Chapter 2: Error Analysis

2.5 Propagation of Error (Multiple variables)(without calculus)

The most general case of error propagation is that in which the derived quantity q depends on several independent variables x, y, . . .[q = q(x,y,...)]. The formula describing this case is kind of ugly, but it's easy to see how it comes about. First the independent variables each vary on their own, thus we must let each one vary independent of the other variables!

Let's assume we want to calculate the uncertainty in the volume of a box which has some base, b +/- s_b, width, w +/- s_w, and height, h +/- s_h. The volume V is:
    V = V(b,w,h) = b w h
and there is uncertainty due to the base's uncertainty, s_b, the width's uncertainty, s_w, and the height's uncertainty, s_h. We can find the uncertainty in the volume due purely to the base's uncertainty as in the previous section:
    S_Vb = |V(b+s_b,w,h) - V(b,w,h)| = (b+s_b)wh - bwh.
Similarly, we can find the uncertainty in the volume due purely to the width's uncertainty:
    S_Vw = |V(b,w+s_w,h) - V(b,w,h)| = b(w+s_w)h - bwh.
And the uncertainty in the volume due purely to the height's uncertainty:
   S_Vh = |V(b,w,h+s_h) - V(b,w,h)| = bw(h+s_h) - bwh.
Now remember that base, width, and height are all independent! So we cannot say the uncertainty in the volume is:
    S_V = S_Vb + S_Vw + S_Vh, this is WRONG!
Because the base, width, and height are independent, the uncertainty in the volume is an independent summation of these components. In science we call this "addition in quadrature".

Now, in general if we have a dependent quantity q which depends on several independent variables x, y, . . .[q = q(x,y,...)], the uncertainty in q is S_q, given by:

. (9)

Where S_qx = q(x+s_x,y,..) - q(x,y,..), S_qy = q(x,y+s_y,..) - q(x,y,..), …

Example: You know the location of some landmark on a trip as 40 miles north (n) and 26 miles east (e) of your starting point, each within a standard error of 0.5 miles. You want to calculate its distance. Then

and

thus

In any case, (9) is quite general, provided only that you have standard deviation estimates s_x, s_y, . . . for all the variables that your result depends on. Let me stress that it doesn't matter how the individual estimates were arrived at: from repeated trials of an experiment, from the manufacturer's specifications for an instrument, or just from your educated guess as to how closely you can read a meter stick.

Also, notice something else that you can learn from the structure of Equation (9). Each of the individual terms that are being combined is the contribution of one of the variables to the standard error of the result. (At one point I called it S_qx.) In practice, it often happens that one of these terms is much bigger than the others. If this is the case, it tells you a couple of things: first, that just figuring that one term - S_qx instead of the messier (9) - will do, at least as a quick approximation; and second, that if you want to improve the experiment, which of your measurements you need to work on.

Section 2-6