Sum and mean of normal random sample
If \(\{X_1, X_2, ..., X_n\}\) is a random sample from a distribution with mean \(\mu\) and variance \(\sigma^2\), we have found formulae for the mean and variance of the sample mean (and of the sum of the sample values).
If the random sample comes from a normal distribution, we can also find the shapes of their distributions.
Sum and mean of a normal random sample
If \(\{X_1, X_2, ..., X_n\}\) is a random sample of \(n\) values from a \(\NormalDistn(\mu, \;\sigma^2)\) distribution,
\[\begin{align} \sum_{i=1}^n {X_i} & \;\; \sim \;\; \NormalDistn(n\mu, \;\sigma_{\Sigma X}^2=n\sigma^2) \\ \overline{X} & \;\; \sim \; \; \NormalDistn(\mu, \;\sigma_{\overline X}^2=\frac {\sigma^2} n) \end{align} \]Samples from other distributions
Whatever the shape of the distribution from which the random sample is selected from, the sample sum and mean are approximately normal when the sample size is large.
Central Limit Theorem (informal)
If \(\{X_1, X_2, ..., X_n\}\) is a random sample of \(n\) values from any distribution with mean \(\mu\) and variance \(\sigma^2\),
\[\begin{align} \sum_{i=1}^n {X_i} & \;\; \xrightarrow[n \rightarrow \infty]{} \;\; \NormalDistn(n\mu, \;\;\sigma_{\Sigma X}^2=n\sigma^2) \\ \overline{X} & \;\; \xrightarrow[n \rightarrow \infty]{} \; \; \NormalDistn(\mu, \;\;\sigma_{\overline X}^2=\frac {\sigma^2} n) \end{align} \]This way of writing the Central Limit Theorem describes how it is interpreted in practice, but the following is a more precise statement of the result.
Central Limit Theorem (more precise)
If \(\{X_1, X_2, ..., X_n\}\) is a random sample of \(n\) values from any distribution with mean \(\mu\) and variance \(\sigma^2\),
\[ Z_n = \frac {\sum_{i=1}^n {X_i} - n\mu} {\sqrt{n}\; \sigma} \quad \xrightarrow[n \rightarrow \infty]{} \quad \NormalDistn(0,\; 1) \]The Central Limit Theorem is the main reason why the normal distribution is so important in statistics. Sample means are approximately normal, whatever the distribution from which the values are sampled. Many other summary statistics from large random samples also have approximately normal distributions.