It would carry us too far afield in the theory of probabilities to demonstrate the relations existing between the mean error e and the probable error r. But we may remark in pass ing that it is shown in works on the theory of errors, that, approximately: r 51, 5, or: (23) f = 2 ?ft 1 [vv] Furthermore, remembering that e and r have reference to the precision attainable from a single observation, we can get the correspond ing errors belonging to the arithmetical mean from m observations by the aid of the principle already used, viz.: that the diminution of error is proportional to the square root of the num ber of observations. Denoting the mean and probable errors of the arithmetical mean from rn observations by ea and re equations (22) and (23) give: (24) E. = (m 1) E191 re 2 NI 3 m (m 1) Mean and Probable Error for Several Un Equation (22), so far as we have yet considered it, applies only to the case of a single unknown x observed directly in times. But a similar expression can also be found for the case of observation equations of the form (4), when solved by means of normal equa tions of the form (9). Having found values of the unknowns x, y, a, from the solution of normal equations, we substitute these values in the original observation equations, and thus obtain numerical values of the residuals v, by which amounts the original observation equa tions fail of being exactly satisfied. With these values of the we compute [vv]. If we then let: u the number of unknown x, y, etc., ap pearing in the original observation equations, it may be shown that we shall have for the mean and probable errors: (26) e = u (22) r = 2 [fel 3 m u For the complete demonstration of these equations we must refer to works on the Method of Least Squares. The equations fur nish average approximate values for the mean and probable errors of any average observed quantity n, as it appears in any observation equation of the form (4). The unknowns x, y, a, are of course determined from the whole group of equations with much greater precision, and therefore with smaller mean and probable errors. In the case of a, for instance, it is shown in works on Least Squares that its mean and probable errors are smaller than those given by (26) and (27) in the proportion of: NI [cc. 2] io 1, .
where [cc. 2] is the final coefficient of a ap pearing in the last reduced equation (13) re sulting from the Gaussian elimination of the normal equations (9). Consequently, denoting by es and re the mean and probable errors of a, we have: '28) -- [vv] [cc. 2] (m ii)' 2 [vv] 29) re --=-3 [cc. 2] (m u) The corresponding mean and probable er rors of x and y can be found by rearranging the observation equations in such a way that each unknown in turn will come out last in the Gaussian elimination.
Weights. It happens sometimes that the quantities n, appearing in the observation equa tions (4), have been determined by the ob servers with unequal precision. For instance, a certain is may have been observed twice, while all the others depend upon a single observation only. In such a case, that particular n is said to have 'double weight? or the weight 2, while all others have the weight 1 only. Obviously, we can treat this case in our reductions by simply writing the doubly weighted equation twice, with all its coefficients and unknowns, among the observation equations, and then proceeding as usual. There is, however, a bet, ter way of dealing with this matter of varying weights; and we can derive it from a consid eration of the rule for the formation of normal equations from observation equations. We may use the following: Rule for Weighting Observation Equa Multiply each observation equation throughout by the square root of its weight and then form normal equations by the usual rule.
It may be seen readily, in the simple case of one equation only having double weight, that this rule will produce precisely the same effect on the normal equations as would result from merely 'writing that particular observation equa tion twice among the other observation equa tions, in the manner just explained. But it is
also shown in works on Least Squares that the above rule for weights holds good when all the observation equations have different weights, and even when these weights are fractional.
To illustrate the foregoing, we shall give the complete solution of a set of observation equa tions resulting from a precise determination of clock error made with the transit instrument of Columbia University Observatory in New York. The observation equations were as fol lows, the numerical terms n being expressed in seconds of time.
x + 0.04y + 1.29s 01.27= 0, x + 0.08y + 1.26z 0.02=0, x+ 1.36y + 2.84s 6.52 = 0, x Q.23y + 1.54z-1.32=0, (2') x + 0.03y + 1.30s + 0.06=0, x 0.07y 1.302 0.02 0, x + 0.14y 1.21z + 0.84=0, x + 0.03y 1.30s + 0.48=0, x 0.01y 1.33s + 0.31 =0, x 1.28y 2.750 4.68 = O.
Each of these equations depends on the observation of a different star; the unknown x is the clock.error, and the other two unknowns y and a relate to certain errors of adjustment in the transit instrument.
The solution now proceeds as follows: Applying the for Normal Equations' gives: + 0.34x 0.63y + 29.43s-9..78=0, (9') 2.63x + 3.59y 0.63s + 15.30 = 0, + 10.00x-2.63y +0.34s-11.22=0.
The method of equations (10) and (11) transforms these into the following: (11') + 2.90y 0.54z + 12s.35 =- 0, 0.54y + 29.42z 9.40=0.
A further application of (12) gives: (13') + 29.32s 7s.10 = 0.
From this we obtain: (14') z = + If we now substitute this value of s in either of equations (11') we obtain: y =-415.25.
These values of y and a can now be substi tuted in either of equations (9'), and we get: x=-0.005.
We now substitute the numerical values thus obtained for x, y and z in equations (2'), and obtain the following residuals, v: V vv 08.125 0.015625 .055 .003025 .075 .005625 .025 .006250 .125 .015625 .005 .000025 .045 .002025 .045 .002025 . .025 .000625 .125 .015625 [vv] .060850 Applying equation (27), remembering that m is the number of residuals, in this case 10, we have: (27') 08.060850 08.061.
4 10-3 This is the «probable of a single ob servation. To obtain the corresponding quan tity for the value of z given by equation (14') we must use equation (29). We have from (13'): [cc. 2]=29.32.
Consequently, from (29) : 2 0.060850 08.011.
(291 rg=3 \I 29.32 (10 3) This result signifies that it is an even chance whether the actual error of our numerical value of z is greater or less than 08.011, so that the probable error becomes a sort of test of the precision of our result.
To obtain the probable error of another un known, as x, we rearrange the equations (2') so as to make x the last unknown, instead of z. Written in this way, the equations are: (2") + 0.04y + 1.29 x 0a.27° 0, etc., etc.
The normal equations are: + 3.59y-0.63z-2.63x + 158.30=0, (9" ) 0.63y + 29.43z + 0.34x 9.78 = 0, 2.63y + 0.340 -I- 10.00x 11.22 = O.
The method of equations (10) and (11) gives: +21.32z 0.12x 7a.10=0, (11") 0.12z+ 8.07x 0.01 =0.
The application of (12): (13") + 8.07x 08.01 =0.
And, consequently: (14") x + This is practically the same as the value al ready obtained, and thus affords incidentally a very complete check on arithmetical accuracy. Values of y and z can of course be obtained as before by substitution.
Applying equations (29) again, we have the probable error of x: 3 (29") = 0.060850 Os.022.
8.07 (10-3) ± Equation (14") shows that the clock was exactly right; and (29") that the precision of this clock-error determination is very high.
Bibliography.Wright add Hayford, justment of Observations); Chauvenet, ical and Practical Astronomy' (Vol. II, Appen dix); Jordan, (Vermessungskunde) (Vol. I).