276 On 'Best' Values of Co7istants in Frequency Distributions 
The intersection point of the two regression Hnes is m = 124-0453, 
m' = 12-7070, which is seen to be very near to the general means. Introducing 
that point into the equations of the hnes, they take the form 
»< = 12-7070 + -1342345 {y^ - 124-0453), 
= 124-0453 + -682455 {x^ - 12-7070). 
From the slopes of the lines we find the value -3027 for the correlation coefficient, 
whereas the method of least squares gives the value -2941. 
Although we have found the material to be decidedly heteroscedastic and the 
weighting of the two series of means rather dif?erent from that of the marginal 
frequencies, we nevertheless see that the resulting regression lines differ very little 
from the ordinary regression hnes, both the deviations of the means and the 
correlation coefficient derived from them being less than their probable errors. 
(8) The conclusions to be drawn from the present investigation are: 
(i) The definition of ' best,' which leads to the method of moments being con- 
sidered 'best' and incidentally to the method of least squares being considered 
'best,' is undoubtedly somewhat arbitrary. If we use Pearson's 'Goodness of Fit' 
test, then the method of moments is not necessarily the 'best,' the best value of 
the constant termed the mean is not necessarily the mean, nor generally the best 
value of the correlation coefficient between two variates that calculated by the 
moments and product moment method. 
(ii) On the other hand the present numerical illustrations appear to indicate 
that but little practical advantage is gained by a great deal of additional labour, 
the values of P are only shghtly raised — probably always within their range of 
probable error. In other words the investigation justifies the method of moments 
as giving excellent values of the constants with nearly the maximum value of P or 
it justifies the use of the method of moments, if the definition of 'best' by which 
that method is reached must at least be considered somewhat arbitrary. 
The present paper was worked out in the Biometric Laboratory and I have 
to thank Professor Pearson for his aid throughout the work. 
