the Choice of Means. 271 



being taken as zero. Now the probable error incident to the 

 Arithmetic Mean is found by Prof. Chauvenet to be *022. 

 There is then no reason for much preferring to 0*01. 



As a second example of the safety of the method, let us 

 take the observations cited by Sir G. Airy at the end of (the 

 later editions of) his c Theory of Errors/ The original obser- 

 vations, which have been submitted to me by the kindness of 

 Mr. Turner, of the Royal Observatory, Greenwich, are in 

 number 636. Arranging them in the order of magnitude and 

 counting from the lowest, we shall find that the 3 18th obser- 

 vation is — 0"03 (where zero is the Arithmetic Mean). The 

 319th observation has the same value. And accordingly that 

 is the value of the Median. Now, according to Sir G. Airy's 

 results, the probable error of the Arithmetic Mean is about 



*57 



— ; : that is, about *022. The value —'03, therefore, is 



a/642 ' ' ' 



not a bad one upon the supposition that the given observa- 

 tions are perfectly normal, that they range under a single 

 probability-curve. 



Upon the supposition, which there is reason for entertain- 

 ing, that the given observations are somewhat "discordant," 

 the value — *03 is just as eligible as zero. It may be remarked 

 that, by Laplace's formula, the probable-error for the Median 

 is about *03. So that upon neither supposition is there much 

 to choose between the two results. 



As an example of the general case, where the method is not 

 only safe but useful, let us take the 684 observations (of transits 

 of Mercury)* discussed by Prof. Newcomb in the i American 

 Journal of Mathematics, 1885. The Arithmetic Mean being 

 taken as zero, I find for the Median —0*45. The error of 

 this result may be found by the formula above given, if we 

 put for the greatest ordinate 30 ; that being the number of 

 observations per unit of abscissa in the neighbourhood of the 

 centre. I find for the Modulus 0'6, and for the Mean error 

 (in the sense of square root of half-modulus) about *55. On 

 the other hand, Prof. Newcomb's method (if I have rightly 

 worked the laborious arithmetic which it involves) gives 

 for the correction of the Arithmetic Mean — * 6, subject to a 

 Mean Error of about *5. In view of so slight a difference 

 between the results and so great a liability to error in the 

 calculation, is there much reason for preferring the laborious 

 to the rough and ready method ? In reducing observations 

 it is possible to go further and not fare any better. 



* They are given in full in the ' Astronomical Papers of the American 

 Ephemeris,' vol. i. 



