416 



Mk de morgan, on the theory of 



I shall now enter upon the consideration of the subject from its first principles, making 

 no use of what has preceded, but treating the observations made as finite in number. 



When we have a number of discordant values from which to choose or construct a 

 result, without any other knowledge than that of such similarity of circumstance of the 

 different values as renders it impossible to prefer one to another, we naturally substitute 

 the average for the true result unknown, upon a number of associations which are all 

 covered by the phrase that this average is given by the observations, one with another. 

 That the proper result should lie deep among the observations seems inevitable; and it is 

 therefore some'^^ kind of average, according to the general notion of the word. The mathema- 

 tician, seeing that the balanced character of error makes it more likely that the sum of errors 

 should be zero than anything else, however little likely in itself, sometimes claims to equate 

 the sum of the errors to zero,, and thence to deduce the most probable result ; and thus he 

 arrives at the average as the best substitute for the truth. But why does he give an exclu- 

 sive attention to the sum of the first powers, when zero is also the most probable value of 

 the sum of the third powers, or of any odd powers ? This question cannot be answered. 

 If small discordance only were supposed, the first power might have an easily understood 

 claim of preference : easily understood, not so easily admitted in full. But the argument 

 just stated, so far as it is valid, applies equally to all amounts of discordance. 



The truth is that the average may stand upon a much stronger base of speculation than 

 is usually given, namely, as that from which we have no reason for departure one way rather 

 than the other. It is not merely the mean value of all the given values: it is also the mean 

 supposition of all possible suppositions as to the mode of obtaining value. This may be shown 

 in the following way. 



A single observation is, before others are made, the most probable truth. If the second 

 observation agree with the first, the common value is the most probable truth : and so on, so 

 long as the observations show no discordance. If then (p (a^, a^., ...) be the most probable 

 result of the discordant observations a,, a^, &c., the function (p is subject to the condition 

 ip {a, a, ...) = a. Now in every case in which this condition must be satisfied, whatever the 

 intent of the process may be, it may be shown that the average 2a :« is the most probable 



• A corndealer might dissent:- for to him the average ought 

 to be something near the highest, if not above it. The harvest 

 is, at the time of reaping, never better than an average, rarely 

 so good. " If the fine weather should last three weeks longer, 

 we may expect an average yield," is the strongest admission 

 ever made in the corn market. It is not until the next crop is 

 so far advanced as to admit of gloomy prediction that we hear 

 of " the abundant harvest with which Providence blessed the 

 fields last year:" and this only as a covert hint not to expect 

 the same again. All words are subject to strange mutations 

 when they come into connexion with prices: as the daily ac- 

 counts of the markets show. "I did not," said the farmer, 

 "get as much as I expected for those calves; and I never 

 thought I should." To " expect" is to " demand." 



The farmers have a right to the word average : for its 

 origin is certainly agricultural. Averia, havings, or posses- 

 sions, was a word applied to things in a lump : thus averia 

 ponderis — whence averdepois — refers to the whole mass of 

 goods sold by common weight, as opposed to the selected arti- 

 cles subject to troy weight. Averia, alone, meant precisely 

 what a farmer now calls stock, that is, all the animals which a 

 farm feeds. Afterwards the word was applied only to horses 

 used in farm labour. Averagium was labour of the farm 

 horses, &c. to which the lord was entitled, and for which the 

 composition was the averpenny, Averagium also is of very 

 old use in the sense of the loss of part of the cargo by sea or 

 land thrown over the whole : and probably this is the use by 

 which the word was brought into common life. 



