Combining measurements with a weighted mean

A weighted mean combines independent measurements of the same quantity. More precise measurements receive more weight. It is the constant-model case of weighted least squares:

The model assumes one common underlying value. It does not describe a changing signal, a trend over time or measurements of different physical quantities.

The mean and its propagated uncertainty

For measurements yᵢ with independent standard uncertainties σᵢ:

Under the Absolute convention, the standard uncertainty of the mean is:

This last formula propagates the measurement uncertainties. It relies on their absolute sizes and independence. Repeating a measurement does not average away a calibration error shared by every observation.

A worked example

Consider these illustrative measurements in the same units:

The weights are 25, 6.25 and 25. The weighted mean is approximately 9.95556, with an Absolute standard uncertainty of 0.13333. The less precise measurement at 10.4 contributes less than either of the other two observations.

For these data, chi-squared is approximately 1.88889. There are two degrees of freedom, so reduced chi-squared is approximately 0.94444. Nominal gives the same mean with standard uncertainty approximately 0.12958.

Why there are two uncertainty answers

Nominal estimates a common uncertainty scale from the scatter about the mean:

Nominal is curve.fit's default, including for Weighted Mean. Choose Absolute when you intend the usual propagated uncertainty based on the entered standard uncertainties. Choose Nominal when their common scale should be estimated from the observed scatter. That is an assumption about the measurements, not a choice between two different averages.

If all uncertainties are equal, the weighted mean is the arithmetic mean. Its Absolute uncertainty is the common measurement uncertainty divided by √N. Its Nominal uncertainty is the sample standard deviation divided by √N, when N is greater than one.

One measurement and zero degrees of freedom

One measurement determines the constant value but supplies no independent information about residual scatter. Absolute therefore retains that measurement's uncertainty. Nominal cannot estimate a residual scale and reports its uncertainty as unavailable. Reduced chi-squared is also undefined when N − 1 is zero. Zero degrees of freedom is expected here; it limits what can be inferred.

Trying the example in curve.fit

Select Weighted Mean, enter values in y and standard uncertainties in δy, and select Table for δy. The x columns are retained but are not used. Every included δy must be finite and positive. No starting guess is needed.

Fit once with Nominal and once with Absolute to compare the reported uncertainties. The horizontal band in the Weighted Mean plot is ±1 standard uncertainty of the mean; it does not show the scatter expected for individual measurements. For control details, see Weighted Mean in Help.

Next: measurement uncertainty assumptions · weighted least squares.