- we are about to measure rectangle sides a and b and we know that a will be about 10 times as big
as b. Because σ2A = b2σ2a + a2σ2b, the second term will be about 100 times as important as the first term if a and b have similar standard deviations, and we can conclude that it is much more important to find a way to reduce σ2b than to reduce σ2a.
Usually we are less interested in the variance of V than in the variance of the mean V, or its square root (the standard error of V). We can simply replace the variances in equation (1) above with variances of means. Using variances of means, propagation of errors allows us to estimate how many measurements of each of the variables a,b,c,. . . would be needed to determine V with some desired level of accuracy, if we have a rough idea of what the expected variances of a,b,c, . . . will be. Typically the variables a,b,c,. . . will have different variances which we can roughly predict after a brief pilot study or before we even start the controlled measurement series. If so, a quick analysis of propagation of errors will suggest concentrating most of our limited time resources on one variable, either with a large number of measurements or with slower and more accurate measurements. For example, above we imagined that a is about 10 times as big as b and therefore concluded that we should focus on reducing σ2b instead of reducing σ2a. Even if we have no way of reducing σ2b, we can reduce σ2b (variance of mean b) by increasing the number of measurements, because the standard error σx=σN-0.5.
Equation (1) and ability to calculate simple partial derivatives will allow one to analyze propagation of errors for most problems. Some problems are easier if equation (1) is recast in terms of fractional standard deviations:
(σv/V)2 = (V-1•∂V/∂a)2•σ2a + (V-1•∂V/∂b)2•σ2b +. . . (2)
Based on equation (1) or (2), here are the propagation of error equations for several common relationships of V to the variables a and b, where k and n are constants:
V=ka+nb: σ2v = k2σ2a + n2σ2b
V=ka-nb: σ2v = k2σ2a + n2σ2b
V=kab: σ2v = (k bσa)2 + (k aσb)2
- or: (σv/V)2 = (σa/a)2 + (σb/b)2
V=ka/b: (σv/V)2 = (σa/a)2 + (σb/b)2
V=kan: σv/V = nσa/a
V=akbn: (σv/V)2 = (kσa/a)2 + (nσb/b)2
Non-Normal Distributions
The most frequent statistics pitfall is also a readily avoided pitfall: assuming a normal distribution when the data are non-normally distributed. Every relationship and equation in the previous section should be used only if the data are normally distributed or at least approximately normally distributed. The more data depart from a normal distribution, the more likely it is that one will be
