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TABLES XVIII XX. 



Criteria for Rejection <>j 'ihitf.yiny Observations. (Probability Inteyr.il < 'Itauvenet's 

 1 1 ml frunn's Methods.) 



These provide ooovenientmethodBof discovering whether outlying i-ms 



may reasonably In- ivjrrl.nl. Most of the criteria fur rejection assume the di.s- 

 tribution of the measured quantitiea to be approximately normal. This assumption 

 is reasonable, however, for many series of observational errors and also of anthrojjo- 

 metric nieasmriiiriifs. There are three methods of investigating rejection in current 

 use, the applicabilities of which depend to some extent on the number of op- 

 tions we have to deal with. In the case of short series the .standard deviation of the 

 series may differ very considerably from that of the parent population and attention 

 must be paid to this. For all but the shortest series, say under 15 cases, it will be 

 ample to consider what the changes in the standard deviation would bo with '1 -~> 

 times the probable error found from the approximate formula 6744U<r/v / 2/i added 

 and subtracted from it. 



The methods we have to consider are : 



(a) The mean and standard deviation having been determined from the sample, 

 we find from Table II of Part I the probability of an individual with as great or 

 greater deviation occurring. For this purpose it is desirable to compute the mean 

 and standard deviation with and without the outlier, or outliers. 



($) The probability of an error (or observation) greater than k<r 



= _L [' 



V27TO-J; 



and accordingly of a deviation greater on either side the mean 



2 [ 



= ~T^- e dx = 1 - a k , 

 v27rJ A 



in our usual notation. If in a sample of size n this be less than a half, then the 

 individual with a deviation exceeding kcr is to be rejected. 



This gives us 



n(I-ad<t, 



2w-l 

 a k > = . 



2n 



This is Chauvenet's criterion*. We determine k from the above relation, and njtct 

 observations lying outside ka. If we have rejected an observation by this criterion, 

 we recalculate <r, reduce n by unity and proceed to find another k and consider 

 whether another observation is to be rejected. It will be seen that the choice of 

 half an individual is somewhat arbitrary and Chauvenet's criterion appears from 

 considerable experience to reject too readily. 



(7) The distribution of the differences between the first and second and 

 the second and third observations in samples from a normal frequency has been 



* W. Chauvenet, .-1 Manual of Practical and Splierical Astronomy, Fourth Edition, Vol. n. p. 565. 



