116 On the Probable Errors of Frequency Constants 
(3) We may now examine some of the resulting error correlations. If r^^x^ be 
the correlation in errors of a;, and gc„ we have by (8) 
N \ N) 
For special cases : 
(a) Quartile and Quartile : r,^^^^ = g = "SSSS, 
(b) Median and Quartile : 7\„g^ = r,,,,^^ = ~ = '5774, 
(c) Median and Decile : r,na =i%nd =—^-= = "1925, 
'■ " 3v3 
(d) Decile and Decile : '"'*t<'« ~ ^ ~ 
(e) Quartile and Decile : r.) „ = „ = ~ = "5774, 
(f) Quartile and Decile : = r,i „ = = "1925, 
3 a/3 
/2 
(g) Median and Quarto-decile : ''w ^f/^, = 'I'lmid^ ~ V 3 ~ "8165. 
It is clearly impossible to treat such correlations as zero *. It is of interest to 
note that they are absolutely independent of the nature of the frequency distribution, 
and approach unity as the correlated grades approach each other. 
Now if we consider the determination of a range from the mean and mean 
square aspect we shall have 
Xi = X + X^a, X2 — X + \.2a, 
where Xj and A,2 are quantities to be determined from the form of the frequency 
distribution when we know its nature. 
Thus - o-^^j = cTj- + Xj-o-o" + 2\i X Mean (&z'S<7), 
Now a-i = <T-jN, a,' = ^{I3.,-I)a- 
and Mean {h7c 8a) = J \/^, a"/N, 
where /Sj and /S.j ai'e the usual fundamental frequency constants. Accordingly 
^x. = 'Jjy{^+ ^> + i V(/3.- 1)1* (14), 
Mean (SxM^ = ^ + 2 + ""'^^ + l^i^'^ ('^'^ -1)1 (l^)- 
* See remark, p. 133, 
