434 
Association of Finger- Prints 
8 a. Method of Calculating the Coefficients of Contingency in Restricted Tables. 
It will be noticed that the Tables I to XX and XLVI to LV, given in the 
Appendix, differ in general character from most correlation tables since the whole 
of the cells in the lower right-hand portion are necessarily empty. Consequentl}' 
the usual method of finding the independent probability numbers for the purpose 
of calculating contingencies is not applicable to those Tables. The method which 
has been employed was suggested to me by Professor Pearson. It is as follows: 
Consider Table VI, Appendix, p. 454, which gives the distribution of small 
loops and whorls for the right hand. Commencing with the 45 hands which con- 
contain 5 small loops each, it will be seen that the independent probability 
number is the same as the observed, since a hand which has five prints of one 
class can have no other. In the next column the distribution of the 211 prints by 
independent probability is not in the ratio of 861 to 497 since 45 of the 861 have 
already been disposed of, but in the ratio of 816 to 497, that is, the numbers in 
the two rows are 1314 and 79'9. Again in the third column from the right con- 
taining 3 small loops, 45 and ISTl of the first row are accounted for and 79"9 of 
the second row ; hence the independent distribution of the 306 in the third 
column is in the proportion of 684*9 : 417"1 : 292; that is, the numbers are 
150"3, 91'6, and 64'1 ; and so on. 
It should be noted that the same independent probability numbers are obtained 
if we commence at the bottom of the first column with the 50 hands each con- 
taining 5 whorls and work horizontally instead of vertically. 
The differences between these independent probability numbers and the 
observed numbers are then used to find the contingency in the same way as in 
the ordinary contingency Table. 
No correction for the number of cells has been applied to the contingency 
coefficients in this type of Table as we have, at present, no appreciation of what it 
should be. 
The complete contingency Table, worked as described, is given below. 
Note on Calculation of Contingency Coefficients. It should be borne in mind 
that in finding the independent probability numbers in all contingency tables as 
well as in calculating the standard deviations, it is assumed that the distribution 
of the marginal totals is in the same ratio as would be the case if the whole 
population were taken ; in other words, that if ng is the total of an array when a 
sample N is taken and nig the total of the corresponding array when the whole 
population M is taken, then it is assumed that 
N 
, . n^ — vig . 
Evidently the correct value of the independent probability number in the (s, s') 
cell of an ordinary contingency table would be 
N , N 
