Estimating the mean of a Normal distribution when the distribution's standard deviation is known

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Occasionally, it is possible that the mean is unknown but the standard deviation is known. For example, the manual of a laser ruler that architects use might specify the standard deviation of the error at a certain measured distance. For a given set of n data values randomly sampled from a Normal distribution, with unknown mean m and known standard deviation s, the distribution of uncertainty of the true mean is calculated from a Normal distribution:

                                                     (1)

This can be rewritten as:

                                              (2)

[This page provides an explanation of the derivation of Equation 1]

This spreadsheet example model lets you generate values for the above uncertainty distribution for m for a data set.

Since the unit Normal distribution has a standard deviation smaller than the Student distribution, Equation 2 provides a narrower range of uncertainty, especially at low n, compared with the situation where the standard deviation is unknown, which makes sense because we know more under these circumstances than when we did not know the standard deviation.

Comparison with the Bayesian approach

The Bayesian derivation of the same formula as Equation 1 is given here.