486 Theory of the Average. [CHAP. xix. 



diminishes inversely as the square root of the number of 

 measurements or observations. (This follows from the second 

 of the above formulae.) Accordingly the probable error of 

 the more extensive statistics here is one half that of the less 

 extensive. Take another instance. Observation shows that 

 &quot;the mean height of 2,315 criminals differs from the mean 

 height of 8,585 members of the general adult population by 

 about two inches&quot; (v. Edgeworth, Methods of Statistics: 

 Stat. Soc. Journ. 1885). As before, common sense would feel 

 little doubt that such a difference was significant, but it 

 could give no numerical estimate of the significance. Ap 

 pealing to science, we see that this is an illustration of the 

 third of the above formulae. What we really want to know 

 is the odds against the averages of two large batches differing 

 by an assigned amount: in this case by an amount equalling 

 twenty-five times the modulus of the variable quantity. 

 The odds against this are many billions to one. 



21. The number of direct problems which will thus 

 admit of solution is very great, but we must confine ourselves 

 here to the main inverse problem to which the foregoing 

 discussion is a preliminary. It is this. Given a few only of 

 one of these groups of measurements or observations; what 

 can we do with these, in the way of determining that mean 

 about which they would ultimately be found to cluster? 

 Given a large number of them, they would betray the posi 

 tion of their ultimate centre with constantly increasing- 

 certainty : but we are now supposing that there are only a 

 few of them at hand, say half a dozen, and that we have no 

 power at present to add to the number. 



In other words, expressing ourselves by the aid of 

 graphical illustration, which is perhaps the best method 

 for the novice and for the logical student, in the direct 

 problem we merely have to draw the curve of frequency from 



