower moment /i 2 . 



Formulae (Ixxi.) give the probable errors of the useful constants j8, /il//n* and 

 $, = p, t /p%. We see that they are small and approach the value zero as p is 

 indefinitely increased. 



(8.) Let us restate the formulae for p indefinitely great, i.e., for the normal curve of 

 frequency 



-'' S 



In this case we have //,> = or, // 3 = 0, //, 4 = So- 4 , /?, = 0, /? 2 = 3, skewness = 0, 

 mean and mode coincide. Several of these zero quantities, however, tend to have 

 definite probable errors. 



We have 



