Standard Error

Concepts Clarification

Population Standard Deviation

The square root of population variance

$$\sigma =\sqrt{{\sigma }^2}=\sqrt{\mathrm{E}[X^2]-{\mathrm{E}}^2[X]}$$

Sample Standard Deviation

The square root of sample variance

$$s=\sqrt{s^2}=\sqrt{\frac{1}{n-1}\sum _{i=1}^n{\left(X_i-\bar{X}\right)}^2}$$

Standard Error

The standard deviation of sample mean $\bar{x}$

$${\sigma }_{\bar{x}}=\frac{\sigma }{\sqrt{n}}$$