Goodness of Fit of Frequency Distributions. 377 



and record that the total frequency in (i.), or 83, was 

 inadequate to determine % 2 , while that in (ii.) was adequate 

 to give % 2 with about 3 per cent, possible error. 



We now consider the standard deviation o£ P. This for 

 a fourfold table is 



<r p =i V .{P 4 (^)-P 2 (^)}. 



The value of P 2 (% 2 ) is not tabled in the Tables for 

 Goodness of Fit, for the simple reason that it is provided 

 in the Tables of the Probability Integral *. In fact, 

 for ?2 = 4 t, 



P =\/- (-e-'^--x'e-^+e-iX 2 \d x 



- x e u '(x d x) 



i ri 



= *V „x e ~ r2T<T x 2 



or ?*{X)=\/1 Ce-^dx = 2xi(l-*)=2x\l-iQ. + a)\, 



where ^(1-f-a) is the probability integral %. 



Now x 2 = -7080 and therefore % = '84143 for (i.), and 

 X 2 = 133*3265 and therefore x=ll"54671 for (ii.). 



We can find P 2 (%) for (i.) at once by interpolating 

 between *84 and *85 in the tables for the probability 

 integral. There results ^(1 + «) = '799945, and there- 

 fore K 1 -*) = '200055 and P 2 (%) ='40011 for (i.), but 

 P 4 ( % ) = -8713. Hence 



<r p = I 2-5480(-8713-'4001) = -6003. 



Thus we deduce '4049 for the probable error of P in (i.). 



* This fact has, of course, been known since the first " goodness of 

 fit" table was calculated, but was overlooked by certain American 

 writers. I therefore gave a proof in a joint paper by Dr. Heron and 

 myself, Biometrika, vol. ix. p. 312. 



+ ' Tables for Statisticians,' p. xxxi. 



X IjOc. cit. p. 2. 



