Sample Mean¶

Definition¶

For random variables $X_1,\ldots,X_n$ and $n\ge1$$1$, their sample mean is

The sum adds the $n$ observations before dividing by $n$. No independence assumption is needed to define this arithmetic average.

Interpretation¶

The sample mean summarizes $n$ observations by their arithmetic average. Before the data are observed, $\bar X_n$ is a random variable with its own expectation, variance and sampling distribution; the observed average $\bar x_n$ is one realized number.

The figure marks eight observations on a number line. Their mean $\bar x=(2.1+3.4+3.9+4.6+5.0+5.8+6.7+8.5)/8=5.0$ is the balance point: the deviations to the left and right cancel, $\displaystyle \sum_i(x_i-\bar x)=0$$0$. A single far-away observation moves the balance point; replacing $8.5$ by $16.5$ raises the mean by $1$ to $6.0$.

Two properties make $\bar X_n$ a useful estimate of the population mean $\mu$. It is centred correctly, $\mathbb{E}\bar X_n=\mu$$\mu$, which needs only linearity of expectation. With independent observations of common variance $\sigma^2$, its variance $\sigma^2/n$ shrinks, so the standard deviation $\sigma/\sqrt n$ halves when the sample size is multiplied by four. Without independence the second property can fail: if all $X_i$ equal $X_1$, then $\bar X_n=X_1$ has variance $\sigma^2$ for every $n$.

These two properties lead to the weak and strong law of large numbers, which say that $\bar X_n$ approaches $\mu$, and to the central limit theorem, which describes the shape of the remaining error. A common confusion is between $\bar X_n$ and $\mu$: the sample mean is computed from data and varies from sample to sample, while $\mu$ is a fixed property of the distribution.

Key Results¶

• If every $X_i$ has mean $\mu$, then $\mathbb{E}\bar X_n=\mu$ without independence.
• If the observations are independent with common variance $\sigma^2$, then $\operatorname{Var}(\bar X_n)=\sigma^2/n$${\sigma }^2/n$.

Examples¶

• For observations $2,4,9$, the observed sample mean is $(2+4+9)/3=5$$5$.
• For $100$ i.i.d. $\operatorname{Exp}(2)$ observations, the sample mean has expectation $1/2$ and variance $1/(4\cdot100)=1/400$$1/400$.

Further Properties¶

For realized observations $x_1,\ldots,x_n$, the observed average is $\displaystyle \bar x_n=n^{-1}\sum_i x_i$$n^{-1}\sum _i{x}_i$. A sample proportion is the sample mean of success indicators.

Prerequisites¶

• Addition and Sum
• Random Variable

Related Concepts¶

• Expectation

Sources¶

• Probability Distributions · supporting concept

Aliases: sample average, empirical mean, X bar Symbols: mu, sigma, square root, sum

Last updated: 20 October 2026 03:02