12 KARL PEARSON 



We now want y^. Remembering that the range is 2, we have* 



„ r (to + 1-5) / ..-.. 



y.=N 1^ ' (xui). 



But o" is small, i.e. '07 would be large for cr and ' therefore o-^ large if equal 

 to '005. Thus l/cr" will be small if it is 200 and m small if only of order 100. 

 We may therefore safely use Stirling's theorem to obtain the T-fiinctions. Thus 



r(m+l-5) _ j2Tr{m+V5) e-<'"+^"^' {m+V5)^'^ 

 r(m+l) V2ir(m+1) e-<™+^> (m+l)'«+i 



_., TO+1-5 /m + l-5\'»+i 

 = e X ■ , X ' ' 



Vm + 1 \ TO + 1 

 e X , 1 + ^ 



Vm + 1 \ TO + 1 

 = Vto+I + 



\/to + 1 s/to + 1 



= v/to+1 jl+ \ l 

 (^ 2(to+1)J 



nearly. 



n/2o- 

 Hence we can take for our frequency curve for r 



y = ^{i-^y^'' ^ (xiv). 



When the sample is very small we must retain the full value 



•Jit - 



i2iL(i-.f(p-') (,.). 



(H 



This agrees in the special case for product-moment r's when o-^ = with 



n— 1 



the form proposed by " Student " in Biometriha, Vol. vi. p. 306 and experimentally 

 justified by him. 



We have next to find the area of the tails of this curve beyond a given value rf, 

 to measure the improbability that with no correlation a random sample could give 

 a value r. We clearly have 



P = 2 {^ ^L-(l -x'f^''' dx= 1^- [^ il -x'Y dx 



= - / - - ^ \ -dil —yfY'^'' 

 N 170-2 {m + l)jrX ^ ' 



\2\\{ \-tY^' 1 L_f'j-/1_ 2^m+l^ 1 



"n/tto-L r 2(to+1) 2(TO + l)J,a;^^^ ^' '^''J ' 

 * Phil. Trans. Vol. 186, a, p. 372. 



t r is to be treated as a quantity without sign, a mere numerical quantity and therefore both tails of 

 the frequency distribution are taken — this is the origin of the first factor 2 in the value of P. 



