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 \( \sigma^2_m \).