276 
PROFESSOR KARL PEARSON, MATHEMATICAL 
NXS'R = n^aicr'ii' --—— (wi — W2) (m'l — m'2) . . (xxx.) 
Let — of the N first individuals and — of the N second individuals have fictitious 
P . 2 
values, then ^—— N and - - N will have their true values. If, now, there is no 
P 9. 
correlation between the fictitious values in the two cases, we have at once : 
nj = ^ N, no = — N, nj 
pq - pq 
From this it follows at once that 
or the second term in (xxx.) vanishes. Thus : 
7* ~ 1 XT ? ~ 1 XT 
- -N, ni=- -N. 
P9 Pi 
_ «1 O'lO’ P' 
N 22' 
(xxx.) his. 
Thus R vanishes with r, and no spurious correlation could arise from the existence 
of fictitious values distributed at random through the correlation table. This result 
might, indeed, (as it often is tacitly) be assumed by some, but it seems very desirable to 
have a definite proof. 
It remains to consider 2 and 2'. We have : 
oi¬ 
ls! 2' = n^al + iioO-'l + n^a-i + nial 
-f 7^1 (7?ii — M)^ + no {nio — M)“ + n^ {m^ — M)“ + ^4 (^4 — 
= (7?, + 773) cri -f- (7?, + 7^4) O'a 
+ Oh + ^ 3 ) -f Oh + ^^4)^>^2 — (^1 + ^2 + 773 + n^) 
2 - = 
7ii + n, 2 , 7i, + n^ , , 77i + 713 n .2 + 7?1 /.^2 
o-I + —^ 0-2 -f - o^h — m.z)- 
N 
N 
N N 
1 \ 1 
Similarly 
= (1 - 7) <^1+Wi + (1 - y-j j (»«, - «‘,r- 
= tr; + 7 (ctI — crj) + (1 - 7)7 (m, — m.y- . 
2 '= = <r'i + 7 (< 7 l - -x'f) + (1 - Aj A 
(xxxi.). 
. (xxxii.). 
Now if the introduction of the fictitious values consisted of anything of the 
nature of a wrong pairing of certain individuals, we should simply have o-j = cr.,, 
a'i= cr'o, nil — nu, m'l = ni'o and, accordingly, 2 = cr,, and 2' = cr',. 
