SECT. 7.] Averages. 443 



far the simplest to calculate ; and, what is more, the process 

 of determining it serves also to assign another important 

 element to be presently noticed, viz. the * probable error. 

 Then again, as Fechner notes, whereas in the arithmetical 

 mean a few exceptional and extreme values will often cause 

 perplexity by their comparative preponderance, in the case 

 of the median (where their number only and not their ex 

 treme magnitude is taken into account) the importance of 

 such disturbance is diminished. 



7. A simple illustration will serve to indicate how these 

 three kinds of mean coalesce into one when we are dealing 

 with symmetrical Laws of Error, but become quite distinct 

 as soon as we come to consider those which are unsym- 

 metrical. 



DYX 



Suppose that, in measuring a magnitude along OBDC, 

 where the extreme limits are OB and OC, the law of error 

 is represented by the triangle BAC: the length OD will 

 be at once the arithmetical mean, the median, and the most 

 frequent length: its frequency being represented by the 

 maximum ordinate AD. But now suppose, on the other 

 hand, that the extreme lengths are OD and OC, and that 

 the triangle ADC represents the law of error. The most 

 frequent length will be the same as before, OD, marked by 

 the maximum ordinate AD. But the mean value will now 

 be OX, where DX = ^DC; and the median will be OY, 



where DF=(l- 



