Audited 26 May 2026·Last updated 27 Jul 2026·7 citations·Tier 1·0 uses

Mean, Median, and Mode Calculator

Free mean, median, and mode calculator: paste a dataset and get the arithmetic mean, median, mode(s), range, and count instantly, with worked steps and theory.

Mean, Median, and Mode Calculator

Paste your numbers as a JSON array (e.g. [1, 2, 3, 4, 5]) or as a plain comma-separated list (e.g. 1, 2, 3, 4, 5). Spaces and line breaks are tolerated. You need at least one value, but the statistics get more meaningful from about ten values upward.
Mean (average)
5
The arithmetic mean — the sum of every value divided by the number of values. The most widely used measure of central tendency, but also the most sensitive to outliers and skew.
Median
5
Mode
5
Range
8
Count (n)
10

Background.

The Quanta mean, median, and mode calculator takes any list of numbers — pasted as a JSON array, copied out of a spreadsheet column, or typed in as a comma-separated line — and returns the three classical measures of central tendency together with the range and the sample size. The arithmetic mean is what most people call the "average": add every value, divide by how many there are. The median is the middle value once the data is sorted from smallest to largest, the single number with as much data above it as below. The mode is the value that occurs most often — and unlike the other two it can be undefined (when every value is unique) or multi-valued (when several values tie for the most frequent).

Knowing all three at once matters because each one tells a different story about the same data, and the gap between them is itself diagnostic. When the mean and median sit on top of each other, the distribution is roughly symmetric and any of the three is a reasonable summary. When the mean is dragged noticeably above the median, the data is right-skewed — a long tail of large values is pulling the average up, the way a single billionaire pulls up the mean income of a small town while leaving the median untouched. When the mean sits below the median, the data is left-skewed — there is a long tail of small values, characteristic of test scores with a ceiling, response times measured against a deadline, or ages at retirement.

The U.S. Census Bureau, the IRS, and almost every economic statistics agency reports income with the median rather than the mean precisely because incomes are heavily right-skewed: the 2024 U.S. real median household income was $80,610 while the mean was over $114,000 — the $33,000 gap is the contribution of the right tail and is exactly the reason "average household income" headlines can be misleading.

This page goes well beyond the arithmetic. It walks through when each measure is the right one — mean for symmetric, interval-scaled data; median for skewed or ordinal data and any time outliers are present; mode for categorical, nominal, or strictly discrete data — and how to read bimodal distributions, where two distinct peaks suggest that two different populations have been pooled into the same dataset (a classic example is the bimodal distribution of body temperature once you mix men and women, or the bimodal distribution of geyser eruption intervals at Old Faithful). It explains the difference between population and sample statistics, why the median is a 50th-percentile quantile and how that generalises to quartiles and percentiles, and why range is a useful but fragile companion to standard deviation. Worked examples cover test scores, house prices (where the median is the only honest summary), response times in user-experience research, and the Old Faithful eruption-interval dataset that statistics professors love because its bimodality jumps off the page.

Whether you arrived here to finish a homework set, sanity-check a spreadsheet column, decide which average to put in a report, or understand why the U.S. Census never reports a mean income, the calculator above and the explainer below cover every variant of the question.

What is mean, median, and mode calculator?

Mean, median, and mode are the three classical measures of central tendency — single numbers that summarise where a dataset "sits" on the number line. The arithmetic mean is the sum of the values divided by the count, written x̄ = (Σxᵢ)/n for a sample of size n. The median is the value at the 50th percentile of the sorted data — the middle observation when n is odd, or the average of the two middle observations when n is even. The mode is the value (or values) with the highest frequency of occurrence. Each measure answers a slightly different question about the data. The mean answers "what value, if every observation were equal to it, would give the same total?" — making it the natural summary whenever totals matter (total revenue, total calories, total rainfall). The median answers "what value splits the dataset in half?" — making it the natural summary whenever rank matters more than magnitude (incomes, house prices, lap times). The mode answers "what is the single most common observation?" — the only one of the three that is meaningful for categorical data like favourite colour, blood type, or device manufacturer. The three measures coincide exactly for a perfectly symmetric, unimodal distribution like the normal bell curve. They diverge as soon as the distribution becomes skewed: for a right-skewed distribution the order is mode < median < mean; for a left-skewed distribution it is mean < median < mode. That ordering is so reliable that the gap between the mean and the median is itself a standard diagnostic for skewness, and Karl Pearson's classical skewness coefficient is literally defined as 3·(mean − median)/standard_deviation. Range, reported alongside the three measures, is the simplest measure of dispersion: max − min. It is intuitive and easy to compute, but it depends only on the two most extreme observations, which makes it very sensitive to outliers — a single mis-typed value can multiply the range by a factor of ten while leaving the median and the interquartile range untouched. For that reason serious work supplements the range with the interquartile range and the standard deviation, both of which use more of the data and are correspondingly more stable.

How to use this calculator.

  1. Paste your dataset into the input box. Either format works: a JSON array like [12, 14, 14, 17, 19, 21, 21, 21, 25] or a plain comma-separated list like 12, 14, 14, 17, 19, 21, 21, 21, 25. Whitespace and line breaks between values are ignored.
  2. Verify the parsed count (n) in the results panel matches the number of values you intended to enter. A mismatch usually means a stray character — a letter, an unmatched bracket, two commas in a row — confused the parser.
  3. Read the primary result (the mean) at the top of the results panel and the median, mode, range, and count in the breakdown. Compare the mean and median: if they're close, your data is roughly symmetric; if the mean is noticeably larger, the data is right-skewed; if it's noticeably smaller, it's left-skewed.
  4. If the dataset has no repeated values the calculator reports "no mode" — every observation occurs exactly once, so no single value is more common than any other. If two or more values tie for the highest frequency the dataset is multimodal and you should plot a histogram rather than trust any single summary number.
  5. Use the range as a quick reasonableness check. A range that is much larger than you expected is a strong signal that an outlier or data-entry error has slipped in. Sort the values and inspect the extremes before drawing conclusions.
  6. For inferential statistics (confidence intervals, hypothesis tests) move on to the standard-deviation calculator — central tendency alone is never the full picture. Always pair a measure of centre with a measure of spread.

The formula.

x̄ = (x₁ + x₂ + ⋯ + xₙ) ⁄ n
x̄ · arithmetic meanxᵢ · data valuesn · count of values

Three short procedures, one each. Mean: sum every value, then divide by n. Formally x̄ = (x₁ + x₂ + … + xₙ) / n, or in summation notation x̄ = (1/n)·Σᵢ₌₁ⁿ xᵢ. The calculator computes this in a single linear pass over the data. Median: sort the values in ascending order. If n is odd, the median is the value at position (n+1)/2 in the sorted list (1-indexed). If n is even, the median is the arithmetic mean of the two values at positions n/2 and n/2 + 1. For example, sorted data [1, 3, 5, 7, 9] has median 5 (the third of five values); sorted data [1, 3, 5, 7] has median (3 + 5)/2 = 4. The median is the special case of a quantile at probability 0.5; the first quartile (Q1) is the same idea at probability 0.25, the third quartile (Q3) at probability 0.75. Mode: count the frequency of every distinct value. The mode is the value with the highest frequency. If every value occurs exactly once there is no mode (some textbooks say "every value is a mode" — the practical interpretation is the same: the mode is uninformative). If two values tie for the highest frequency the distribution is bimodal; three or more, multimodal. This calculator returns the smallest modal value as a single number for downstream use and surfaces a multimodal warning in the breakdown text. Range: simply max(values) − min(values). It is the simplest measure of dispersion, computed in a single pass, and equal to the width of the smallest closed interval that contains every observation. Internally the calculator parses your input as JSON first, falls back to splitting on commas if JSON parsing fails, coerces every token to a number, and rejects any input containing a non-numeric token (so a typo like "5.0o" raises an error instead of being silently dropped). Empty datasets are rejected because mean, median, and mode are all undefined for n = 0.

A worked example.

Example

A small ten-value dataset that nicely shows all three measures at once. The sum is 1 + 2 + 3 + 4 + 5 + 5 + 6 + 7 + 8 + 9 = 50, so the mean is 50 / 10 = 5.0. The data is already sorted; with n = 10 (even), the median is the average of the fifth and sixth values, both of which happen to be 5 here, so the median is also 5.0. The value 5 appears twice while every other value appears once, so the mode is 5. The range is 9 − 1 = 8. All three measures of central tendency agree exactly because the data is symmetric around 5 and unimodal. Contrast that with what happens if you replace the largest value with an outlier — say, change 9 to 90. The new sum is 131, so the mean jumps to 13.1, but the median is still 5 (it only depends on the middle ranks, not the magnitudes), and the mode is still 5. That single substitution is the entire story of why economists report median income rather than mean income: one billionaire in a town of a hundred people can push the mean income up by tens of thousands of dollars while leaving the median income unchanged. The mean answers a question about totals; the median answers a question about typical experience; for skewed real-world distributions those are usually different questions with different answers.

values Json[1, 2, 3, 4, 5, 5, 6, 7, 8, 9]

Frequently asked questions.

What is the difference between mean, median, and mode?
All three are measures of central tendency — single numbers that summarise where a dataset sits — but they answer different questions. The mean is the arithmetic average: sum of values divided by count. The median is the middle value once the data is sorted. The mode is the most frequently occurring value. For a perfectly symmetric, unimodal dataset (like the normal bell curve) the three coincide. For skewed data they diverge in a predictable order: in a right-skewed distribution mode < median < mean, and in a left-skewed distribution mean < median < mode. The mean uses every value, so it is sensitive to outliers; the median uses only ranks, so it is robust; the mode uses only frequencies, so it works on categorical data where mean and median are meaningless.
When should I use the mean versus the median?
Use the mean when the data is roughly symmetric, when there are no extreme outliers, and when the underlying quantity is on an interval or ratio scale where totals matter — heights, weights, temperatures, exam scores out of 100, daily rainfall. Use the median when the data is skewed, when outliers are present or possible, or when the variable is ordinal (rankings, Likert-scale responses). House prices, household incomes, response times in software latency analysis, hospital lengths of stay, and lifetime earnings are all classic median-not-mean variables — every one of them has a long right tail of high values that drags the mean up and away from the typical experience. The U.S. Census Bureau, the IRS, the OECD, and most national statistical agencies report median household income rather than mean household income for exactly this reason.
How do outliers affect the mean and median differently?
The mean is extremely sensitive to outliers; the median is almost completely immune. Concretely: take the dataset {1, 2, 3, 4, 5}. Mean = 3, median = 3. Now replace the 5 with 1,000. New mean = 202; new median = 3. A single outlier moved the mean by a factor of nearly seventy while leaving the median unchanged at its original value. This robustness comes from the fact that the median depends only on ranks, not magnitudes — it is the 50th percentile, and percentiles are unchanged by any monotonic transformation of the extremes. This is exactly why income inequality discussions use the median: in 2024 U.S. real median household income was about $80,610 while the mean was over $114,000. The $33,000 gap is entirely the contribution of the top of the income distribution. Using the mean as the "typical" income would systematically misrepresent what most households actually earn.
When is the mode the right measure of central tendency?
The mode is the right choice — and often the only valid choice — for nominal categorical data, where the values are labels rather than numbers. "Most common blood type," "most-purchased product colour," "most-used smartphone OS," and "most common diagnosis in a clinic" are all mode questions; mean and median are simply undefined on data that has no natural order. The mode is also useful for strictly discrete numeric data with a small set of possible values — die rolls, shoe sizes, number of children per household, exam scores out of 10 — and for any distribution where the most-likely outcome is itself the question of interest (the mode of a probability distribution is its peak). The mode is less useful for continuous data, because in a truly continuous sample every observation is essentially unique and the empirical mode is noisy; for continuous data, people usually plot a kernel density estimate and locate its peak instead.
What does it mean when a dataset is bimodal?
A bimodal dataset has two distinct peaks in its frequency distribution — two values (or two regions of values) that occur more often than the values immediately surrounding them. Bimodality is almost always a sign that the dataset is a mixture of two underlying populations that should not have been pooled. Classic real-world examples: human body temperature is bimodal when men and women are combined into one sample because the mean differs slightly between sexes; the eruption intervals of the Old Faithful geyser in Yellowstone are sharply bimodal with peaks near 55 minutes and 80 minutes (the geyser has two distinct eruption regimes); reaction-time data in a perceptual experiment is often bimodal because some trials are processed by a fast "automatic" route and others by a slower "controlled" route. When you see bimodality, do not collapse the data to a single mean — instead, plot a histogram, identify the two populations, and analyse them separately. Reporting a single mean of a bimodal distribution is one of the most common ways to summarise data badly.
Why is median household income lower than mean household income?
Because the household-income distribution is heavily right-skewed: most households cluster near the middle of the income range, but a small number of households earn very large amounts that pull the mean upward without affecting the median. In 2024, the U.S. real median household income was $80,610 (Census Bureau, Current Population Survey ASEC) while the mean was substantially higher, over $114,000. That ~$33,000 gap is the entire contribution of the right tail — the top 5–10% of earners — to the average. The median is the more honest summary of "typical" household income precisely because it is immune to the long tail; the mean is the right number only if you are computing a total (e.g., total national income equals mean income times number of households). Every major statistical agency reports income as a median for this reason.
Can a dataset have no mode?
Yes. If every value in the dataset occurs exactly once, no single value is more common than any other, and the mode is undefined (or, equivalently, every value is a mode — the practical interpretation is the same: the mode carries no information). This is the typical situation for continuous measurements: heights, weights, voltages, and reaction times measured to high precision rarely have exact ties, so the empirical mode is meaningless. For continuous data the conventional substitute is to plot a histogram or a kernel density estimate and identify the value where the smoothed frequency is highest — that smoothed mode is well-defined and useful even when no two raw observations are exactly equal. The Quanta calculator reports the smallest modal value when there is a tie and surfaces a "no mode / multimodal" flag in the breakdown when the data has no unique mode.
What is the range and how does it relate to standard deviation?
The range is the simplest measure of dispersion: max − min. It tells you the width of the smallest closed interval that contains every observation. The range is intuitive and easy to compute by hand, which is why it is taught first, but it has two serious weaknesses: it depends only on the two most extreme observations, so it ignores the bulk of the data, and it is extremely sensitive to outliers (a single mistyped value can multiply it by ten). Standard deviation, in contrast, uses every observation and measures the typical distance from the mean — it is more stable and more informative, especially for inferential statistics where it feeds into confidence intervals, t-tests, and z-scores. Use the range as a quick reasonableness check ("is the spread roughly what I expect?") and then compute the standard deviation for any serious analysis. For skewed data, the interquartile range (Q3 − Q1) is a third option that is both robust and uses more of the data than the range.
How do mean, median, and mode help diagnose skewness?
By their relative ordering. For a perfectly symmetric, unimodal distribution the three measures coincide. For a right-skewed distribution (a long tail of large values, like income or house prices) the order is mode < median < mean: the mean is pulled up by the tail, the mode sits at the dense bulk on the left, and the median sits between them. For a left-skewed distribution (a long tail of small values, like test scores with a ceiling effect, or age at retirement) the order reverses: mean < median < mode. Karl Pearson's classical skewness coefficient is defined directly from this gap as Skew = 3·(mean − median)/standard_deviation, giving a single signed number whose sign matches the direction of skew and whose magnitude grows with severity. So just by computing mean, median, and mode and comparing them you get a free first diagnostic of distribution shape — no histogram required.
What is the difference between a sample mean and a population mean?
A population mean (denoted μ, Greek mu) is the average over every member of the population of interest — a fixed but usually unknown number. A sample mean (denoted x̄, x-bar) is the average over a finite sample drawn from that population — a random variable whose value depends on which sample you happened to draw. The two are computed with the same arithmetic — sum divided by count — but they play different roles. The sample mean is what you actually compute from your data; the population mean is what you usually want to estimate. The Central Limit Theorem tells you that for sufficiently large samples the sample mean is approximately normally distributed around the population mean with standard deviation σ/√n, which is the basis of every confidence interval and t-test you will ever see. The same distinction applies to median and standard deviation: there is a population version (a parameter) and a sample version (an estimator). This calculator returns sample statistics — the values computed from the data you supplied.

References& sources.

  1. [1]NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5 — Measures of Location and Variability. Authoritative definitions of mean, median, mode, and range with worked examples; co-published by the U.S. National Institute of Standards and Technology.
  2. [2]Casella, George and Berger, Roger L. Statistical Inference, 2nd edition. Duxbury / Cengage, 2002 — the standard graduate-level reference for descriptive and inferential statistics. Chapter 5 covers sample mean, sample median, and order statistics rigorously.
  3. [3]OpenStax. Introductory Statistics. Rice University, 2022 — peer-reviewed open-access textbook. Chapter 2 covers measures of the location and centre of data (mean, median, mode) and Chapter 2.7 covers measures of spread (range, variance, standard deviation).
  4. [4]U.S. Census Bureau. "Income in the United States: 2024." Report P60-282, September 2025 — official source for U.S. median and mean household income, used in the income-skew example on this page.
  5. [5]Khan Academy. "Mean, median, and mode review." Statistics and probability — high-quality free explainer aligned to the same definitions used by NIST and OpenStax, useful as a student-facing companion.
  6. [6]Wilkinson, Leland. "Dot Plots." The American Statistician 53(3): 276–281, 1999 — peer-reviewed paper (American Statistical Association) on visualising central tendency and dispersion, including the canonical Old Faithful bimodal-eruptions example cited above.
  7. [7]Pearson, Karl. "Contributions to the Mathematical Theory of Evolution, II: Skew Variation in Homogeneous Material." Philosophical Transactions of the Royal Society A 186: 343–414, 1895 — the original peer-reviewed derivation of the (mean − median) skewness coefficient referenced in the skewness FAQ.
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