SECT. 9.] Theory of the Average. 473 



crowded up towards the centre, when it is small they are 

 thus crowded up. The smaller the modulus in the curve 

 representing the thickness with which the shot-marks 

 clustered about the centre of the target, the better the 

 marksman. 



9. There are several ways of determining the modulus. 

 In the first of the cases discussed above, where our theo 

 retical knowledge is complete, we are able to calculate it a 

 priori from our knowledge of the chances. We should 

 naturally adopt this plan if we were tossing up a large 

 handful of pence. 



The usual a posteriori plan, when we have the measure 

 ments of the magnitudes or observations before us, is this : 

 Take the mean square of the errors, and double this; the re 

 sult gives the square of the modulus. Suppose, for instance, 

 that we had the five magnitudes, 4, 5, 6, 7, 8. The mean of 

 these is 6: the errors are respectively 2, 1, 0, 1, 2. There 

 fore the modulus squared is equal to ; i.e. the modulus is 



J2. Had the magnitudes been 2, 4, 6, 8, 10; representing 

 the same mean (6) as before, but displaying a greater disper 

 sion about it, the modulus would have been larger, viz. ^8 

 instead of J%. 



Mr Galton s method is more of a graphical nature. It 

 is described in a paper on Statistics by Intercomparison 

 (Phil. Mag. 1875), and elsewhere. It may be indicated as 

 follows. Suppose that we were dealing with a large number 

 of measurements of human stature, and conceive that all the 

 persons in question were marshalled in the order of their 

 height. Select the average height, as marked by the central 

 man of the row. Suppose him to be 69 inches. Then raise 

 (or depress) the scale from this point until it stands at such 



