# 4.1.1 - Population is Normal

If the population is normally distributed with mean $$\mu$$ and standard deviation $$\sigma$$, then the sampling distribution of the sample mean is also normally distributed no matter what the sample size is. When the sampling is done with replacement or if the population size is large compared to the sample size, then $$\bar{x}$$ has mean $$\mu$$ and standard deviation $$\dfrac{\sigma}{\sqrt{n}}$$. We use the term standard error for the standard deviation of a statistic, and since sample average, $$\bar{x}$$ is a statistic, standard deviation of $$\bar{x}$$ is also called standard error of $$\bar{x}$$. However, in some books you may find the term standard error for the estimated standard deviation of $$\bar{x}$$. In this class we use the former definition, that is, standard error of $$\bar{x}$$ is the same as standard deviation of $$\bar{x}$$.

Standard Deviation of $$\boldsymbol{\bar{x}}$$ [Standard Error]

$$SD(\bar{X})=SE(\bar{X})=\dfrac{\sigma}{\sqrt{n}}$$