Mean, Median & Mode Calculator
Enter a list of numbers to get the mean, median and mode (including multiple modes), plus the geometric and harmonic means, trimmed mean, midrange, range and skewness — with the sorted list, frequency table and every step shown.
Written by Suhaib Hassan. Definitions follow the NIST/SEMATECH Engineering Statistics Handbook. Our methodology.
It sorts your list, sums it for the mean, picks the middle value(s) for the median, and counts how often each value occurs to find every mode.
Averages disagree on skewed data. This page shows all of them side by side, plots the frequencies, and tells you which one to report.
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nth root of the product
n ÷ Σ(1/x)
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(min + max) ÷ 2
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3 × (mean − median) ÷ s
Frequency distribution and averages
Bars show how often each value (or range of values) occurs. Modes are highlighted; the dashed lines mark the mean and median.
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Step-by-step working
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| Value | Frequency | Share of data | Status |
|---|
Three kinds of average
Each mean answers a different question. Pick the one that matches how your numbers combine.
Arithmetic mean
The everyday average. Best for values that add up — heights, marks, weights — but pulled by extreme values.
Geometric mean
For quantities that multiply — growth factors, index ratios. Requires every value to be above zero.
Harmonic mean
For averaging rates over equal amounts — such as speed over equal distances. Requires every value to be above zero.
Skewness: how mean, median and mode compare
When a distribution is lopsided, the three measures separate in a predictable way (though the ordering is a guide, not a law).
Symmetric
Mean ≈ median ≈ mode. The centre of mass sits at the peak, as in a bell-shaped distribution.
Right-skewed (positive)
Mean > median > mode is typical. A long tail of high values, such as incomes or house prices, drags the mean upward.
Left-skewed (negative)
Mean < median < mode is typical. A tail of low values, such as age at death in high-income countries or scores on an easy exam, pulls the mean down.
Which average should you report?
The right choice depends on the scale of measurement and how skewed the data is.
| Type of data | Sensitivity to outliers | Best measure | Example |
|---|---|---|---|
| Nominal (categories) | Not applicable — no numeric order | Mode only | Most common shoe size or party choice |
| Ordinal (ranked) | Low | Median | Satisfaction on a 1–5 scale |
| Interval / ratio, skewed | High for the mean | Median (or trimmed mean) | House prices, salaries, response times |
| Interval / ratio, symmetric | Moderate | Arithmetic mean | Repeated lab measurements, heights |
| Growth rates or ratios | Moderate | Geometric mean | Average annual growth over several years |
Worked example: a skewed set of household incomes
Twelve household incomes in thousands of dollars: 38, 42, 45, 48, 52, 55, 58, 62, 65, 70, 95, 650. (Load "Skewed incomes" from the examples menu to see it above.)
Mean distorted
mean = 106.67 ($106.7k)
11 of 12 households (91.7%) earn less than this “average”, because one very large value inflates it.
Median robust
median = (55 + 58) ÷ 2 = 56.5
Half of households earn less and half more, so the median describes a typical household far better here.
Where each average is used
Economic statistics
Official income and house-price figures are usually reported as medians because a few very large values would distort the mean.
Governing measure: medianRetail & inventory
A shop ordering stock wants the most common size, not the average size — the mode identifies the size that sells most.
Governing measure: modeGrading & measurement
Exam marks are usually summarised with the mean, while two well-separated modes can reveal two distinct groups in a class.
Governing measure: mean, modesWhat are the mean, median and mode?
The mean is the sum of the values divided by how many there are. The median is the middle value once the list is sorted — or the mean of the two middle values when there is an even number of them. The mode is the value that occurs most often; a data set can have one mode, several, or none.
How to calculate them
- Mean: add all values, then divide by the count.
- Median: sort the values. With n values, take the value at position (n + 1) ÷ 2, averaging the two middle values when n is even.
- Mode: count each value; the highest count wins. If every value occurs once there is no mode.
- Trimmed mean: drop the lowest and highest 10% of values (rounded down), then average the rest.
By Suhaib Hassan · Reviewed 28 September 2026 · How CalculatePilot verifies formulas →
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods — 1.3.5.1 Measures of location
- NIST/SEMATECH e-Handbook of Statistical Methods — 1.3.5.11 Measures of skewness and kurtosis
- ISO 3534-1:2006, Statistics — Vocabulary and symbols — Part 1: General statistical terms and terms used in probability.
Frequently asked questions
What if a data set has more than one mode?
Then it is bimodal (two modes) or multimodal (more). This calculator lists every value that ties for the highest frequency. If every value appears exactly once, there is no mode.
Why does an extreme value change the mean but not the median?
The mean uses every value's size, so one very large number pulls it up. The median depends only on the middle position in the sorted list, so making the largest value larger changes nothing.
What is the difference between arithmetic, geometric and harmonic means?
Arithmetic averages values that add, geometric averages values that multiply (such as growth factors), and harmonic averages rates over equal amounts. For positive, unequal values they always satisfy harmonic ≤ geometric ≤ arithmetic.
How do I find the median with an even number of values?
Sort the list and average the two middle values. For 2, 4, 6, 8 the middle values are 4 and 6, so the median is 5.
How does the value:frequency format work?
Type each distinct value followed by a colon and how many times it occurs, for example 5:2, 8:4 means two 5s and four 8s. The calculator expands it into a full list before working everything out.
What is a trimmed mean?
A trimmed mean discards a fixed share of the smallest and largest values before averaging. Here 10% is removed from each end, rounded down, so it resists outliers while still using most of the data.
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