492 Theory of the Average. [CHAP. xix. 



the same apparatus of calculation as in the former case. We 

 take the initial average as the probable position of the true 

 centre or ultimate average : we estimate the probability that 

 we are within an assignable distance of the truth in so doing 

 by calculating the error of mean square ; and we appeal 

 to this same element to determine the modulus, i.e. the 

 amount of contraction or dispersion, of our derived curve of 

 facility. 



The same general considerations will apply to most other 

 kinds of Law of Facility. Broadly speaking, we shall come 

 to the examination of certain exceptions immediately, 

 whatever may have been the primitive arrangement (i.e. 

 that of the single results) the arrangement of the derived 

 results (i.e. that of the averages) will be more crowded up 

 towards the centre. This follows from the characteristic of 

 combinations already noticed, viz. that extreme values can 

 only be got at by a repetition of several extremes, whereas 

 intermediate values can be got at either by repetition of 

 intermediates or through the counteraction of opposite ex 

 tremes. Provided the original distribution be symmetrical 

 about the centre, and provided the limits of possible error be 

 finite, or if infinite, that the falling off of frequency as we 

 recede from the mean be very rapid, then the results of 

 taking averages will be better than those of trusting to 

 single results. 



28. We will now take notice of an exceptional case. 

 We shall do so, not because it is one which can often 

 actually occur, but because the consideration of it will force 

 us to ask ourselves with some minuteness what we mean in 

 the above instances by calling the results of the averages 

 better than those of the individual values. A diagram will 

 bring home to us the point of the difficulty better than any 

 verbal or symbolic description. 



