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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:
(1)
where t(n-1) is a Student-t distribution with (n-1) degrees of freedom.
[This page provides an explanation of the derivation of Equation 1]
is the unbiased single point estimate of the
true standard deviation (calculated by STDEV(
) in Excel), given by:
The Student-t distribution is unimodal and symmetric about zero. The
formula therefore centres 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 
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 Student(n) distribution has a standard deviation
of 
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:
This 
spreadsheet example model
lets you generate values for the above uncertainty distribution for m
for a data set.
Comparison with the Bayesian approach
The Bayesian derivation of Equation 2 is given here.