122 Or the Probable Errors of Frequency Constants 
(5) The question now arises as to whether we could still further reduce the 
above inaccuracies by combining these results in pairs in the best proportions, say 
\s and \g> where Xs + A.s= 1. Let the corresponding median value be nisi,', then 
and making this a minimum for varying X^, X,,- we lind 
As 
— 2 [hm-gSms'] 
[8/»jj8/;v] 
— 2 [S/HsS?/ts'] 
— 2 [SnigSms'] 
.(29). 
Table IV supplies the needful values of the a^'s and what we require are the 
mean products 
N 
But ni = N — Vi, Jia = N — ft2, thus finally 
[V,S/%] = ^^^^ (30), 
where corresponds to :(\ the outermost of the paired grades. 
This leads us at once to 
[8ms8m.'] = <T^,,^Hs/Hs' (31) 
which much simplifies the calculations, if we remember that cr,„^ and correspond 
to the grade further from the median. 
Hence for correlation of m^, vis' we have 
= (^2)- 
Tables V and VI give first the mean product-moment [S??isS/«s'] f-'id secondly 
the standard deviation o-,,,^^.- and the coefficients Xg, Xg^ of iDgg' = Xgntg + Xg'iUg'. 
It is clear from Table VI that the lowest value of the probable error of the 
median will be obtained by deducing the median from the two 1st deciles and from 
the two 3rd deciles and taking '34388 of the first plus '65612 of the second value; 
the value of o-,,,^., will then be l'0468cr/\/iV, i.e. only about 5 more inaccurate 
than that found by the arithmetic mean. We may search about this value for 
I* It is convenient to write [5/»j,5Hij,'] to sij^nify mean vaiue of the product 5»i,5h/,/. 
