Consider the case in which a single physical quantity, y, is some function of the 's: y = y(1, ..., M). The "best" value for y is then y* = y(i*). For example y could be the path radius of an electron circling in a uniform magnetic field where the measured quantities are 1 = , the period of revolution, and 2 = v, the electron velocity. Our goal is to find the error in y given the errors in . To first order in (i - i*) we have
A well-known special case of Eq. (12), which holds only when the variables are completely uncorrelated, is
In the example of orbit radius in terms of and v this becomes
in the case of uncorrelated errors. However, if is non-zero as one might expect, then Eq. (12) gives
It is a common problem to be interested in M physical parameters, y1, ..., yM, which are known functions of the i. In fact the yi can be thought of as a new set of i or a change of basis from i to yi. If the error matrix of the i is known, then we have
In some such cases the ðyi / ða cannot be obtained directly, but the ði / ðya are easily obtainable. Then
Suppose one wishes to use radius and acceleration to specify the circular orbit of an electron in a uniform magnetic field; i.e., y1 = r and y2 = a. Suppose the original measured quantities are 1 = = (10 ± 1)µs and 2 = v = (100 ± 2) km/s. Also since the velocity measurement depended on the time measurement, there was a correlated error = 1.5 × 10-3 m. Find r,r, a, a.
Since r = v / 2 = 0.159 m and a = 2v / = 6.28 × 1010 m/s2 we have y1 = 12 / 2 and y2 = 2 2 / 1. Then ðy1 / ð1 = 2 / 2, ðy1 / ð2 = 1 / 2, ðy2 / ð1 = -22 / 12, ðy2 / ð2 = 2 / 1 . The measurement errors specify the error matrix as
Eq. 13 gives
Thus r = (0.159 ± 0.184) m
For y2, Eq. 13 gives
Thus a = (6.28 ± 0.54) × 1010 m/s2.