# Estimator vs real probability - what is the standard deviation?

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.

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