144 ESSAY ON PROBABILITIES. 



the common average is the most probable truth, and its 

 weight is as many times the weight of one observation as 

 there are observations in all : or, the average of n 

 observations each of which has the weight w, is entitled 

 to the same confidence as one observation made under 

 circumstances which give it a weight nw. Of this we 

 have an example in p. 137-:, and I now give another, 

 detached from the method there given of finding the 

 weight. 



EXAMPLE. The weight of each observation being 

 18, what is the probability that the average of 50 

 observations lies within '05 of the truth. The weight 

 of this average is 18 X 50 or 900, the square root 

 of which is 30. Now -05 X 30 is 1*5, which, being t, 

 H is '966, so that the probability required is about 23 to 

 1 in favour of the event. The mean risk of error of such 

 an average is -fg-g- divided by 30, or ^-ff-^ less than 

 01 ; by which it is meant that if repeated sets of 50 

 observations each were made, the errors of these sets, 

 neglecting their signs, would not average so much as 

 01 x 2 or -02 each. 



Let us now suppose that positive and negative errors 

 are not equally likely. Hitherto, the absolute truth 

 has been the most likely ; that is, though the pro- 

 bability of any one observation giving mathematical 

 truth was infinitely small, yet so was that of any given 

 error being exactly attained, and the infinitely small 

 probability of the first case was greater than that of 

 the second.* Let us now suppose that errors equal to 

 or near to P are more probable than any other, and so 

 that, x being the truth, any observation is equally likely 

 to exceed or to fall short of (not #) but x + P. This 

 is equivalent to describing a curve of the following 

 figure in place of that in p. 132. Here A B is to be 



* To compare the proportions of these indefinitely small probabilities, 

 say of absolute truth and of either the error +e, or that of e exactly, 

 take H' from table I., corresponding to 0, and to e multiplied by the square 

 root of c. Thus, c being 1, the probabilities of an error and an error 2 

 are as I'l to -02, or as 55 to 1. 



