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Let $G$ be the whole. Let $F \subset G$. I have estimated a probability $\hat p = 0.195$ for a random variable $X \in F$. The real $p$, however, is $0.2$. I want to compute the standard deviation $\sigma$ of $\hat p$ knowing the real $p$. I don't understand what is asked in the first place. As far as I understand standard deviation is the average deviation from the expected value when I continuously perform my experiment. A first thought that comes to mind is compute $\sigma$ for $p$ and $\hat p$ and look at the difference. I'm really confused on this.

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Recall $\hat{p} = \dfrac{X}{n}$ where $X$ is a binomial random variable based on $n$ trials and success probability $p$. Then recalling that $\text{Var}(X) = np(1-p)$,

$$\text{Var}\left(\hat{p} \right) = \text{Var}\left(\dfrac{X}{n}\right) = \dfrac{1}{n^2}\text{Var}(X) = \dfrac{np(1-p)}{n^2} = \dfrac{p(1-p)}{n}$$

and since is the variance of $\hat{p}$, take the square root to obtain the standard deviation.

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  • $\begingroup$ Thank you! I wasn't aware $\hat p$ was defined like that. But yeah, obviously. That is how obtained it. $\endgroup$ – Marc Dec 5 '19 at 22:20

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