On the Choice of Means. 269 



of each cluster, together with its given weight (or inverse 

 mean-square-of-error) . We know by the Law of Error that 

 if f were the real point, then the probability that the Arith- 

 metic Mean of any cluster of errors ranging under any (one 

 and the same) facility- curve would occur at the point x 



1 -a-x)* 

 (between x and x + Ax) is Axx -7=- e c2 , where c 2 = twice 



Virc 

 the mean-square-of-error of the facility- curve under which 

 each member of the cluster ranges. Hence, looking only at 

 the Arithmetic Mean and weight of the given cluster, x 1} os 2 , 

 &c, we may say that the a posteriori probability of its having 



I -(g-aQ2 



resulted from the real point P is proportioned to .- e & • 



\Zirc 



where x is the Arithmetic Mean, and c the weight of the 

 cluster. Thus the Probability-Curve, with its peculiar faci- 

 lities for calculation, again makes its appearance. We can 

 apply the rules primarily appropriate to observations obeying 

 the typical Law of Error , to find both the most probable and 

 the most advantageous Mean. And thus we find that the Arith- 

 metic Mean is the best solution for a single cluster (and the 

 Weighted Arithmetic Mean for several clusters) . The Method 

 of Least Squares is seen to be our best course when we have 

 thrown overboard a certain portion of our data — a sort of 

 sacrifice which has often to be made by those who sail upon 

 the stormy seas of Probability. 



But this explanation must be received with some qualifica- 

 tions. The analogy of insurance shows that when we do not 

 utilise all our information, it may be a delicate question what 

 part of it it is best to utilise. Take the case put by Dr. Venn"*, 

 of an insurance company which is dealing with a consumptive 

 Englishman in Madeira. The ideally best plan would be to 

 use the statistics for consumptive Englishmen resident in 

 Madeira. But failing that, it may be a nice question whether 

 it is better to refer the customer to the category of con- 

 sumptive persons, or of Englishmen, resident in Madeira. 

 Similarly in our case it might be better, instead of directing 

 our exclusive attention to the Arithmetic Mean and the cor- 

 related characteristic of the inverse mean-square-of-error, to 

 prescind in the same sense some other Mean, and in particular 

 the Median, with its correlate the Greatest Ordinatef. The 

 Median of any cluster fluctuates according to a Probability - 



* ' Logic of Chance/ ch. viii. 



t Laplace, Theorie Analytique, Supplement 2, sect. 2. See an illustra- 

 tion in my paper on Problems in Probabilities, Phil. Mag. Oct. 1886. 



