Empirical Proof of the Law of Error. 339 



In this table the first three columns are the same as in the 

 last table ; the fourth column contains the sum of the expe- 

 rienced observations from zero up to the point indicated by 

 the corresponding entry in the second column. For instance, 

 182 in the fourth column means that between zero and the 

 point '362 Modulus of Probability-curve, that is, *2 of a 

 second, there actually occurred 182 observations. The next 

 three columns give the computed values of the number of 

 observations up to the critical point of each rival curve. The 

 Third Exponential, on account of the difficulty which its inte- 

 gration presents, has been omitted. The eighth column 

 contains the computed number of observations up to certain 

 points for the Probability-curve. The ninth column contains 

 the ecart or difference between the computed and experienced 

 numbers for the Probability-curve. The tenth column gives 

 the Modulus which measures the probability of a certain ecart 

 occurring. It is deduced from the formula \/2p(l— _p)N, 

 where p is the proportion of the observations which, according 

 to theory, should occur within the point {e. g. for the point 

 •724, ^ = '69); N is 470. The last column gives the odds (to 

 one) in favour of the Probability-curve ; the odds that, if the 

 Probability-curve held good, the ecart would be at least as 

 large as it is. The last four rows give the corresponding 

 results for the Rival curves, their ecart, Modulus of ecart, 

 Patio of ecart to Modulus, and thence deduced odds (to one) 

 against the Rival Curves. 



It is noticeable that the Right Line comes out unscathed 

 from this ordeal. The results would doubtless have been 

 more striking had the number of observations been greater. 

 This is the case in the next example (table, p. 341), which 

 consists of the height-measurements of 683068 Italian recruits 

 tabulated by Signor Perozzo*. The Median and Arithmetic 

 mean being coincident at the point or compartment 1 '62 metres, 

 I have thought it legitimate to suppose the curve folded about 

 the central ordinate; so that, measuring from that centre as 

 zero, we shall now have between and "005 (metre) the num- 

 ber of observations which Signor Perozzo puts at 1*62 — that 

 is, between 1*61.5 and 1*625. Between '005 and '015 of the 

 new arrangement we shall have the sum of the observations 

 registered for 1*61 and 1*63 ; and so on. 



By actual counting I_find for the Mean Error *053. 

 Whence the Modulus = s/ it x Mean Error = '094. By means 

 of this datum the second column is graduated and the critical 

 points are selected. After the explanation of the preceding 

 table, further comment is unnecessary. It is noticeable that 

 * Annales de Demographies 1878 ; Annali di Statistica, 1878, vol. ii. 



