MISCELLANEA. 
Note on the Probable Error of the Coefficient of Correlation in the 
Variate Difference Correlation Method. 
By a. RITCHIE-SCOTT, B.Sc. 
In Biometrika, Vol. x. p. 278, Dr Anderson of Petrograd has given the following expression 
for the probable error of the correlation coefficient between differences of the mth order. 
Probable error of j^r^j = -67449 
= -67449 Ll,^ j-J- [n-m + 2(n-m-l)( V 
Jn - m [n - m[_ \m + IJ 
„, / m.TO-1 \^ / m . m - 1 . m - 2 \^ , ~|) 
+ 2 (?i - m - 2 . f, + 2 (w - m - 3 ) + etc. [ , 
where m = order of difference, 
n — size of original population (i.e. number of years in series of observations etc.). 
In this formula the population is supposed to be reduced by unity at each difference taken. 
It is, however, found more convenient in practice to keep the population constant by taking in 
an additional observation at each successive operation of taking the difference. If we denote 
this constant population by n' the expression within the brackets reduces to 
—An' + 2{n'-l) -] + 2 2 = -) + 2(w' - 3) -) + etc. 
w |_ ^ \m+lj \m+l.m+2J \m + 1 . m + 2 . m + 3/ J 
\ nJ\m+\J \ n' J \m + I .m + 2J 
f ■, 3\/ m.m-l.m-2 \^ 
+ 21-- + etc. 
\ n J \m +l.m+2.m + 3/ 
^ / m Y_j_ / m .m - 1 V^ / m.m-l.m-2 \ 
~ { \m + 1 ) \m + 1 . m + 2/ \m + 1 . m + 2 . m + 3/ ^ ^"j 
_ 1 _ ^ (/ Y 2 / m.m - 1 Y ^ 2 / m.m-l.m-2 Y + etc I 
n' \\m + 1/ ^ \m + 1 . m + 2/ \m + 1 . m + 2 . m + 3/ ^ ^'J 
= 2^(m)-l-m 
The series <^ (m) and/(wi) have been summed as follows by Professor Pearson: 
f m \^ f m .m - \ \^ / m.m-l.m-2 \^ 
d) (m) = 1 + - + -5 + s ., + etc. 
^ ^ ' \m + 1/ \m + 1 . m + 2/ \m + 1 . m + 2 . m + 3/ 
