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:

\mu=Normal\Big(\bar{x},\sigma \Big/ \sqrt{n}\big) (1)

This can be rewritten as:

\mu=Normal(0,1)*\sigma \Big/_{\sqrt{n}}+\bar{x} (2)

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

The model Estimate Mean for Normal Distribution When StDev is Known lets you generate values for the above uncertainty distribution for m for a data set.

The links to the Estimate Mean for Normal Distribution When StDev is Known software specific models are provided here:

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.