Karl Pearson and Egon S. Pearson 
153 
the slice between A", and X„ has for its volume on dY 
J x, 
the slice is therefore given by the normal curve : 
Ordinate = const, x e" '^^ • 
It seems therefore not unreasonable after the surface of revolution is stretched 
and slid into a correlation surface to assume the slice to be still approximately a 
normal curve. Unfortunately the determination of the best mean and standard 
deviation for normal material given in broad categories does not admit of very easy 
solution. What we need is the difference between the means of a columnar array 
and of a marginal frequency as a multij^le of the standard deviation of the latter. 
We shall obtain results dififeiing more or less from each other according to the 
individual broad category we take as the basis of comparison between as the 
standard deviation of the sth slice and cr„ the standard deviation of the marginal 
frequency. In fact the range of any broad category or of any combination of broad 
categories, except the tail categories, can be made a means of linking up o-., and cr,,. 
A little experience, however, shows (a) that it is undesirable to find the ag of 
any array from a category of small frequency, and (b) that for arrays of small total 
frequency symmetrical tripartite divisions as far as feasible are the best*. The last 
column in Table XVI shows the system selected for each of our columnar arrays. 
Take, for example, s = 5, the columnar array may be taken on the base of 3' 
and 4' categories as 
l' + 2' 9j (-335 
3' + 4' 361 and compared with J 421 
5' + 6' + 7' 24 244 
Totals 69 1000 
as the corresponding marginal distribution. The corresponding proportional 
1304 '3350 
frequencies up to the dichotomic planes are: .q^^)-^ 'T^nd .i^-'^qq • The distances of the 
mcanf from the two dichotomic planes in the first case are 
- 1-1245(7, and +-39 10 <r,„ 
and in the second case 
- -4261(7,, and +-6935(7,^, 
where ctj is the standard deviation of the normal curve assumed to represent the 
columnar array 5. Accordingly the range of 3' + 4' categories 
= 1-5155(7, = l-1196(7y, 
which gives (7, in terms of a,j. 
The probable error of a standard deviatiou found in this way is discussed in Biometrika, Vol. xiii. 
p. 129. 
t Found from the Probability Integral Table. 
