Mean Median Mode Range Calculator

Calculate mean, median, mode, range, geometric mean, sum, count, minimum, maximum, quartiles, variance, standard deviation, and detailed statistical analysis from any dataset.

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What Are Mean, Median, Mode, and Range?

Mean, median, mode, and range are the four most-taught descriptive statistics — quick numerical summaries that describe a dataset without listing every value. The mean is the arithmetic average. The median is the middle value once the data is sorted. The mode is whichever value appears most often. The range is the difference between the largest and smallest value. Together they answer the two questions almost every analysis starts with — "where is the centre?" and "how spread out is it?"

This calculator goes far beyond those four numbers — it also computes geometric mean, harmonic mean, trimmed mean, sum, count, quartiles (Q1, Q2, Q3), IQR, percentiles, population and sample variance, standard deviation, coefficient of variation, skewness, kurtosis, and the 1.5×IQR outlier check, then draws five interactive charts so you can see the distribution. Related tools include the probability calculator, percentage calculator, and scientific calculator.

How the Four Core Measures Work

Mean (Average)

Add every value, then divide by how many you added. The mean uses every observation, which makes it sensitive to extremes — a single outlier can drag it far from where the rest of the data lives.

Median (Middle)

Sort the values from low to high. With an odd count, the median is the middle entry; with an even count, it's the average of the two middle entries. The median ignores extreme values, making it the right summary for skewed data.

Mode (Most Common)

The value that occurs more often than any other. A dataset can have no mode (every value unique), one mode (unimodal), two modes (bimodal), or several (multimodal). The mode is the only summary that works on non-numeric data like categories.

Range (Spread)

The simplest measure of spread — maximum minus minimum. Range is intuitive but fragile: it depends on only two values, so a single extreme observation determines the whole answer. For most analyses, IQR or standard deviation is a more stable choice.

6 Ways to Use This Statistics Calculator

Summarise a dataset

Paste any list of numbers and get every descriptive statistic in one click — mean, median, mode, range, quartiles, and more.

Measure spread

Compare variance, standard deviation, IQR, and coefficient of variation to see how tightly values cluster around the average.

Detect outliers

Use the 1.5×IQR rule to automatically flag values that may be data-entry errors or extreme observations worth investigating.

Diagnose distribution

Skewness, kurtosis, the histogram, and the box plot together tell you whether your data is normal, skewed, or heavy-tailed.

Show your working

The step-by-step tab walks through every formula with your actual numbers — ideal for homework, lab reports, or teaching.

Export a report

Copy results to the clipboard, download a CSV, share a link, or print a clean PDF-ready analysis with charts and step-by-step working.

Best Practices for Descriptive Statistics

Never report a centre without also reporting a spread. "The mean is 50" tells you almost nothing on its own — was the standard deviation 2 or 200? Always pair a measure of central tendency (mean or median) with a measure of variability (standard deviation or IQR) so the reader has both halves of the story.

Look at the data before averaging it. A histogram or box plot takes seconds to scan and can immediately reveal a skew, a second peak, or a typo that an aggregate number would silently absorb. If the distribution is skewed, prefer median and IQR over mean and standard deviation; if it's roughly symmetric, the mean and standard deviation are fine.

Decide upfront whether your dataset is the whole population or a sample. Sample variance divides by n − 1 (Bessel's correction) so it produces an unbiased estimator of the true population variance, while population variance divides by n. Mixing the two is one of the most common statistics mistakes — both results are shown above so you can pick correctly.

Why Descriptive Statistics Matter

Business

Sales reports, KPI dashboards, customer-satisfaction scores, and conversion-rate experiments all reduce hundreds of rows to a few summary numbers. Median order value, percentile response time, and standard deviation of revenue are the daily language of operations.

Finance

Asset returns, portfolio volatility, value-at-risk, and Sharpe ratio all depend on mean and standard deviation. Outlier-resistant medians appear in real-estate price reports and household income statistics for exactly the same reason.

Research & Science

Lab measurements, clinical trial endpoints, and reproducibility studies require precise reporting of mean ± standard deviation (or median and IQR for skewed data). Confidence intervals build directly on these summaries.

Machine Learning

Feature scaling, normalisation, gradient clipping, and anomaly detection all rely on mean, standard deviation, and quantiles. Inspecting these per-feature before model training catches bugs that no validation curve will reveal.

Where Statistics Gets Tricky

Skewed distributions

Income, housing prices, response times, and counts of rare events all skew right. The mean gets pulled toward the long tail and stops describing a typical observation — the median is almost always the better summary.

Bimodal data

A bimodal dataset has two peaks. Reporting just the mean hides this entirely (it lands between the peaks where nothing actually is). Always check the histogram before reducing to a single number.

Outliers vs. signal

The 1.5×IQR rule is a starting point, not a license to delete data. Sometimes outliers are typos; sometimes they're the most interesting observations in the dataset. Investigate before excluding.

Small samples

With fewer than 30 observations, descriptive statistics are still useful but inferential statistics (t-tests, confidence intervals) need to switch to small-sample (Student's t) methods. Variance estimates are also much noisier.

The Core Statistical Formulas

Mean

x̄ = Σxᵢ ÷ n

Sum of all values divided by the count.

Median (odd n)

median = x₍ₙ₊₁₎⁄₂

Middle value of the sorted dataset.

Median (even n)

median = (x₍ₙ⁄₂₎ + x₍ₙ⁄₂₊₁₎) ÷ 2

Average of the two middle values.

Range

range = max(x) − min(x)

Difference between largest and smallest.

Population variance

σ² = Σ(xᵢ − x̄)² ÷ n

Average squared deviation (whole population).

Sample variance

s² = Σ(xᵢ − x̄)² ÷ (n − 1)

Unbiased estimator for a sample.

Standard deviation

σ = √σ² · s = √s²

Square root of variance — same units as data.

IQR

IQR = Q3 − Q1

Width of the middle 50% of values.

Outlier bounds

[Q1 − 1.5·IQR, Q3 + 1.5·IQR]

Anything outside is flagged as an outlier.

Geometric mean

ⁿ√(x₁·x₂·…·xₙ)

Right summary for growth rates and ratios.

Harmonic mean

n ÷ Σ(1/xᵢ)

Right summary for rates (km/h, $/unit).

Skewness

γ₁ = m₃ ÷ σ³

Third standardised moment — tail direction.

Common Statistics Mistakes

1
Reporting only the mean for skewed data
House prices and incomes have long right tails. The mean gets pulled upward by a few millionaires and no longer describes a typical observation. Quote the median (and IQR) instead.
2
Mixing population and sample variance
Using ÷ n when you should use ÷ (n − 1) underestimates variance. Population variance applies when the dataset is the entire population; sample variance applies to a subset drawn from a larger population.
3
Ignoring bimodal distributions
If two distinct subgroups are mixed in one dataset, the overall mean falls between the two peaks where almost no observations actually live. Always plot the data first.
4
Deleting outliers reflexively
Outliers can be data-entry errors or they can be the most interesting observations. Investigate every flagged value before excluding it from analysis.
5
Confusing variance and standard deviation
Variance is in squared units (dollars-squared, seconds-squared) and rarely interpretable on its own. Standard deviation lives in the original units — quote it whenever you want intuition about spread.
6
Forgetting to label scales
"The mean is 72" means different things if the data is exam scores out of 100 versus heart-rate in BPM. Always include units when reporting any descriptive statistic.

Real-Life Uses of Statistics

Education

Class averages, percentile rankings, grade distributions, and standardised test scoring all rely on mean, median, and percentiles to make pass/fail decisions fair across cohorts.

Health & Sport

Heart-rate variability, blood pressure ranges, body composition, and athletic performance benchmarks summarise long records as mean ± SD or median (IQR) for clinical reporting.

Quality Control

Manufacturing tolerances use process-capability indices built on standard deviation; control charts flag any sample whose mean drifts more than 3σ from the historical centre line.

Data Science

Every model starts with descriptive statistics: per-column mean, std, min, max, quartiles, missing-value counts. The summary table catches data-quality problems before any modelling begins.

Methodology you can verify

Every result above is computed live from your dataset using textbook formulas — sample variance uses Bessel's correction (n − 1), percentiles use linear interpolation (the same convention as Excel's PERCENTILE.INC and NumPy's numpy.percentile), and outliers are flagged by the 1.5×IQR rule. Read more on the methodology and editorial policy pages.

Frequently Asked Questions

The mean — also called the arithmetic average — is the sum of every value divided by how many values there are. For the dataset {10, 2, 38, 23, 38, 23, 21} the sum is 155 and the count is 7, so the mean is 155 ÷ 7 ≈ 22.143. The mean uses every observation, which makes it the most informative single summary when the data is roughly symmetric, but also the most sensitive to outliers — a single extreme value can pull the mean far from where the rest of the data lives. For skewed distributions like incomes or housing prices, the median is usually a better description of a typical observation.

The median is the middle value of the sorted dataset. To find it, sort the values from low to high; if the count is odd, the median is the middle entry, and if it's even, the median is the average of the two middle entries. For {10, 2, 38, 23, 38, 23, 21}, the sorted list is {2, 10, 21, 23, 23, 38, 38} and the middle (fourth of seven) value is 23. The median is unaffected by the magnitude of extreme observations — it changes only if you move a value across the middle of the dataset — which is why it's preferred for skewed data and income statistics.

The mode is the value that appears most often. A dataset can have one mode (unimodal), two modes tied for the most occurrences (bimodal), several tied modes (multimodal), or no mode at all when every value is unique. For {10, 2, 38, 23, 38, 23, 21}, both 23 and 38 appear twice while every other value appears once, so the dataset is bimodal with modes 23 and 38. The mode is the only descriptive statistic that also works on non-numeric data — you can find the mode of a list of categorical labels like "most common eye colour."

The range is the simplest measure of spread — the difference between the maximum and minimum values. For {2, 10, 21, 23, 23, 38, 38}, the range is 38 − 2 = 36. Range is intuitive and quick but very fragile: it depends on only two of your observations, so a single typo at either extreme will dominate the answer. For a more stable measure of spread, prefer the interquartile range (IQR), standard deviation, or coefficient of variation, all of which this calculator computes automatically.

Variance is the average squared distance of each value from the mean. Squaring the deviations turns negatives into positives and weights large deviations much more than small ones. Population variance σ² divides by n; sample variance s² divides by n − 1 (Bessel's correction) to produce an unbiased estimate of the population variance when your data is only a sample. For {10, 2, 38, 23, 38, 23, 21} with mean ≈ 22.143, the population variance is about 152.69 and the sample variance is about 178.14. Because variance is in squared units (e.g. dollars-squared) it's rarely interpreted alone — most analyses report standard deviation instead.

Standard deviation is the square root of variance, returning the spread measure to the original units of the data. A small standard deviation means most values cluster tightly around the mean; a large one means they're spread out. Like variance, there are population (σ) and sample (s) versions — the calculator shows both. For roughly normal data, about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three (the famous "68-95-99.7 rule"). Pair standard deviation with the mean whenever you report a centre, since the two together describe both location and spread.

A bimodal dataset has two values that tie for the highest frequency — two equal peaks in the distribution. Bimodal data often appears when two distinct sub-populations are mixed in the same dataset (e.g. patient ages in a clinic that treats both children and seniors). When data is bimodal, reporting only the mean is misleading because the mean lands between the two peaks where almost no observations actually live. Always plot a histogram before averaging to spot bimodality, and consider splitting the data into the two underlying groups for separate analysis.

Quartiles divide a sorted dataset into four equal parts. Q1 (the first quartile) is the 25th percentile, Q2 is the median (50th percentile), and Q3 is the 75th percentile. There are several quartile conventions in use — this calculator uses linear interpolation between the surrounding values, which is the same method as Excel's PERCENTILE.INC and NumPy's numpy.percentile (interpolation='linear'). The interquartile range (IQR) is Q3 − Q1 and captures the middle 50% of the data. IQR is more robust than the full range because it ignores the most extreme values on both ends.

An outlier is a value that lies far from the rest of the dataset and may be a typo, a measurement error, or a genuinely unusual observation. The standard 1.5×IQR rule flags any value outside the interval [Q1 − 1.5·IQR, Q3 + 1.5·IQR] as a potential outlier. The Charts tab on this page highlights outliers in red on the distribution bar chart and marks them as separate dots on the box plot. Outliers are worth investigating before deciding whether to keep or remove them — sometimes they're the most interesting observations in the dataset, and reflexively deleting them can hide real effects.

Use the median when the data is skewed, contains outliers, or when you want a robust description of a "typical" value. The mean is pulled toward the long tail of a skewed distribution (which is why median household income and median house price are reported rather than the mean — both are right-skewed by a small number of very high values). Use the mean when the data is roughly symmetric, when you need every observation to contribute proportionally, or when you'll plug the average into a formula that assumes the arithmetic mean (e.g. total = mean × count). Reporting both, along with the standard deviation and IQR, is the safest practice in any analysis.

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