Probable in a Correlated System of Variables. 165 



Now let % s be the value found for the sample, and x the value 

 required marking the system of deviations of the observed 

 quantities from a group-system of the same number accu- 

 rately representing the general population. 



Then : 



2 = g { (» l '— m ) 2 \ _ g f nJ—m a —ft) 9 1 

 X m / ( m s + fi J 



if we neglect terms of the order (/*/>»«) 3 . 

 Hence : 



Now ^ s must, I take it, be less than %, for otherwise the 

 general population distribution or curve would give a better 

 fit than the distribution or curve actually fitted to the 

 sample. But we are supposed to fit a distribution or curve 

 to the sample so as to get the " best " values of the constants. 

 Hence the right-hand side of the above equation must be 

 positive. If the first term be negative then it must be less 

 than the second, or the difference of % and % s is of the 

 order, not of the first but of the second power of quantities 

 depending on the probable errors of the sample. On the 

 other hand, if the first term be positive, it means that there 



sative correlation between — and — -' or that when 



isne 



the observed frequency exceeds the theoretical distribution 

 given by the sample (m'>m,), then the general population 

 would fall below the theoretical distribution given by the 

 sample (m<w?«), and vice versa. In other words the general 

 population and the observed population would always tend to 

 fall on opposite sides of the sample theoretical distribution. 

 Now this seems impossible ; we should rather expect, when 

 the observations exceeded the sample theoretical distribution, 

 that the general population would have also excess, and vice 

 versa. Accordingly, we should either expect the first term 

 to be negative, or to be very small (or zero) if positive. In 

 either case I think we may conclude that % only differs from 

 X' by terms of the order of the squares of the probable errors 

 of the constants of the sample distribution. Now our argu- 



