Means and Standard Deviations of Estimators

If $X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}$ is the mean and $S_X$ the standard deviation of a random sample of size $n$ from a population with mean $\mu$ and standard deviation $\sigma$, then

$$\begin{split} E(X{\mkern -13.5 mu}\overline{\phantom{\text{... ...rline{\phantom{\text{X}}}) & = \frac{s_X}{\sqrt{n}} \end{split}$$

If $\widehat{P}$ is the sample proportion for a random sample of size $n$ from a population with population proportion $p$, then

$$\begin{split} E(\widehat{P}) & = p \\ \mathop{\rm sd}\n... ...ehat{P}) & = \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}} \end{split}$$

If $X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}_1$ and $X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}_2$ are the means and $S_1$ and $S_2$ the standard deviations of two independent random samples of sizes $n_1$ and $n_2$ from populations with means $\mu_1$ and $\mu_2$ and standard deviations $\sigma_1$ and $\sigma_2$, then

$$\begin{split} E(X{\mkern -13.5 mu}\overline{\phantom{\text{... ... & = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \end{split}$$

If $\widehat{P}_1$ and $\widehat{P}_2$ are the sample proportions for independent random samples of size $n_1$ and $n_2$ from populations with population proportions $p_1$ and $p_2$, then

$$\begin{split} E(\widehat{P}_1 - \widehat{P}_2) & = p_1 - p_... ...t{p}_1)}{n} + \frac{\hat{p}_2 (1 - \hat{p}_2)}{n}} \end{split}$$