If we can consider the probability of correlated events to be zero, we can rewrite the variance of these series of measurements as (equating \( M=N \)) $$ \begin{equation} \sigma^2_N\approx \frac{1}{N}\left(\langle f^2\rangle - \langle f \rangle^2\right)=\frac{\sigma^2_f}{N}. \tag{3} \end{equation} $$ We note that the standard deviation is proportional to the inverse square root of the number of measurements $$ \begin{equation*} \sigma_N \sim \frac{1}{\sqrt{N}}. \end{equation*} $$