Median

A median is a point with at least half the probability on either side.

A median of a weighted probability distribution. With positive weights summing to one, at most half the weight lies strictly below a weighted median and at most half lies strictly above it. The lower weighted median chooses the first observed value whose cumulative weight reaches one half; the convention matters when the cumulative mass is exactly one half.

For ordered distinct values with positive weights summing to one, choose the first value whose cumulative weight reaches one half. The preceding cumulative weight is strictly below one half, and the remaining mass is at most one half, proving the displayed conditions. Stopping on equality chooses the lower value when a cumulative weight is exactly one half. Sorting followed by a scan takes $O(n\log n)$ time; median of medians partitioning with a residual weight target gives deterministic linear time.

For a finite first absolute moment, the minimizers of expected absolute loss are the medians. With a density, differentiating the loss gives $2F(a)-1$, so every point where the cumulative distribution function equals one half minimizes it. A density gap may give a whole minimizing interval. In contrast, finite squared loss is uniquely minimized by the expectation.

The sample median is a central value of an ordered statistical sample. The usual numerical convention selects the middle order statistic for odd sample size and the arithmetic mean of the two central order statistics for even sample size:NIST uses this convention. ^[ref]. For $(0.7,0.1,0.9,0.4,0.2)$, the sample median is $0.4$; for $(0.1,0.2,0.4,0.7)$, it is $(0.2+0.4)/2=0.3$, which need not be an observed value.

More generally, every value between the two central order statistics in an even statistical sample is a median of its empirical distribution function. The generalized-inverse sample quantile at $p=1/2$ selects the lower central order statistic in that case, so it can differ from the midpoint convention. Ties are allowed in all these definitions.

The sample median measures the center using sorted positions. Changing the largest observation from $0.9$ to $9$ in the five-value example leaves the sample median at $0.4$, while the sample mean rises from $0.46$ to $2.08$. Its sampling distribution still depends on the population probability distribution: the mean $1/2$ found for five observations from a uniform distribution is a special calculation, and a sample median's expected value need not equal the population median in general.

For a concrete counterexample, three independent and identically distributed random variables with rate-one exponential distribution have sample median $X_{(2)}$ with expected value $1/3+1/2=5/6$, by independent exponential order-statistic spacings. The population median instead solves $1-e^{-m}=1/2$, giving $m=\log 2\approx 0.693147$. Thus even the middle order statistic can have mean different from the population median.

The coordinatewise median of vectors applies the univariate sample median separately to each coordinate. It is easy to compute but depends on the chosen coordinate axes.