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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:







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:




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}} 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:

 Crystal Ball



Note that in Crystal Ball we can write equation 1 as the following Student distribution:




Comparison with the Bayesian approach

The Bayesian derivation of Equation 2 is given here.



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