Statistics

Finding the Mean, Median, Mode, and Spread of a Dataset

This calculator finds the mean, median, mode, range, variance, and standard deviation of any list of numbers. Paste or type your numbers into the box below — separated by commas, spaces, or new lines, in any combination — and every common descriptive statistic is calculated at once.

Mean is the arithmetic average. Median is the middle value when the numbers are sorted (or the average of the two middle values, for an even-sized list). Mode is whichever value (or values) appear most often — if every number appears exactly once, there’s no mode.

Standard deviation measures how spread out the numbers are from the mean — a small standard deviation means the numbers are clustered close together; a large one means they’re spread widely. This calculator reports it two ways: population standard deviation (when your numbers are the entire group you care about) and sample standard deviation (when your numbers are a sample used to estimate a larger population) — the sample version is slightly larger, correcting for the fact that a sample tends to underestimate the true spread of the full population.

The Formula

$$\vC{\text{Mean}} = \frac{\sum \vA{\text{Numbers}}}{\vB{\text{Count}}}$$ $$\vD{\text{Population Variance}} = \frac{\sum (\vA{\text{Number}} - \vC{\text{Mean}})^2}{\vB{\text{Count}}}$$ $$\vD{\text{Sample Variance}} = \frac{\sum (\vA{\text{Number}} - \vC{\text{Mean}})^2}{\vB{\text{Count}} - 1}$$ $$\text{Standard Deviation} = \sqrt{\vD{\text{Variance}}}$$

Population or sample, matching whichever variance is used.

The only difference between the population and sample versions is dividing by the count itself versus the count minus one (known as Bessel’s correction) — everything else is identical.

Worked Example

The numbers 2, 4, 4, 4, 5, 5, 7, 9:

1. Count: $\vB{8}$. Sum: $\vA{40}$. Mean: $\vA{40} \div \vB{8} = \vC{5}$.
2. Sorted: 2, 4, 4, 4, 5, 5, 7, 9 — median is the average of the two middle values (4 and 5): $(4 + 5) \div 2 = 4.5$.
3. Mode: 4 (appears three times, more than any other value).
4. Range: $9 - 2 = 7$.
5. Population standard deviation: 2.0. Sample standard deviation: ≈2.14.

Key Factors to Consider

• Mean and median can tell very different stories when a dataset has outliers or is skewed. A single very large or very small value pulls the mean toward it, while the median stays anchored to the middle of the sorted data regardless of extreme values — this is exactly why median household income, for example, is commonly reported instead of (or alongside) mean income, since a handful of very high incomes would otherwise skew the mean upward.
• Choosing population versus sample standard deviation depends entirely on what your numbers actually represent, not on preference. If the entered numbers are the ENTIRE group you care about (every student in one specific class), use population. If they’re a SAMPLE meant to estimate a larger group (a survey of some customers meant to represent all customers), use sample — picking the wrong one systematically under- or over-states the true variability.
• A dataset can have no mode, one mode, or multiple modes, unlike mean and median which are always single values. If every value in a list appears exactly once, there’s simply no mode — and if two or more values tie for the most frequent, a dataset can genuinely be “bimodal” or “multimodal,” which is itself often a meaningful signal about the underlying data.
• Standard deviation is most intuitively interpreted for roughly bell-shaped (normal) distributions, where roughly 68% of values fall within one standard deviation of the mean. For data that’s heavily skewed or has a very different shape, standard deviation is still a valid spread measure, but the intuitive “68-95-99.7 rule” percentages don’t apply as cleanly.

Common Mistakes

• Always using population standard deviation (or always using sample) out of habit. Which one is correct depends entirely on whether the entered numbers are the whole group or just a sample of it — picking the wrong one systematically over- or under-states the true spread, regardless of how carefully the rest of the math is done.
• Treating the mean as “the typical value” when the data is skewed. A handful of unusually large or small numbers pulls the mean away from where most of the data actually sits — the median is often the more honest “typical value” for skewed data like incomes or home prices.
• Forgetting that mode can be empty, singular, or plural. Assuming every dataset has exactly one mode leads to confusion when a list has no repeated values (no mode) or multiple values tied for most frequent (multimodal) — both are normal, expected outcomes, not errors.

Useful to Know

• Need to standardize a single value against a known mean and standard deviation instead of computing them from a list? Z-Score Calculator converts one value into a z-score and percentile.
• Working with just two numbers and want their percentage difference rather than a full descriptive-statistics breakdown? Percent Error Calculator and Percentage Calculator cover that simpler case.
• Curious how likely a particular outcome is, rather than just describing a dataset you already have? Probability Calculator calculates the odds of single and combined events.