SECT. 25.] Theory of the Average. 489 



measurements of a wall with a tape, and that the average of 

 these was 150 feet. The scrupulous surveyor would give us 

 this result, with some such correction as this added, pro 

 bable error 3 inches . All that this means is that we may 

 assume that the true value is 150 feet, with a confidence that 

 in half the cases (of this description) in which we did so, we 

 should really be within three inches of the truth. 



The expression for this probable error is a simple multiple 

 of the modulus: it is the modulus multiplied by 4769... 

 That it should be some function of the modulus, or E.M.S., 

 seems plausible enough ; for the greater the errors, in other 

 words the wider the observed discrepancy amongst our 

 measurements, the less must be the confidence we can feel 

 in the accuracy of our determination of the mean. But, of 

 course, without mathematics we should be quite unable to 

 attempt any numerical assignment. 



25. The general conclusion therefore is that the de 

 termination of the curve of facility, and therefore ultimately 

 of every conclusion which rests upon a knowledge of this 

 curve, where only a few observations are available, is of 

 just the same kind as where an infinity are available. The 

 rules for obtaining it are the same, but the confidence with 

 which it can be accepted is less. 



The knowledge, therefore, obtainable by an average of a 

 small number of measurements of any kind, hardly differs 

 except in degree from that which would be attainable by an 

 indefinitely extensive series of them. We know the same 

 sort of facts, only we are less certain about them. But, on 

 the other hand, the knowledge yielded by an average even 

 of a small number differs in kind from that which is yielded 

 by a single measurement. Revert to our marksman, whose 

 bullseye is supposed to have been afterwards removed. If 

 he had fired only a single shot, not only should we be less 



