Discordant Observations, 367 



In this table the first row is obtained by taking at random 

 ten digits from a page of Statistics, counting for ten. The 

 second row consists of the squares of these numbers. The 

 third row was thus formed from the second : — I took 25 

 random digits, and divided their sum by 25 ; then multiplied 

 this mean by 1 0. I similarly proceeded with 49 (fresh) digits, 

 and so on. It will be noticed how the defective precision of 

 the fourth and seventh observations makes itself felt. It was, 

 however, a chance that they both erred as far as they could, 

 and in the same direction. 



In the light of these distinctions I propose now r to examine 

 the different methods of treating discordant observations. For 

 this purpose the methods may be arranged in the following 

 groups : — 



I. The first sort of method is based upon the principle that 

 the calculus of probabilities supplies no criterion for the cor- 

 rection of discordance. All that we can do is to reject certain 

 huge errors by common sense or simple induction as distin- 

 guished from the calculation of a posteriori probability. 



II. Or, secondly, we may reject observations upon the 

 ground that they are proved by the Calculus of Probability 

 to belong to a much worse category than the observations 

 retained. 



III. Or, thirdly, we may retain all the observations, affecting 

 them respectively with weights which are determined by 

 a posteriori probability. 



IV. In a separate category may be placed a method which, 

 as compared with* the simple Arithmetical Mean, reduces the 

 effect (upon the Mean) of discordant observations — the method 

 which consists in taking the Median f or " Centralwerth" % of 

 the observations. 



I propose now to test these methods by applying them in 

 turn to all the hypotheses above specified. 



I. (a) The first method — which is none other than Airy's, 

 as I understand his contribution § to this controversy — is 

 adapted to the first hypothesis. Upon the second hypothesis 

 (/3) the first method is liable to error, which, as will be shown 

 under the next heading, is avoidable. (7) Upon the third 

 hypothesis the first method is not theoretically the most 

 precise ; but it may be practically very good. 



II. Under the second class I am acquainted with three 



* This is pointed out by Mr. Wilson in the Monthly Notices of the 

 Astronomical Society, vol. xxxviii., and by Mr. Galton, Fechner, and 

 others. 



t Cournot, Galton, &c. 



% Fechner, in Abhandl. Sax. Ges. vol. [xvi.]. 



§ Gould's Astronomical Journal, vol. iv. pp. 145-147. 



