The Sampling Distribution of the Sample Mean \(\bar{X}\)

Distribution of \(\bar{X}\)

1. When \(X\) follows a Normal Distribution with Known Population and Variance

If a random sample of size \(n\) is taken from a normal distribution with mean, \(E(X)=\mu \) and variance, \(Var(X)={{\sigma }^{2}}\), then the sample mean, \(\bar{X}\) , follows a normal distribution with mean \(\mu \)and variance \(\frac{{{\sigma }^{2}}}{n}\) .

\(X\sim N\left( \mu ,{{\sigma }^{2}} \right)\Rightarrow \bar{X}\sim N\left( \mu ,\frac{{{\sigma }^{2}}}{n} \right)\) and \({{X}_{1}}+{{X}_{2}}+...+{{X}_{n}}\sim N(n\mu ,n{{\sigma }^{2}})\)

2. When \(X\) follows Any Distribution with Known Population and Variance

Central Limit Theorem, CLT

If a random sample of size \(n\), where \(n\) is large (\(n>30\)), is taken from a any distribution with mean, \(E(X)=\mu \) and variance, \(Var(X)={{\sigma }^{2}}\), then by CLT,

\({{X}_{1}}+{{X}_{2}}+...+{{X}_{n}}\sim N(n\mu ,n{{\sigma }^{2}})\) approximately,

Sample mean of \(X\), \(\overline{X}\sim N\left( \mu ,\frac{{{\sigma }^{2}}}{n} \right)\) approximately.

Note, if \(X\) is already normally distributed, then there is no need to use Central Limit Theorem.