Karl Pearson 
121 
Thus we see how rapidly, if h be small with regard to <r x , the value of the 
quantity 
and therefore of r x( j , its square root, approaches unity. 
(6) It is desirable to illustrate the approach of r xC to unity as we increase 
the number of groups. I have therefore worked out this approach for three and 
more symmetrical groups when (a) we use the approximate formula (xv), (b) we 
still suppose the classes to have their ranges equal but the frequencies to be given 
by a normal curve, (c) we assume the frequencies in the classes and not the class 
ranges to be equal, the frequency being supposed normal, (d) we suppose the 
frequencies to increase by 50°/ 0 at each stage, i.e. to be as 1, 3/2, 9/4, 27/8, etc. 
These cases will be sufficient to indicate what sort of frequencies we should 
take for few classes in order to get the highest correlation between variate and 
its class-mark. I shall after these theoretical investigations consider a few actual 
cases of " broad " categories. 
Values of r xC for Various Groupings*. 
No. of 
Classes 
(a) 
Equal Ranges, Any 
Frequency. 
Formula (xv) 
(6) 
Equal Ranges, 
Normal Frequency 
(c) 
Equal 
Subfrequencies, 
Normal Frequency 
(d) 
Increasing 
Subfrequencies, 
Normal Frequency 
3 
•817(/i = 2o-) 
■859 
•891 
•876 
4 
•901 (A = l-5o-) 
•915 
•928 
•912 
5 
•938 <7i = l-2o-) 
•943 
•947 
•928 
6 
•957 {h = (r) 
•960 
•959 
•939 
8 
•978 (A =-75o-) 
•977 
•972 
10 
•985 (A = -6<t) 
•985 
•979 
12 
•990 (h = -bo-) 
•989 
•984 
u 
•992 (A = fo-) 
•992 
■987 
16 
•994 (A = fo-) 
•994 
•990 
20 
•996 r/i = : 3<7) 
■996 
•992 
It is clear from the fourth column that nothing whatever is gained by 
exaggerating the frequencies of the extreme or tail sections. Further : 
(a) Equal frequencies are better than equal ranges up to about six classes. 
After six classes it is better to take equal ranges. 
(b) After six classes the approximate formula (xv) is amply sufficient in the 
case of equal ranges to obtain the value of r x g . 
Thus we have the general rule that up to six broad categories it is desirable to 
make the frequencies of those classes approximately equal, but beyond this greater 
* The number of classes given in (a) and (fc) is based upon the range of 6<r covering for such data as 
we usually deal with, i.e. 500 to 1000 cases, the total frequency. 
Biometrika ix 16 
