Empirical Proof of the Law of Error. 333 



of errors, abstraction faite du signe*. Only we should take ac- 

 count that the mean from which we measure our errors above 

 and below is itself liable to error. When we take the Greatest 

 Ordinate as the mean, this sort of correction is not possible. 



It appears that the mean error above and that below the 

 Arithmetic Mean can only be different when the Arithmetic 

 Mean and the Median are different. Accordingly an equally 

 good test of symmetry may be afforded by comparing the 

 Arithmetic Mean and the Median and estimating the proba- 

 bility of the observed difference occurring if the curve were 

 really symmetrical. Take, for example, the heights of 25,878 

 American recruits recorded in the Report of the International 

 Statistical Congress, vol. ii. p. 748. The Arithmetic mean is 

 there given as 68'2. For the Median I find 68*15. For the 

 Modulus-squared, which tests the difference to be expected 

 between these Means, we have (by adding the squares of the 



2Se 



n 



Modulus applicable to each Mean separately) — %- + sp2 1 5 



where the e's are errors measured from the Arithmetic Mean, 

 P is the greatest ordinate (the number of observations per 

 unit of abscissa at the densest part of the given group). Here 

 2Se 2 , as ascertained by actual counting (as well as by induc- 

 tion from other anthropometrical statistics J) is 13 nearlv. 

 n= 25,878. A nd P is 4054. Whence for the sought Mo- 

 dulus w T e have \/ '0005 + *0008 = *04 nearly. And the observed 

 difference (between the Median and Arithmetic Mean) is*05: 

 that is, very slightly greater than the corresponding Modulus ; 

 so that no great significance attaches to the observed indica- 

 tion of asymmetry. 



As a second example, take the anthropometrical statistics 

 cited by Mr. Merriman in his ' Method of Least Squares ' 

 (1885), art. 136. There the number of observations is 18,780. 

 The Arithmetic Mean, as calculated by Mr. Merriman, is 



67*24. The Median, as estimated by me, is 67*28. , as 



deduced from Mr. Merriman's results, is 12*25. And P is 

 3000 nearly. Hence, for the Modulus appertaining to the 

 difference between the Median and Arithmetic Mean, we have 

 •04 ; the same as the observed difference. Accordingly the 

 indication of asymmetry is insignificant. 



* Theor. Anakjtique, Book ii. art. 19 ; adapted by Todhunter, ' History 

 of Probability,' art. 1006. See below, p. 342. . 



t See the formula quoted from Laplace in my paper " On Problems in 

 Probabilities " in the Philosophical Magazine, October 1886. 



X See " Methods of Statistics," Journal of the Statistical Society, 

 Jubilee volume (1885), p. 195. 



