ON THE 'BEST' VALUES OF THE CONSTANTS 
IN FKEQUENCY DISTKIBUTIONS. 
By KIRSTINE SMITH. 
(1) If we attempt to fit the normal or Gaussian curve to a system of observa- 
tions, we almost invariably determine the constants x and a of the equation 
by the ynethod of moments. This method of moments has been extended by Thiele, 
Pearson, Lipps and others to obtain the constants involved in various skew 
frequency curves and series. It is an undoubtedly utile and accurate method; 
but the question of whether it gives the ' best ' values of the constants has not been 
very fully studied. It is perfectly true that if we deal with individual observations 
then the method of moments gives, with a somewhat arbitrary definition of what 
is to be a maximum, the 'best' values for a and x in the above equation to the 
Gaussian. Pearson* has shown that the method of moments agrees with the 
method of least squares in the case where the distribution is given by a high 
order parabola, and accordingly the method of moments is hkely to give a very 
good result, when an expansion by Maclaurin's Theorem would closely give a 
frequency function. But the method of least squares itself can now-a-days hardly 
be spoken of as more than a utile and accurate method of fit, indeed its utihty, 
owing to the cumbersome nature of the equations which frequently arise, is often 
far less than that of the method of moments. 
Gauss' original proof that the probability of the observed individual results 
was a maximum when x and a have been determined by moments has led to the 
extension of the conception that for grouped data, and for other results than the 
Gaussian curve, the 'best' values of the constants must be given by the lowest 
possible moments. This is of course not true For example, if we had as fre- 
quency curve 
1 (x - xY 
and used individual observations, then the Gaussian 'best' value for x would be 
that found by determining the point for which the third moment coefficient 
* Biometrika, Vol. I. pp. 267-70. 
