136 Influence of " Broad Categories" on Correlation 
Illustration IV. I take now an extreme case, namely the Table: 
A. 
a i 
a- 2 
a 3 
Totals 
h 
4983 
761 
49 
5793 
1166 
1661 
572 
3399 
h 
30 
248 
530 
808 
Totals 
6179 
2670 
1151 
10,000 
I call this an extreme case because the table is only 3 x 3, and the frequencies 
and the subranges are very unequal. 
We have x s /a x = --6172, +'7011, + 1 '687 1 , 
y t /<T y = --6750, +-7100, +1-8531. 
Whence I find S (j^) = '694,222, r mQ% = "8332, 
S(*2f). -712,753, r tC =-8m. 
I then worked out the mean square contingency and found for crude numbers 
^=5146-6. 
Corrected for number of cells <j> 2 = "51426. 
Whence C = --5827. 
Therefore r xy = -5827/08332 x -8442) = -8284. 
The actual table has been constructed in round numbers as the distribution 
of a Gaussian frequency surface for r = '80, the di visions being at x/(r x =-^-'S 
and + 1'2 and y/<r y = + '2 and +1'4, and the correlation is hardly likely to differ 
by a unit from "80. Considering the marked inequality of the subfrequencies 
the corrected contingency approaches closely to the correlation — at least the 
approach is sufficient for any argument likely to be drawn from the data in a 
3 x 3-fold table. 
Illustration V. It is profitable to show the amount of error which will be 
introduced by taking a marked case of non-Gaussian frequency. When looking 
out for such cases many years ago, I found one of the most representative 
instances in the case of barometric heights. I take the following table from 
the memoir by Dr Lee and myself*, and I have arranged the 3 x 3-fold table 
so as to give very unequal frequencies. We have thus in this case non-Gaussian 
frequency and very unequal frequencies. We find 
x s = - 1-1479, --1581, + 9161, 
y t = - 1-2797, --2940, +-8424, 
S ^ = -763,093, r y Cy = -8736. 
* Phil. Trans. Vol. 11)0, A, p. 453. 
