What is the standard way to find averages of measurement values that contain statistic and systematic uncertainties?

How to find the average value of several measurement results if we know their statistical and systematics uncertainties? There might be that there is no such thing as best way to do it, but it would be helpful to know how is this done in different areas of physics/science.

• "the standard way to..." (emphasis added) Which one? Especial with regard to the systematics you have to take care to be sure you understand how the uncertainty would effect your result. – dmckee May 11 '12 at 15:28
• – Pygmalion May 11 '12 at 15:43

If all of your measurements are completely independent, then I would add the statistical and systematic errors in quadrature to find the effective uncertainty on each measurement, $$(\sigma^i_\text{effective})^2 = (\sigma_\text{stat}^i)^2 + (\sigma_\text{syst}^i)^2,$$ and find the error-weighted mean as suggested by Dai, $$\left< x \right> = \frac{\sum x_i / (\sigma_\text{effective}^i)^2}{\sum (\sigma_\text{effective}^i)^{-2}},$$ where the uncertainty on the mean is $$(\sigma_\text{effective}^\text{mean})^{-2} = {\sum (\sigma_\text{effective}^i)^{-2}}.$$ If your measurements are statistically independent but share a common systematic uncertainty — for example, you have several experiments that use a related technique to determine an overall normalization — then treating the systematic uncertainties as independent would overestimate the total uncertainty. In that case you might weight the measurements by their statistical uncertainty, $$\left< x \right> = \frac{\sum x_i / (\sigma_\text{stat}^i)^2}{\sum (\sigma_\text{stat}^i)^{-2}}, \quad\text{with statistical uncertainty } (\sigma_\text{stat}^\text{mean})^{-2} = {\sum (\sigma_\text{stat}^i)^{-2}},$$ and assign a separate systematic uncertainty $\sigma_\text{stat}^\text{mean}$ to the mean based on ordinary error propagation from the individual measurements. For example, if your systematic uncertainty is $\sigma_\text{syst}^i/x_i = 1\%$ for all of your measurements, you could reasonably assign $\sigma_\text{syst}^\text{mean}/\left<x\right> =1\%$.

Real measurements usually have a table or a section in the documenting paper entitled "Error budget" which tabulates the authors' estimates for the systematic uncertainties, and combinations of real, independent measurements will have some correlated and some uncorrelated quantities. The proper way to combine such measurements is to construct a covariance matrix. This is a nontrivial task which makes up a long chapter in many PhD theses. For one thoughtful discussion, see §10 of the introduction to the PDG's Review of Particle Physics.

The mean of a distribution sampled $N$ times is

$$\bar{x} = 1/N \sum x_i$$

What you want though is the weighted mean

$$\bar{x} = \frac{ \sum w_i x_i }{ \sum w_i }$$

Here, in order to account for the measurement uncertainties, the weights should be the inverse of the variances (inverse of the standard deviations $\sigma_i$ squared, obviously), or

$$w_i = 1/\sigma_i^2$$