156 ESSAY ON PROBABILITIES. 



duced. But even in such a case we have no right to 

 say that the average is preferred to the result of the 

 method of least squares ; for the former is then a par- 

 ticular case of the latter. Let three observations give 

 9, 11, and 16, the average of which is 12. The 

 errors, taking this average as the truth, are 3, I and 4, 

 the sum of the squares of which is 26. This is the least 

 possible sum of the squares. To try this, assume 11 as 

 the most probable truth : the errors are then 2, 0., and 

 5, the sum of the squares of which is 29 : assume 13, 

 and the errors are 4t, 2, and 3, the sum of the squares 

 of which is 29- To avoid introducing fractions, I 

 have only assumed whole numbers, but if I take 12*1 

 or 11 '9, I find the sum of the squares of the errors to 

 be in the first case 26-03, and in the second 26-03. 

 So that when one result only is in question, a direction 

 to take the average is equivalent to a direction to 

 make the sum of the squares of the errors the least 

 possible. 



Let us now suppose two results of observation, say 

 that we wish to know the fraction which A is of B, 

 where both A and B are subject to errors, the positive 

 and negative being equally likely. Suppose, for ex- 

 ample, that we ask for the proportion of the population 

 of a country which is buried in a year. Statistical 

 returns will furnish the population of each year, and 

 the burials, both subject to errors. There are now 

 obviously two ways of taking an average ; I may 

 either divide the average burials by the average popu- 

 lation, or find the proportion which the first is of the 

 second in each year, and take the average of the pro- 

 portions. One not versed in mathematics would sup- 

 pose that these must give the same results, but any 

 simple instance will show the contrary. The average 

 dividend and the average divisor do not give the average 

 quotient. For instance, let dividends be 12, 13, and 

 17, and divisors 20, 22, and 30. The fractions -J, 

 -LJ, and -J,are'6, -591, and '567, the average of which 

 is '586. The average dividend is 14, the average 



