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THE AMERICAN NATURALIST [Vol. XLV 



rant, but in the left lower quadrant, at the point marked X. 

 Here, as examination will show, it would receive the same sub- 

 script that it has now, and would count as negative, exactly as 

 it now does. Again, suppose the pair 36 by 31 were similarly 

 transposed; it would still fall in the left upper quadrant, at the 

 point marked Y, where it would receive the same subscript as at 

 present and count as positive, just as at present. And so of all 

 other cases; the value of a pair is not altered in any way by 

 changes in the way it is entered in the table. In making the 

 table, therefore, the pairs may be entered only once and quite at 

 random, or in any way that is convenient. 



2. With regard to the mean and standard deviation, the ap- 

 parent advantage of symmetrical tables is that they give us the 

 actual mean of all the individuals; it is to this mean that our 

 correlation must refer. But this actual mean can readily be 

 obtained from the tables in which each pair is entered but once, 

 in. any way that happens to be convenient. It is merely neces- 

 sary to add together the sums of the rows and of the columns 

 of the table. Thus in Tab 1 .' TT the number of individuals having 

 the length 35 is not 17 (sum from the row beginning. with 35), 

 nor 6 (sum from the column headed 35), but 23 (sum from both 

 the row and the column) and so for all other classes. It will be 

 well to illustrate by an example certain of the steps in the com- 

 putation. Table II shows a correlation table of single entry, as 

 prepared for computation of the coefficients of correlation and 



After finding the sums of the rows (given in column A at the 

 right) and of the columns (given in B, underneath), we place 

 the latter sums (B) by the side of A, in the proper places (as 

 at B'), then add the two sets, giving the sums shown in the 

 column C at the right. These are the same sums that we should 

 get from a symmetrical table; adding these we get the total 

 number of individuals (250 in Table II). Now from this 

 column C we find the approximate mean in the usual way; it 

 lies in this case at the length 37 (with 38 individuals). Through 



find the correlation in the usual way. In so doing (1) we make 

 use always of the sums in the column C in finding mean, stan- 

 dard deviation, etc. (2) We use for both horizontal and ver- 

 tical axes of reference in computing the correlation in all cases 



