THEORY OF STATISTICS. 
than, less than, or equal to the ratio (8, (Anyi Dae). The 
whole of the contingency table can be analysed into a series of 
elementary groups of four frequencies like the above, each one 
overlapping its neighbours so that an rsfold table contains 
(r—1) (s—1) such “tetrads,” and the associations in them all can 
be very quickly determined by simply tabulating the ratios like 
(428,) (Ani 8.) (4,801) (4dnsi Bor), ete., or perhaps better, 
the proportions (4.,.8,)/{ (AnB,) + (4118), ete., for every pair 
of columns or of rows, as may be most convenient. Taking the 
figures of Table II. as an illustration, and working from the 
rows, the proportions run as follows :— 
For rows 1 and 2. For rows 2 and 3. 
1768/2714 0-651 946/1061 0-892 
807/2194 0-368 1387/1825 0-760 
189/935 0-202 746/1034 0-721 
47/100 0-470 53/69 0-768 
In both cases the first three ratios form descending series, but 
the fourth ratio is greater than the second. The signs of the 
associations in the six tetrads are accordingly— 
The negative sign in the two tetrads on the right is striking, 
the more so as other tables for hair- and eye-colour, arranged in 
the same way, exhibit just the same characteristic. But the 
peculiarity will be removed at once if the fourth column be placed 
immediately after the first : if this be done, 7.e. if “red ” be placed 
between “fair” and “brown ” instead of at the end of the colour- 
series, the sign of the association in all the elementary tetrads 
will be the same. The colours will then run fair, red, brown, 
black, and this would seem to be the more natural order, consider- 
ing the depth of the pigmentation. 
11. A distribution of frequency of such a kind that the 
association in every elementary tetrad is of the same sign 
possesses several useful and interesting properties, as shown in 
the following theorems. It will be termed an isotropic dis- 
tribution. 
(1) In an isotropic distribution the sign of the association is 
the same not only for every elementary tetrad of adjacent Jrequen- 
cies, but for every set of four frequencies in the compartments 
common to two rows and two columns, e.g. (4,B,), (dn.,B,) 
(4dnBrio) (drei pBrig): 
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