: THEORY OF STATISTICS. 
The theorem last given for evaluating P for an aggregate of 
tables is illustrated by the experimental data of Tables C and D. 
The values of x2 for the 350 fourfold tables of Table B were 
added together in pairs, giving 175 pairs. According to theory 
the resulting frequency-distribution for the totals of pairs of x's 
should be given by differencing the column of the P-table for 
n’=3. The results of theory and observation are compared in 
the first pair of columns of Table C. Testing goodness of fit, 
grouping the values of x2 7 and upwards, 2" is 8, x? is 5-53, and 
P is 0:60. 
Grouping the values of x? for the 350 experimental tables 
similarly in sets of three and summing, we get the observed 
distribution on the right of Table C, and the theoretical distribu- 
tion by differencing the column of the P-table for n»’=4. 
Grouping values of x? 8 and upwards, and testing goodness of fit 
between theory and observation, »’ is 9, x2 is 2°18, and P 0-97. 
TABLE C.— Theoretical Distribution of Totals of x? (calculated from Independ- 
ence-values) for Pairs and for Sets of Three Tables with 2 Rows and 2 
Columns, compared with the Actual Distributions given by Experimental 
Tables. mn’ must be taken as 3 in the first case, and 4 in the second. 
Pairs of Tables. Sets of 3 Tables. 
Sum of 
x°’s. 
Expectation. Observation. Expectation. Observation. 
0-1 68-9 67 23°1 21 
1-2 41-8 46 26°5 26 
2-3 25°3 22 21°0 | 22 
3-4 15-4 19 15-1 19 
4-5 93 10-4 
5-6 56 7°0 
6-7 34 46 
7-8 2:1 : 30 ' 
8— 3:2 : 5-3 
Total 2 9 q 
Table D makes a similar comparison for the values of x? 
calculated from independence, for 100 pairs of 4 x4 tables. 
Here there are 9 algebraically independent &’s for each table of 
the pair, and consequently n’ must be taken as 19. Differencing 
the P-table for n’ =19, the expected distribution is obtained, which 
is shown in the first column of Table D, the observed distribution 
386 
175-0 175 116°¢ 11%