ten the case, our measure of precision is α95, then the rule of thumb is only approximately true because the t’s of Table 1 are only approximately equal to 2.0.
Propagation of Errors
Sometimes the variable of interest actually is calculated from measurements of one or more other variables. In such cases it is valuable to see how errors in the measured variables will propagate through the calculation and affect the final result. Propagation of errors is a scientific concern for several reasons:
- it permits us to calculate the uncertainty in our determination of the variable of interest;
- it shows us the origin of that uncertainty; and
- a quick analysis of propagation of errors often will tell us where to concentrate most of our limited time resources.
If several different independent errors (ei) are responsible for the total error (E) of a measurement, then:
E2 = e12+e22+. . . +eN2
As a rule of thumb, one can ignore any random error that is less than a quarter the size of the dominant error. The squaring of errors causes the smaller errors to contribute trivially to the total error. If we can express errors in terms of standard deviations and if we have a known relationship between error-containing variables, then we can replace the estimate above with the much more powerful analysis of propagation of errors which follows.
Suppose that the variable of interest is V, and it is a function of the several variables a, b, c, . . .: V=f(a,b,c,...). If we know the variances of a, b, c, . . ., then the variance of V can be calculated from:
σ2V = (∂V/∂a)2•σ2a + (∂V/∂b)2•σ2b +. . . (1)
Thus the variance of V is equal to the sum of the product of each individual variance times the square of the partial derivative. For example, if we want to determine the area (A) of a rectangle by measuring its two sides (a and b): A=ab, and σ2A = (∂A/∂a)2•σ2a + (∂A/∂b)2•σ2b = b2σ2a + a2σ2b. Propagation of errors can be useful even for single-variable problems. For example, if we want to determine the area (A) of a circle by measuring its radius (r): A=πr2, and σ2A = (∂A/∂r)2•σ2r = (2π r)2σ2r.
Why analyze propagation of errors? In the example above of determining area of a circle from radius, we could ignore propagation of errors, just convert each radius measurement into an area, and then calculate the mean and standard deviation of these area determinations. Similarly, we could calculate rectangle areas from pairs of measurements of sides a and b, then calculate the mean and standard deviation of these area determinations. In contrast, each of the following variants on the rectangle example would benefit from analyzing propagation of errors:
- measurements a and b of the rectangle sides are not paired; shall we arbitrarily create pairs for calculation of A, or use propagation of errors?
- we have different numbers of measurements of rectangle sides a and b. We must either discard some measurements or, better, use propagation of errors;
