\ X V] Introduction cxxxv 



of the rarity of a particular ratio concocted with the sample, but may be dangerous 



if intfipirt. il us ;i me.-isure of the rarity of the sample itself*. 



That "Student" himself has not laid too great emphasis on his test i, I think, 

 Hear, but the emphasis used by others must lead us to be cautious in iu 

 application. 



While " Student's " analysis follows the lines indicated above of the probability 

 of his ratio in the case of a sample drawn from a normal parent population, he 

 uses it in the examples he gives for a somewhat different purpose, where ite 

 application needs some consideration. 



Let u and v be two variates, each of which follows the normal law, then their 

 difference u v will also follow a normal curve with mean u v and standard 

 deviation \/oi a -I- o- 2 2 2p<ri<rt, which latter is the standard deviation of the difference, 

 <r tt _ r , if p be the correlation coefficient of u and v. 



Accordingly if we take samples from these populations with means niu, m v and 

 standard deviations S M , s v and correlation r, then 



m u -m v and _ = Vs M * + s v z - 2rs u s, 



will follow in their frequencies the two curves used by " Student " to obtain his 

 ratio distribution, and if we write 



/ v 



then x' will follow the law of distribution in samples of n given by 



"Student" tacitly takes u = v, or he assumes the mean difference of the popu- 

 lation from which he is sampling to be zero. He is therefore measuring the 

 probability of the ratio x' on the assumption that u and v are taken at random f 

 from the same parent population. If the ratio x' gives a very small chance of 

 occurrence, he assumes that on his hypothesis u and v are not drawn from the same 

 parent population. But with " Student " u and v are not independent samples of 

 necessity as in the test (e') for two samples (see pp. cxxxvii et seq.). 



* A cephalic index among Englishmen of 80-0 is not uncommon, but if we say it has arisen from 

 a skull length of 210 mm. and a skull breadth of 168 mm. we recognise that we are dealing with a very 

 exceptional case on two counts. That is the non-rarity of a ratio is not sufficient to justify us in 

 considering the individual whom it characterises as of common occurrence. 



f Actually, however, this is not so, in for example his Illustration I ; his two populations are linked 

 by a high correlation due to individual reaction to soporifics. If he gets a high in u , he will get a high m r , 

 and if he gets a low m u he will have a low . According to the test, if u=v, the most probable value 

 of / is m u ; but if v and u are correlated, this is not so ; it will be u + pv, (m w - )/ , for the means in 

 samples follow the regression line of the parent population. 



