Mean and average errors

The arbitrary choice of the probable error for comparing the errors which are committed with equal facility in different sets of observations, appears most natural because the probable error occupies the middle plaoe in a series arranged according to order of magnitude so that the number of errors less than the fictitious probable error, is the same as those which exceed it. There are other standards of comparison. In Germany, the favourite method is to employ the mean error, which is defined as the error whose square is the mean of the squares of all the errors, or the error which, if it alone were assumed in all the observations indifferently, would give the same sum of the squares of the errors as that which 

Let m denote the mean error, then, by integration as on page 343, 

From (8) and (9), preceding section, the mean error, m, which affects each single observation is given by the expression 

The mean error must not be confused with the mean of the errors, or, as it is sometimes called, the average error,1 another standard of comparison defined as the mean of all the errors regardless of sign. If a denotes the average error, we get from page 235, 

The average error measures the average deviation of each observation from the mean of the whole series. It is a more useful standard of comparison than the probable error when the attention is directed to the relative accuracy of the individual observations in different series of observations. The average error depends not only upon the proportion in which the errors of differ- 

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