Median Calculator

Definition

Let x₁, …, xₙ be a finite sample of real numbers, and let x₍₁₎ ≤ x₍₂₎ ≤ … ≤ x₍ₙ₎ be their ordered values (order statistics). The sample median Me is defined in two branches: when n = 2k+1 is odd, Me = x₍k+1₎; when n = 2k is even, Me = (x₍k₎ + x₍k+1₎)/2. That is the average-of-two-middles convention used in descriptive-statistics textbooks and in the NIST/SEMATECH e-Handbook of Statistical Methods. The median is the empirical ½-quantile (second quartile Q₂).

This page’s calculator accepts a list of any length in one text field, parses numbers (space, semicolon, comma as separator or decimal), sorts ascending, and applies the definition above. Computation: numeric sort, then either the middle element or the average of the central pair. It is not a density estimator or a weighted median — every successfully parsed value has weight 1.

In teaching practice it helps to separate the mathematical formula from floating-point limits: this calculator shows the browser’s floating-point computation (about 15–16 significant digits), not a paper proof of symbolic identities. When inputs are extreme, check the domain first, then the floating-point range.

Historical and conceptual context

The median as a “middle value” appears in nineteenth-century descriptive statistics beside the mean and the mode (measures of location). Francis Galton and later authors stressed resistance to extremes when describing incomes, heights, or test scores. In estimation theory the sample median estimates the population median; under a Laplace distribution it is more efficient than the mean. In robust statistics the median is an M-estimator with a high breakdown point (50%).

Conceptually the median is not the arithmetic, geometric, or harmonic mean. It is also not the midrange (average of min and max). It does not sort “by eye” — it needs a full ordering or an equivalent selection algorithm. This calculator does not compute confidence intervals or sign tests; it shows the point median of the entered list.

The sibling “median of five numbers” page has fixed n = 5 and five fields. Here n is free: a school list, grades, measurements. For the mean of the same list use the arithmetic-mean page.

Formal properties

Input: n ≥ 1 finite real numbers. Output: Me ∈ ℝ. Monotonicity: if every xᵢ rises (or does not fall), the median does not fall. Translation: Me(xᵢ + c) = Me(xᵢ) + c. Positive scaling λ > 0: Me(λx) = λ Me(x). For λ < 0 the order reverses and the median scales by λ (after sorting the scaled values).

Robustness: changing fewer than half the observations cannot move the median outside the range of the remaining values (breakdown point 50% for odd n in the classical sense). The mean has breakdown 0% — one infinite observation destroys the result. For 1 1 2 3 100 the median is 2 and the mean is 21.4.

Link to quantiles: Me = Q₀.₅. For odd n the median coincides with the middle element. For even n averaging the two middles is one of several software conventions; this calculator uses that classical average.

Full derivation

Step 1. Collect tokens from the list field and parse finite numbers into L = [y₁, …, yₙ], n ≥ 1.

Step 2. Order them: x₍₁₎ ≤ … ≤ x₍ₙ₎ (comparison sort on the number).

Step 3. If n is odd, let k = (n−1)/2 (zero-based). Then Me = x₍k+1₎ = L_sorted[k]. Example: n = 5, list 12 7 9 3 18 → 3, 7, 9, 12, 18 → Me = 9.

Step 4. If n is even, let i = n/2 − 1 and j = n/2. Then Me = (x₍i+1₎ + x₍j+1₎)/2. Example: n = 4, list 4 10 2 8 → 2, 4, 8, 10 → Me = (4+8)/2 = 6.

Step 5. Existence of the ordering follows from the total order on ℝ. The average of two middles is well-defined in ℝ. The browser result is a floating-point number.

Special cases

Examples: 12 7 9 3 18 → 9; 4 10 2 8 → 6; 10 12 11 1000 → 11.5; 2 5 9 11 20 → 9; 1 1 2 3 100 → 2; 10 10 10 10 50 → 10; 3 6 9 12 → 7.5; −2 0 4 8 10 → 4. Singleton: one number is its own median. All equal: Me equals that value.

Empty list: no result. Non-numeric tokens are skipped. Decimals with a period or a comma are equivalent. Zeros and negatives enter the sort like any other value.

When n = 2, the median is the average of both elements — the same as the midrange for a two-point sample. For larger n midrange and median diverge.

Numerical and computational notes

The calculator sorts the values as numbers, not as text. The sum of two middles divided by 2 is exact for typical school data; at extreme values remember the limited floating-point precision. Display formatting (en-US / pl-PL) does not change the internal value.

Parser: spaces and semicolons always separate; several commas without spaces treat the comma as a separator; a single comma inside a token is the decimal mark. The axis diagram marks the points and the median.

The calculator does not weight observations, does not auto-drop outliers, and does not compute an interquartile range — those belong to other tools.

Related notions and sources

Related: arithmetic mean, five-field median, min/max and range, quartiles, percentiles, mode. In estimation theory — efficiency of the median versus the mean under different distributions.

Sources: NIST/SEMATECH e-Handbook — Percentiles / measures of location; Wikipedia — Median; MathWorld — Statistical Median; intro-stats textbooks (order statistics). Educational tool — not a textbook substitute or a full research-sample analysis.