Sample Mean

probability-theory

Definition

Sample Mean

For a random sample $X_1,X_2,\dots ,X_n$, the sample mean is defined as:

$$\overline{X_n}=\frac{1}{n}\sum _{i=1}^n{X}_i$$

Properties

Unbiasedness

The sample mean is an unbiased estimator for the expectation of each identically distributed random variable:

$$\mathbb{E}[\overline{X_n}]=\mathbb{E}[X_i]\forall i\in \{1,2,\dots ,n\}$$

Variance

The variance of a sample mean is given by:

$$\mathbb{V}ar(\overline{X_n})=\frac{{\sigma }^2}{n}$$

Normal

If a sample is obtained from a Gaussian distribution, then the sample is also normal-distributed.

$$X_1,X_2,\dots ,X_n\stackrel{iid}{\sim }\mathcal{N}(\mu ,{\sigma }^2)⟹\overline{X_n}\sim \mathcal{N}\left(\mu ,\frac{{\sigma }^2}{n}\right)$$