Median (measure of central tendency)

How the median is computed

To compute the median of a finite list, first sort the values from smallest to largest. If the list length n is odd, the median is the middle entry at position (n+1)/2 in the sorted sequence. If n is even there are two common conventions: take the arithmetic mean of the two central values, or choose one of the two central values (for example, the smaller or the larger). Using the mean of the middle pair makes the median a unique real number even when the dataset contains only integers; the alternative integer-valued convention keeps the median equal to an observed datum.

For continuous distributions the median is the value m for which the cumulative distribution function equals 0.5. In practice, for grouped or binned data one can interpolate between class boundaries to estimate the median. For weighted samples, a weighted median generalizes the idea by finding a cutpoint that balances total weight on either side.

Properties and robustness

The median is robust: it is far less affected by extreme values or heavy-tailed observations than the arithmetic mean. This makes it a preferred measure of central tendency for skewed distributions, such as income or property values, where a small number of large observations can distort the mean. Statistically, the sample median is an estimator of the population median and, under mild regularity conditions, is consistent and asymptotically normal.

Variants, applications and examples

• Simple example: the median of {2, 5, 7} is 5; the median of {2, 5, 7, 9} is (5+7)/2 = 6.
• Weighted median: useful when each observation carries a different importance or frequency.
• Geometric median and spatial median: extensions used in multivariate data to minimize total distance to all points.
• Median filters: a common nonlinear smoothing tool in signal and image processing that removes impulsive noise without blurring edges as much as a linear average.

The median is widely used in descriptive reports (median household income, median house price), in robust statistical procedures, and in decision rules that seek a representative central outcome less sensitive to outliers than the mean. It should be reported together with dispersion measures (interquartile range, median absolute deviation) to convey spread.

History, notation and notable distinctions

The notion of a middle or dividing value has long been part of descriptive analysis; the modern formal use of the term "median" and its systematic study were established in the 19th and early 20th centuries as statistics developed as a discipline. Notation varies: the median of a variable X is often written med(X) or ~X. Unlike the mean, which minimizes squared deviation, the median minimizes the sum of absolute deviations and therefore embodies a different optimality criterion.

Related topics and further reading: probability theory, statistics, number, finite, odd number, even number, mean, ordinal-valued, sample, biased, statistical estimator.