i am trying to refresh my knowledge in error analysis and stumbled over an interesting question.
Suppose i have a radioactive compound and i want to measure the standard deviation of the decay count. To do that, i have two possible strategies
Strategy 1: I measure the decay count $\nu$ for a fixed amount of time $N$ times. I assume the timing to be exact. I then calculate the sample mean $\bar{\nu}$ and the SDOM using $$ \bar{\nu} = \frac{\sum_{i=1}^{N}{\nu_i}}{N} \qquad \text{and} \qquad \sigma_{\bar{\nu}} = \sqrt{\frac{1}{N(N-1)} \sum_{i=1}^{N}{(\nu_{i}-\bar{\nu})^{2}}} $$
Strategy 2: I treat the $N$ measurements as one big measurement and derive the error by taking the square root of the total count and deviding by $N$. See below $$ \sigma = \frac{\sqrt{\sum_{i=1}^{N}{\nu_i}}}{N} $$
My question now is: Shouldn't $\sigma_{\bar{\nu}}$ in strategy 1 and $\sigma$ in strategy 2 ideally lead to one and the same solution? And shouldn't the two expressions for the errors then be equal algebraically? I tried deriving one expression using the other but did not get anywhere really. Maybe it only holds true for $N \rightarrow \infty$? Thank you for your help in advance!