Standar error and confidence interval
T. Yamato
July, 27, 2021
The mean and standard error of $N$ measurements of qantytiy $A_i :(i=1, …, N)$ are \(\left<A\right> = \frac{1}{N} \sum_{i=1}^{N}A_i, \: \sigma_e=\frac{\sigma}{\sqrt{N}},\) , respectively, where $\sigma$ is the standard deviation of this measurements: \(\sigma^2 = \frac{1}{N} \sum_{i=1}^{N}(A_i - \left<A\right>)^2.\) For instance, the confidence interval (CI) of the conficence level 95 % for the real mean $\overline{A}$ is \(\left[\left<A\right>-1.96s, \left<A\right>+1.96s \right]\), where $s$ is the estimate of $\sigma_e$. In other words, we can expect that the real mean is found in the interval between \(\left<A\right>-1.96s\) and \(\left<A\right>+1.96s\) with the probability of 95%.