For a given set of *n* data values randomly sampled from an assumed Normal distribution, with unknown mean m and *unknown* standard deviation s, the distribution of uncertainty of the true mean is calculated from a Student-t distribution:

{\mu=t(n-1).\Big(\frac{\widehat{\sigma}}{\sqrt{n}}\Big)+\bar{x}} |

(1)

where t(n-1) is a standard Student-t distribution with (*n*-1) degrees of freedom. [This page provides an explanation of the derivation of Equation 1].

{\widehat{\sigma}} is the unbiased single point estimate of the true standard deviation (calculated by STDEV( ) in Excel), given by:

{\widehat{\sigma}=\sqrt{\frac{\displaystyle\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}}{n-1}}} |

The standard Student-t distribution is unimodal and symmetric about zero (in the standard student distribution, the mode = 0). The formula therefore centers the uncertainty distribution of the value of the true mean m around the sample mean x which is the "best guess". It also has a spread that increases with the standard deviation _{{\widehat{\sigma}}
{\widehat{\sigma}}
} and decreases with the square root of the sample size *n*. The Student-t distribution looks quite like a unit Normal distribution but flatter, with greater spread than the unit Normal distribution: a Standard Student(0,1,*n*) or Student(*n*) distribution has a standard deviation of {\sqrt{\nu/(\nu-2)}} compared with a standard deviation of 1 for the unit Normal distribution:

**Figure 1** Examples of the Student-t distribution

In fact, the larger *n* gets, the closer the Student-t distribution approaches a unit Normal distribution (i.e. Normal(0, 1)). So, for large *n* (greater than 20 is usually fine), Equation 1 is very well approximated by:

{\mu \approx Normal(0,1)\Big(\frac{\widehat{\sigma}}{\sqrt{n}}\Big)+\bar{x}} |

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

The links to the Estimate Mean & StDev for Normal Distribution When Neither Known software specific models are provided here:

*Comparison with the Bayesian approach*

The Bayesian derivation of Equation 2 is given here.