Mean, median, and mode are the three ways statisticians describe the “middle” of a set of numbers — and once you see them side by side, they’re simple. This guide explains each one with worked examples, adds range and standard deviation, and shows when to use which. For instant answers with full working, use our free Mean, Median, Mode & Range Calculator.

The Three Measures of Central Tendency

“Central tendency” just means the typical or central value. The mean, median, and mode each capture it differently, and they can give different answers for the same data — which is exactly why it helps to know all three.

How to Find the Mean (Average)

Add all the values, then divide by how many there are:

Mean = sum ÷ count

Example: for 3, 7, 7, 19, 4 → sum = 40, count = 5, so the mean = 40 ÷ 5 = 8. The mean is the everyday “average,” but a single very large or very small value (an outlier) can drag it away from the typical value.

How to Find the Median

Sort the numbers and take the middle one. With an odd count, it’s the single middle value; with an even count, average the two middle values.

Example (odd): sorted 3, 4, 7, 7, 19 → median = 7. Example (even): 4, 7, 7, 19 → median = (7 + 7) ÷ 2 = 7. Because the median ignores how extreme the outliers are, it’s the better “typical” value for things like income or house prices.

How to Find the Mode

The mode is the value that appears most often. In 3, 7, 7, 19, 4 the mode is 7. A set can have no mode (all values unique), one mode, or several modes (multimodal). The mode is the only measure that also works for non-numerical data — like the most common shirt size sold.

How to Find the Range

The range shows how spread out the data is:

Range = maximum − minimum

Example: 19 − 3 = 16. It’s quick but sensitive to outliers, since it depends only on the two extreme values.

Standard Deviation and Variance

Standard deviation measures the average distance of values from the mean — small means tightly clustered, large means widely spread. Variance is simply the standard deviation squared. There are two versions: the population formula (divide by n) when your data is the whole group, and the sample formula (divide by n − 1) when it’s a sample of a bigger group. The calculator reports both so you can pick the right one.

Mean vs Median vs Mode: When to Use Each

• Use the mean for roughly symmetrical data without big outliers — test scores, measurements.
• Use the median when data is skewed or has outliers — income, house prices, response times.
• Use the mode for the most common category or value — popular sizes, frequent ratings.

Reporting more than one often tells a richer story than any single number.