Karl Pearson and David Heron 
235 
it is obvious that Mr Yule's mean character will always coincide with his median, 
or his 
M = mean = character of individual with minimum value of variate +^R. 
a- = (standard deviation)- = ~ ^1 + 
or, if n be at all large, 
o- = iZ/(2 V3). 
It will be clear that for large numbers 
maximum value — minimum value of variate 
° = Mi ■ 
a relation entirely opposed to the practical independence of range and standard 
deviation in variates with which we are familiar*. But another difficulty at once 
arises; in actual practice the subranges have all sorts of different values, and we 
may know one or more of them. Which one of these subranges is to be taken as 
the standard unit and have the range expressed in terms of it ? Some numerical 
illustrations will emphasise the extraordinary difficulties, not to say contra- 
dictions, of Mr Yule's process of treating subranges as equal units in determining 
correlation. 
For example, the Registrar-General gives ages of Husband and Wife from 15 
to about 100. Hence by Mr Yule's method : 
Mean Age of Husband = 5 7 '5 years, 
Mean Age of Wife = 5 7 '5 years, 
Standard Deviation, Husband = 18 - 7639 years, 
Standard Deviation, Wife = 187639 years. 
The actual values are : 
Mean Age of Husband = 42-8306 years, 
Mean Age of Wife = 40-5838 years, 
Standard Deviation, Husband = 13'0649 years, 
Standard Deviation, Wife = 12 - 6813 years. 
Thus the Yulean values may differ by 30 °/ 0 to 50 °/ 0 from the true values. 
To assume that the skew distribution is Gaussian will give much better results 
than this. Let us illustrate it on the very skew Husband and Wife, Barometer, 
and Ivy Leaf data. In dealing with actual data, we have to express the means of 
a variety of arrays in terms of a known subrange common to them all, e.g., bay 
colour in horses or hazel eyes in men. We will apply in succession Mr Yule's 
hypothesis and the normal curve to the above data. 
* For such frequencies as occur in practice it is much safer to take the range about 6 times rather 
than 3 - 5 times the standard deviation. 
