Answer to Question #5456 Submitted to "Ask the Experts"

Q

When averaging two analytical results reported as Result +/- 2-sigma uncertainty, I know that I have to average the uncertainty terms using SQRT (μ1 squared + μ2 squared + ...... μ(n)squared) divided by n. My question is: do I use the 1-sigma uncertainty in this equation or the reported 2-sigma uncertainty? The source I have implies it should be the 2-sigma uncertainty.

A

As the expression you quote implies, the uncertainty in the arithmetic average is obtained by calculating the square root of the sum of the squares of individual uncertainties and dividing the result by the number of data points, n. (As an aside, we might note that implicit in this approach is that all the data points being used have been measured with approximately the same degree of precision; if the uncertainties in all the data points are essentially equal, the final expression for the uncertainty in the mean reduces to the familiar quantity of the individual data point uncertainty divided by the square root of n (i.e., σ/n^0.5). If there is considerable dispersion among the uncertainties, σ_i, of the data points, the generalized expression that represents the uncertainty in the mean ends up being given by (1/Σ(σ_i²))^0.5; this latter generalized expression yields the same result as the more familiar expression when data point uncertainties are equal.)

If you desire the 2-sigma uncertainty in the final result you may, as your source implies, use the 2-sigma uncertainties in each of the quantities used in computing the average. You can get the same results by doing the calculation using the 1-sigma uncertainties and multiplying the resulting propagated 1-sigma uncertainty by 2.

This makes mathematical sense since, if you considered doing the calculation using the individual 2-sigma values, each term in the summation of squares would have the coefficient 4 (i.e., 2 squared), which can be taken from the sum as a common factor, and when you take the square root of the sum with the factor the result is 2 (square root of four) times the propagated 1-sigma value. Good luck.

George Chabot, PhD, CHP