With the assumption that the average measurements i are also defined as iid stochastic variables and have the same probability function p , we defined the total average over m experiments as
\overline{X}=\frac{1}{m}\sum_{i} \overline{x}_{i}.and the total variance
\sigma^2_{m}=\frac{1}{m}\sum_{i} \left( \overline{x}_{i}-\overline{X}\right)^2.These are the quantities we used in showing that if the individual mean values are iid stochastic variables, then in the limit m\rightarrow \infty , the distribution for \overline{X} is given by a Gaussian distribution with variance \sigma^2_m .