14 
PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 
Probable error of r = '67449^. 
'67449 [ (a + d)(c -f V) t ( ? (a + c)(d + b) ( , %(ci + b) (<7 + c) 
“ 7^7 1 “ + V* N* + ^ IP - 
+ 
ad — be 
N 3 
ah — ed ac — bd 11 
a). 
where y 0 , i/q, and xjj 2 are readily found from Equations (xlix.), (xlvii.), and (xlviii.). 
Thus the probable error of r can be fairly readily found. It must be noted in using this 
formula, that a is the quadrant in which the mean falls, so that h and k are both 
positive (see fig., p. 2). In other words, we have supposed a -f c > b + d and 
a + b > e + d. Our lettering must always be arranged so as to suit this result 
before we apply the above formula. 
§ (5.) To Find a Physical Meaning for the Series in r, or for the e of Equation (xxi.). 
Return to the original distribution ° c j of p. 2. If the probabilities of the two 
characters or organs were quite independent, we should expect the distribution 
ci -j- b cc -f- c 
AT a + b b -f d 
N N 
N N N 
AT c + d a + c 
-j^y c -j- d b "4“ d 
N N 
N N 
Now re-arranging our actual data we may put it thus : 
a + b b + d ad — be 
a | b 
c I d 
+ b a + c ^ ad — be 
N N 
N 
N 
c + d a + c ad — be 
N N 
N 
N 
N N 
N 
, T c + d b + d , ad — be 
N —-4-- 
N N “ N 
Accordingly correlation denotes that - ^ - has been transferred from each of the 
second and fourth compartments, and the same amount added to each of the first and 
third compartments. If rj = (ad — 6c)/N 2 , then y is the transfer per unit of the total 
frequency . The magnitude of this transfer is clearly a measure of the divergence of 
the statistics from independent variation. It is physically quite as significant as the 
correlation coefficient itself, and of course much easier to determine. It must vanish 
with the correlation coefficient. We see from (xxi.) that 
y = e X HK, 
or we have an interpretation for the series in r of (xix.). 
Now. obviously any function of y, just like y itself, would serve as a measure oi 
the divergence from perfectly independent variation. It is convenient to choose a 
function which shall lie arithmetically between 0 and 1. 
