K. Pkarson 217 



All this liiis nothiiig in il pfculiar to heredity, it is simply an application 

 of the higher theory of statistics. 



If we ask liow closel}' tho above linear fuuction gives the probable Q-value 

 froni the known P-value; the answer is : Ailinirably, for those eharacters which 

 have beeil actually tested. I have 21 tables inv(jlviiig three eharacters for diverse 

 relatimiships in man and eaeli i'nibracing upwards of 1000 pairs. I choose one, 

 the eiibit, perfectly at randimi and it will be seen at once that, within tiie limits 

 of randoni sanipling, uo function conid give the probable character of Q for a given 

 value of F with greater efficiency. 



The line here dealt with is the so-called regression line ; it is a purely 

 Statistical res\iit, and has no relation to any biological theory or hy|)othesis as 

 to heredity. Its deterniinatiou depends on the so-called coefficient of correlation 

 of P and Q, which we will write r^,, and their variabilities a-p and cr,^. In addition 

 we must know the niean, the modal or, what is ofteu sufficient, the median values 

 of P and Q. 



Now suppose that instead of one relative P we have any number P,, Po, P3 ..., 

 and let the correspouding Statistical constauts be 'V,7, »V29' 'V;.'/ ••• °'Pi' °'ih' °^v> — 

 Theu the theory of statistics shows us that, W -p.^ be the most probable deviation of 

 Q from the type of its generation, and hj,^, li^,,^, lip,... be the observed deviations of 

 Pi, P„, P3 ... from the types of their respective generations, then : 



p., = J.^"h,, + J.^'K.. + J.^'^K, (i). 



y'i f' ?'i 



where J^, J.^, J^ ... are known expressions involving only the ?y/s, and can be 

 calculated as soon as the latter have been found from Observation. 



Further the variability of Q about its most probable value, when we know the 

 P's, takes on the average the value a^ x A, where A is another weil-knowu and 

 calculable function of the r^,/s. 



The above is in no sense a biological theory, it is based on no data whatever 

 except the actual statistics ; it is merely a convenient Statistical mcthod of 

 expressing the observed focts. If the facts are there it expresses thein up to 

 a certain point, — the most probable or most frequent value of tlie iudividiial 

 given his relatives. It cannot possibly within this ränge be upset by any 

 bypothesis of heredity, Mendeiian or otherwise, for it is based on no biological 

 assumptions whatever. It is merely a convenient description of Statistical tabula- 

 tions. 



I now proeeed to limitations based on actual experience. 



(i) The correlation coefficients between relatives are positive. 



There is overwhelming evidence in favour of this in insects, animals and plants, 

 for (Hrect relationship back to great-great-grandparents and in a considerable 

 ränge of collateral relationships — fraternal and avuncular. 



Biometrika 11 28 



