The middle value that extreme outliers barely touch
The median splits ordered data into equal upper and lower halves. Unlike the mean, a few extreme values cannot yank it away—so median income often describes a center that billionaire spikes leave alone. That robustness is why it anchors modern descriptive statistics.
For a finite list, arrange values from smallest to largest; with an odd count, the middle entry is the median—six is the median of a seven-number list when it sits fourth. With an even count, convention usually averages the two central values. For a population, any value such that at least half the mass is at most that value and at least half is at least that value qualifies; medians need not be unique. The median is a 2-quantile, also the second quartile, fifth decile, and fiftieth percentile. It works for ordered grades from F to A even when scores are not numeric.
Compared with the arithmetic mean, the median resists skew and dubious outliers, which is why robust statistics prizes it. Extreme results need not even be known: if a few subjects never finish a timed task, a median completion time can still be computed. Notation varies—med(x), a tilde over x, μ₁/₂, or M—so authors must define their symbol. Sample medians are reasonably efficient estimators; for large normal samples the median’s variance is about fifty percent larger than the mean’s, corresponding to roughly sixty-four percent efficiency relative to the minimum-variance mean.
For continuous densities a median m satisfies the probability of being at most m equal to one half. Symmetry simplifies special cases: a symmetric unimodal law puts median at the mode; a symmetric law with mean μ has median μ; a normal with mean μ and variance σ² has mean, median, and mode all equal to μ. A uniform on [a, b] has median (a + b)/2. The Cauchy distribution may lack a mean yet still has median equal to its location parameter x₀. Power-law tails with exponent a greater than one yield an explicit median in terms of a and the minimum support point. The popular slogan that the mean always sits “further into the tail” than the median is not generally true—only that the two cannot drift arbitrarily far apart under stated inequalities.
Source: Median
