Xbar

Xbar, written as x-bar, is the symbol commonly used in statistics to denote the sample mean of a set of observations. It serves as a measure of central tendency for the data and is denoted by x̄. If a sample contains n observations x1, x2, ..., xn, the x-bar is computed as the average: x̄ = (1/n) Σ x_i.

As an estimator, x̄ estimates the population mean μ. If the underlying population has finite variance σ^2 and the sample consists of independent and identically distributed observations, then E[x̄] = μ, so x̄ is an unbiased estimator. The variance of x̄ is Var(x̄) = σ^2/n, meaning the precision of the estimate improves with larger samples. For normally distributed populations, x̄ is itself normally distributed with x̄ ~ N(μ, σ^2/n). By the central limit theorem, this approximate normal behavior holds for large n even when the population is not normal.

In practice, x̄ is used to construct confidence intervals for μ. When the population standard deviation σ is known, a z-interval x̄ ± z_{α/2} (σ/√n) is used; when σ is unknown, a t-interval x̄ ± t_{α/2, n-1} (s/√n) is employed, where s is the sample standard deviation. X̄ also appears in quality control, where the X-bar chart monitors the process mean over time.

Variations include treating x̄ as a vector for multivariate data, in which case it represents the mean of each variable. While simple and interpretable, x̄ is sensitive to outliers and may be less robust than alternative measures of central tendency.