\.\\l", L1"- A ] Infmiltn-tiun clxxii 



population. Hence after drawing at random a doublet or triplet, thenc must be 

 returned tn "tin- l>ag" before ;i second is drawn. In a sample uf it we might treat. 

 \n doublets or $n triplets as independent random samples of 2 or ,% but it would 

 not be legitimate to take all possible doublet or triplet arrangements out of the 

 and count them as \n (n- 1) or ^n(n- l)(n - 2) independ- -nt samples of 2 

 In random sampling in doublets from an indefinitely large population, no individual 

 would be likely to repeat itself \(n 1) times in \n (n - 1) drawings. 



The standard error of p from M triplet drawings if M be considerable is given by 



m(-)_ I 



l -^ ........ 



(iii) Samples of Four, n = 4. 



Here we have, if we put p = s\na, 



2 



r = {cota + a(l - cot* a)] .................................... (xxi), 



7T 



cr r 2 =l-2cot 2 a + 2acot 3 a-r 2 ................................. (xxii), 



2 



f*9 =- {cota + 6 cot 3 a +a(l 3 cot 2 a 6 cot 4 a)} ......... (xxiii), 



7T 



/i/ = 1 4 cot 2 a 6cot 4 a+a(6cot 3 a + 6cot 5 a) ............ (xxiv). 



From these formulae Table XXXI C has been calculated. The table of ordinates 

 (Table XXXII), under the heading of n = 4, indicates that the frequency is expressed 

 by truncated J-curves, starting at p = with the rectangle, which means that 

 samples of 4 from an indefinitely large normal population without correlation are 

 equally likely to give a correlation of any intensity from 1 to + 1. 



_It may be of interest to note that in the case of samples of 4, the chance that 

 the correlation coefficient of a sample should lie between and + r takes the form 



m(+r) (1 p 2 )$ fcos~ 1 ( rp) rp 

 - - 



(xxv), 

 M 



while the chance of a value between and r is 

 m (~ r > C 1 ~ P 2 ) f f r P 



M TT (1-tV (l-rV)*I 



Hence the chance that the coefficient of correlation in a sample of 4 should lie 



wi (+ r) 4- m (- r) /l-*\* 



-- = 



(XXV11) - 



And the excess of positive over negative coefficients in M samples of 4, 



/ i \ f /i ^ cos" 1 p + 2p Vl p*\ 

 = m(+ 1) m( 1) = M( 1 -- -J ...(xxvni). 



