Actual, Mean and «Probable» Errors.— The values obtained for the unknowns by the Method of Least Squares are not the true values, but only the most probable values ob tainable under the circumstances. They are subject to error; and we shall here consider three different quantities technically known as 1. The actual error, defined as the diver gence from the truth, and designated by the symbol 6.
2. The mean error, defined as an error of such magnitude that its square is the mean of the squares of all the actual errors A belong ing to the several observations. It is desig nated by the symbol e. If there are m (18) ee E,LAI• 3. The probable error, whose magnitude is such that it is an even chance whether any random actual error p belonging to any ob servation is bigger or smaller than the prob able error. It is designated by r.
To determine the value of e for the case of a single unknown x, observed m times, we pro ceed as follows. Let: the arithmetical mean of the observed values of the unknown x, equation (17).
q the unknown and unknowable true value of x.
Then, as we have seen, x. is the most prob able value of x, according to the Method of Least Squares, and [vv] will be a minimum if we substitute that value of x in equations (15). This substitution gives: ze = VI, X. ± •-= Vo, Xo na = etc.
But if, instead of x,, we substituted the true quantity xe q, we should get a series of true errors A, instead of the residuals v. This would give: xe-hq+n, = Al, = A2, xe-i-q+ne= As, etc.
A subtraction between these last two sets of equations gives: Al = th+q, (19) q, As = vs-1-q, and there will be en such equations. Squaring and adding them gives: [A A ] [vv] +me+2[v]q.
But, from the principle of the arithmetical mean, .
[v]0, so that : (20) [AA] [ vv) +me .
From the equation of definition (18) : [AA] =m ea ; so that, re m Now, the value of q' can never be known; but we can get an approximation to it easily enough. It can be shown by the theory of probabilities that if we continue observing any quantity, the mean error of the arithmetical mean of the observed values will decrease in proportion to the square root of the number of observations.
To demonstrate this principle, let us con sider: Two observed quantities, ni, n,,affected with actual errors, Ah Ale and mean errors, e, . e,.
Now let N be the sum of ti, and n, or: N= n, + and designate by A' and e' the actual and mean errors of N. We shall have: A2; and squaring this equation: A' As . Al A2 A, + 2 Al A2.
If k observations have been made to deter mine n, and the above becomes by summa tion: [A' Al [Aa + [Aa LS,2] + 2 [A2 A2].
In this equation, the last term, 2 [A1 All may be considered zero, because it will dis appear, or nearly disappear, in the general average, on account of positive and negative errors being equally probable, a priori. But the squared terms remain, being always posi tive, and we have: [A' A'] = [As Al] + [A2 A2]. Consequently, by our equation of definition (18), this becomes: e= EL e, + E2 Eq.
This equation brings out the important prin ciple that the square of the mean error of the sum of two quantities is equal to the sum of the squares of their respective mean errors. The principle can of course be extended so as to include three, or any greater number, say m, of observed quantities.
Now let us suppose the n's to have been ob served with equal accuracy, so that el and el will be equal, and we have: e' e'= 2 el el, or for m n's: e,, or e'= yrs e, This last equation shows that the mean error 9 of the sum of ns equally precise quantities is Vm times the mean error of one quantity. But the arithmetical mean of m observations is lim times their sum, and its mean error there fore e' /In, or, according to the above: E Mean error of arithmetical mean = - Nas which establishes the principle enunciated above, that the mean error of the arithmetical mean decreases in proportion as the square root of the number of observations increases.
In accordance with this principle, .re will be more accurate than any individual average x in the proportion of Vess to 1. Consequently, since q is the error of re and e a sort of aver age value for the error of any a-, we may take as a good approximation for es — .
Substituting this value of q' in equation (21), we get: (m ee = Ivo], or: (22) e _= [vv] Nita — 1 and equation (22) will enable us to calculate approximately the mean error e from the known residuals v.