Mean Median Mode Calculator

Type or paste a list of numbers and the calculator returns the mean, median and mode at once, along with the count and the sum. Separators are flexible: commas, spaces, tabs and line breaks all work, and anything that is not a number gets skipped, so a spreadsheet column pastes in without cleanup. Read the three averages together rather than one at a time. When they land close together the data is roughly symmetric; when they pull apart, the gap between them is itself a clue to the shape of the distribution.

How each measure is calculated

The mean is the total divided by how many values there are:

mean = (x₁ + x₂ + … + x_n) ÷ n

The median is found by sorting the list and taking the middle value; with an even count there is no single middle, so the two central values are averaged. In a sorted list of n values the median sits at position (n + 1) ÷ 2. For five values that points at the third value exactly; for six values it points halfway between the third and fourth, which is why an even count forces the average of the two neighbours. The mode is the value that appears most often, so it is found by counting how many times each distinct value occurs and keeping whichever count is highest. A list can have one mode, several tied modes, or none worth reporting.

The default list — 12, 7, 3, 7, 9 — makes a compact worked example. The sum is 12 + 7 + 3 + 7 + 9 = 38 and the count is 5, so the mean is 38 ÷ 5 = 7.6. Sorting gives 3, 7, 7, 9, 12, and the third of the five values is the median: 7. The value 7 appears twice while every other value appears once, so the mode is also 7. Three measures, two different answers — which is why it pays to look at more than one.

Where the three averages came from

The arithmetic mean is the oldest of the three in practice, though it was not always trusted. Combining several measurements into a single number by adding them and dividing was resisted for a long time, because a single careful reading seemed safer than a blurred pile of them. Although the Greeks knew the idea, it was not generalised beyond two values until the sixteenth century, and astronomy is where the habit changed: Tycho Brahe took the mean of repeated observations to cut the errors in his estimates of the positions of celestial bodies, and Simon Stevin's decimal notation of 1585 made the arithmetic of doing so far less painful. The English word "average" has nothing to do with arithmetic. It descends from the Arabic ʿawār, meaning defect or damage, which travelled through Mediterranean sea trade into the Latin avaria used in twelfth- and thirteenth-century Genoa for losses at sea. Under the maritime rule still called general average, cargo thrown overboard to save a ship was paid for by all the owners in proportion to their stake, and that idea of spreading one party's loss across everyone with a share in the voyage is what eventually attached the word to a single levelled-out figure standing in for a whole set of numbers.

The median arrived later and under several names, though the underlying idea is much older: the Talmud already handled a set of divergent appraisals by taking the middle of them. Edward Wright, in his 1599 book Certaine Errors in Navigation, may have been the first to describe choosing the middle of a set of readings, in a passage on compass navigation. Roger Joseph Boscovich leaned on the same idea in 1757 while fitting a line to observations, and in 1774 Pierre-Simon Laplace argued explicitly for the middle value, on the criterion that it minimises the expected magnitude of the error. The term itself is younger: the French mathematician Antoine Augustin Cournot wrote of the valeur médiane in 1843, Gustav Fechner put the median to work in psychology and social measurement, and Francis Galton settled on the English word "median" in 1881 after trying "middle-most value" and "medium" in earlier years.

The mode carries the newest label of the three. Karl Pearson introduced it in 1895, writing that he had found it convenient to use the term "mode" for the value at which a frequency curve reaches its peak — the most common reading in a set. The umbrella phrase that covers all three, "central tendency", is younger still, dating only from the late 1920s, long after each individual measure was in daily use.

When the median beats the mean

The mean is sensitive to outliers; the median is not. Picture a ten-person company where nine people earn 45,000 and the founder takes 500,000. The mean salary is 905,000 ÷ 10 = 90,500 — a figure nobody at the company actually earns — while the median is 45,000, which describes the typical employee perfectly. Any skewed distribution behaves this way: incomes, house prices, insurance claims, city populations. A single very large value drags the mean toward itself while barely moving the median, because the median responds only to the rank of a value, not its size.

Official statistics lean on the median for this reason. The US Census Bureau reports median household income, and the UK Office for National Statistics quotes median weekly pay, precisely because a handful of very high earners would drag the mean well above what a typical household sees. When a headline says "average", it is worth checking which measure was used — the same figures can look prosperous or precarious depending on the choice.

Multimodal lists and lists with no mode

If two values tie for the highest frequency the data is bimodal, and the calculator lists every tied value: 2, 4, 4, 6, 6, 8 has modes 4 and 6. This shows up when data mixes two distinct groups, such as adult shoe sizes drawn from both men and women, where two peaks reflect two underlying populations rather than one. When no value repeats at all, the result reads "no repeats" rather than pretending every value is a mode — with continuous measurements such as weights or reaction times, exact repeats are rare and the mode stops being informative. The mode is also the only one of the three that works on categories as well as numbers: the most common blood type or the best-selling colour has a mode but no mean.

Assumptions worth knowing

The mean here is the arithmetic mean, the everyday sum-and-divide. It is one of the three classical Pythagorean means; the geometric mean, which multiplies the values and takes a root, suits growth rates and compounding, while the harmonic mean suits average speeds and other rates, and on the same positive data neither of them can exceed the arithmetic mean, matching it only when every value in the list is identical. The calculator treats the pasted list as the complete data set, so nothing is weighted, grouped or interpolated. Schoolwork on grouped data goes the other way: reading a median off a cumulative frequency graph, as UK GCSE maths does, returns an estimate rather than an exact figure, because the individual figures inside each class interval have already been discarded. Results here are computed at full floating-point precision and the displayed figures are rounded to at most four decimal places, so a value shown as a round number may still carry hidden fractions when it feeds into a later calculation.