400 THE GRAMMAR OF SCIENCE 



We have now the outHne of a numerical method of 

 appreciating correlation. Its stages are briefly the 

 following : — 



(a) Measure or count some character in pairs of organs 

 of the same individual or different individuals, for 500 to 

 1000 cases. 



(d) Prepare a correlation table, by selecting successively 

 organs of a given size in one member of the pair and 

 forming the array of the organs in the other member of 

 such pairs. 



(c) Add up the columns and rows of this table, so as 

 to form a column and row of totals as on pp. 394 and 

 399 ; the means and standard deviations of the column 

 totals and the row totals give the types and variabilities 

 of the two organs in the whole population. Let these be 

 M^, M,, and a^, cr,. 



(d) Find the means of each separate column array ; 

 then the best fitting straight line drawn through these 

 means when plotted (as in the diagram, p. 395) is the 

 line of regression for the second organ on the first, 

 and its slope = raja^. Similarly the line of regres- 

 sion for the first organ on the second may be found 

 from the plotted means of the rows, and its slope is 



Mathematical theory tells us that these two lines 

 intersect in the point which corresponds on the diagram 

 to the population means of the two organs, and further 

 that r can be found in the following manner. Let 

 M, +-r be the value of the first organ in one member of 

 the pair, and M., +j' the value of the second organ in the 

 other member of the pair ; thus x and y are the deviations 

 from their means of the two organs of an associated pair ; 

 then take the average value for all associated pairs of the 

 product xy.y and divide this by the product of the 

 standard deviations a^Y-a,,; the result is r, the correlation 

 coefficient. 



{e) An examination of the diagram (p. 395) shows us 

 that if we have a first organ of magnitude ni^ = M^ + x, 

 then the most probable value of the second organ {i.e. the 



