The Monte Carlo method clearly yields approximate results. The accuracy
deppends on the number of values that we use for the average. A
possible measure of the error is the ``variance'' defined
by:

and

The ``standard deviation'' is . However, we should expect that the error decreases with the number of points , and the quantity defines by (271) does not. Hence, this cannot be a good measure of the error.

Imagine that we perform several measurements of the integral, each of
them yielding a result . Thes values have been obtained
with different sequences of random numbers. According to the central
limit theorem,
these values whould be normally dstributed around a mean
. Suppouse that we have a set of of such measurements
. A convenient measure of the differences of these measurements is
the ``standard deviation of the means'' :

and

Although gives us an estimate of the actual error, making additional meaurements is not practical. instead, it can be proven that

This relation becomes exact in the limit of a very large number of measurements. Note that this expression implies that the error decreases withthe squere root of the number of trials, meaning that if we want to reduce the error by a factor 10, we need 100 times more points for the average.