~ 6 



clxvi Tables for Statisticians and Biometridans [XXXI a ~ A , LI 



From the above formulae Table XXXP has been dressed, which gives the con- 

 stants of the U-curves for the distribution of r in samples of three. The first page of 

 Table XXXII (p. 185) gives the ordinates of 10 of these U-curves. As the differences 

 of the argument p are '1, it is needful occasionally for certain values of p to use the 

 above formulae for the constants. Tables of the Complete Elliptic Integrals have 

 been computed by Legendre's Traite des Fonctions Elliptiques, Tom. n. The pro- 

 bability integrals for n = 3 * are 



m(+r)_I( cos" 1 r / /I r 2 cos~ x ( rp)\) 



T~ -2J 1 ~pr + T -v r^y~~F ;j 



cos" 1 r / / 1 r 2 cos" 1 (rp) 



i-t v^wu I / ., / J- I \JW V ' f / I I / ' V 



1 nil A / v ' ' |L (xvi\ 



A r* I * A/-I o<> i /( \ AV1 /' 



A fTT- ' \ V *" 1 _. / V / 



M 2 ( ITT TV l-r*p 2 



Again, the integral of the area from r to + r 



m (+ r) + m ( r) cos" 1 r 1 r 2 /, cos" 1 (rp)\ 



- 1 - - <""> 



m(+r)-m(-r) ( l\-r z \ ... 



-V = H 1 -vr^y) ............... (xvm) - 



Whenr-l, m(+ >~ m P . ...(xix). 



Thus if we take M samples of three from a normal population, we can easily 

 determine p by simply counting the number of positive and negative correlations. 



The values of m(+) and m( ) for each value of p are given in the last two 

 columns of Table XXXP. 



The determination of the sign of the correlation in the case of triplets is not so 

 easy as in the case of doublets. If #1, yi, x z , y%, x 3} y s be the three pairs, the sign of 

 their correlation depends on that of their product-moment 



- i (%2/i + ^22/2 4- a&ys) - i fa + ^ + a) 3 (#1 + 2/a + ?/3>, 

 or, if y = I (T/I + 2/2 + 2/3), 



on the sign of fa - #1) (y a - y ) + fa - a?i) (3/3 - ) 



Now if we take #1, # 2 . #3 in ascending order of magnitude x^ x\ and ^3 oc\ 

 will both be positive. Hence if y z and ?/ 3 are both greater than y, the correlation 

 will be positive; if they are both less, it will be negative. If y-^ y and y$ y are 

 of opposite sign, then we must consider the numerical values of the four quantities 

 #2 - i, 2/2 y, 3 \ and y 3 y; but as a rule it is not needful to multiply them 

 out. 



The student must be careful of one point in approaching the correlation of a 

 parent population by consideration of the correlations of doublets or triplets. He 

 must not take out n cases and form all the possible doublets or triplets from these. 

 The theory is based upon random sampling from an indefinitely large normal 



* I owe these to Mr E. C. Fieller. 



