4.2. Propagation of Errors

It is occasionally necessary to propagate errors to determine the total error that results from performing mathematical operations using several numbers that have associated errors. A very complete discussion of error propagation is presented at: http://www.itl.nist.gov/div898/handbook/mpc/section5/mpc55.htm. The discussion below is purely practical...

For addition and subtraction of A and B to produce C:

For multiplication and division of A and B to produce C:

Thus, determining the combined error on the equation (A+B)/C requires using the first equation to get the error associated with (A+B). This result, in turn, is plugged into the second equation to get the total error. As an example, consider the following equation:

First, one must determine the error on the 15+1200 by taking the square root of the squares of the standard deviations:

Next, one determines the error on the division. First, it is necessary to calculate the relative standard deviations:

and

Combining these yields:

Finally, one must convert the relative standard deviation into an absolute value. Solving the equation, ignoring the errors, yields 46.74, so:

Thus, the final result is 46.73 ± 3.61.