In this section, we apply the formalism for the statistical error to the case of a Gaussian energy distribution with independent measurements. While it is clear that no real physical system is described by such a distribution, we nevertheless feel it is instructive to carry out the calculation because it allows us to derive a simple, closed-form expression for the statistical error. In addition, we expect that the behavior of this ``toy'' system will be qualitatively the same as that of a real system.
With independent measurements, all correlation times are zero, and the expression for the relative error (14) is simplified to be:
For this example, we will use a continuous probability distribution which is
symmetric about E = 0 and has a standard deviation 
.
The quantities needed for the error analysis are then easily computed by
integrating 
with the appropriate function.
For this example, we will calculate the statistical error in the energy and
specific heat that the quantities we need are
By inserting these into (15) we find that the expression for
the square of the relative error in 
is
Because the average value of E is zero at 
, the relative
error is not well behaved. Let us therefore consider the
absolute statistical error, 
For 
, the error reduces to the expected result 
. For small values of 
, the error increases moderately, but
as 
gets larger, the error in E begins to increase dramatically due
to the exponential term 
and continues to rise
without limit. The error in the specific heat has nearly the same
form as the error in the energy:
While the explicit form will be different for real systems, the qualitative
behavior of the error should remain: For a region around 
, the
errors will increase only moderately, then will begin to rise rapidly as
increases.
