The mean, median, mode, and range are four ways to describe a set of numbers. Three of them — the mean, median, and mode — are measures of central tendency: they tell you what a "typical" value looks like. The fourth, the range, measures how spread out the data is. On this page you will find a clear definition of each, exactly how to compute it (including the tricky even-count median), when to use each one, worked examples, common mistakes, and an interactive quiz to practice.
What are the mean, median, mode, and range?
Each measure answers a different question about your data:
- Mean — the average. It balances all the values into one number.
- Median — the middle value once the numbers are in order.
- Mode — the value that appears most often.
- Range — the spread, from the smallest value to the largest.
The table below shows what each one is and exactly how to compute it. Notice that only the mode is about frequency, and only the range describes spread rather than a central value.
| Measure | What it is | How to compute |
|---|---|---|
| Mean | The average value | Add all the numbers, then divide by how many there are |
| Median | The middle value in order | Sort the numbers; pick the middle one. If there are two middle numbers, average them |
| Mode | The most frequent value | Find the value that appears most often. There can be none, one, or several |
| Range | The spread of the data | Subtract the smallest value from the largest value |
How to find the mean, median, and mode
Let's compute all four on one data set so you can see how they differ. Take the numbers 4, 8, 8, 5, 3, 8, 6 (seven values).
Mean. Add them up: 4 + 8 + 8 + 5 + 3 + 8 + 6 = 42. There are 7 values, so the mean is 42 ÷ 7 = 6.
Median. Sort first: 3, 4, 5, 6, 8, 8, 8. With 7 values the middle one is the 4th, so the median is 6. (Always sort before you pick the middle — this is the most common mistake.)
Mode. The value 8 appears three times, more than any other, so the mode is 8.
Range. The largest value is 8 and the smallest is 3, so the range is 8 − 3 = 5.
One data set, four different answers — that is why we have four measures.
When to use each measure
The measures don't compete — each is best for a different kind of question.
- Mean is the everyday "average" and is great when the values are fairly even, like average test scores or daily temperatures. Its weakness: a single very large or very small value (an outlier) can pull it far from the middle.
- Median is the middle value, so outliers barely move it. That's why house prices and incomes are usually reported as medians — one mansion won't distort the typical value.
- Mode is the only measure that works for things you can't average, like the most common shoe size sold or the favourite colour in a class. A data set can have one mode, several modes, or no mode at all.
- Range doesn't describe a typical value at all — it tells you how spread out the data is. A small range means the values are close together; a large range means they are far apart.
Worked examples: step by step
Example 1. All four measures on one data set
Data set: 7, 3, 7, 10, 3, 6, 3
Step 1 — Sort the numbers. 3, 3, 3, 6, 7, 7, 10.
Step 2 — Mean. Add them: 3 + 3 + 3 + 6 + 7 + 7 + 10 = 39. There are 7 values, so the mean is 39 ÷ 7 ≈ 5.57.
Step 3 — Median. With 7 values the middle one is the 4th in the sorted list: 3, 3, 3, 6, 7, 7, 10. The median is 6.
Step 4 — Mode. The value 3 appears three times, more than any other, so the mode is 3.
Step 5 — Range. Largest minus smallest: 10 − 3 = 7.
Answer: mean ≈ 5.57, median = 6, mode = 3, range = 7.
Example 2. Median with an even number of values
Data set: 12, 5, 8, 20 (four values — an even count).
Step 1 — Sort. 5, 8, 12, 20.
Step 2 — Find the two middle values. With an even count there is no single middle number. Here the two middle values are 8 and 12.
Step 3 — Average them. (8 + 12) ÷ 2 = 20 ÷ 2 = 10. The median is 10 — even though 10 is not one of the original numbers. That is normal for an even-count median.
Example 3. A data set with no mode
Data set: 2, 5, 9, 11, 14.
Every value appears exactly once, so no value repeats. This data set has no mode — that is a valid answer, not an error.
By contrast, the data set 4, 4, 7, 7, 9 has two modes (4 and 7) because both appear the most often. A set with two modes is called bimodal.
Common mistakes to avoid
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Finding the median without sorting the numbers first.
The median is the middle value of the numbers in order. Always sort from smallest to largest before you pick the middle. In 7, 2, 9 the median is 7 (sorted: 2, 7, 9), not 9.
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For an even number of values, picking just one of the two middle numbers.
With an even count there is no single middle. Take the two middle values and average them. For 5, 8, 12, 20 the median is (8 + 12) ÷ 2 = 10, not 8 or 12.
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Assuming every data set must have exactly one mode.
A data set can have no mode (if nothing repeats) or several modes (if two or more values tie for the most frequent). Both are correct answers — don't force a single mode.
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Confusing the mean and the median, and treating them as the same thing.
The mean is the average (add up and divide); the median is the middle value in order. They are often different, especially when there is an outlier. For 2, 3, 4, 100 the mean is 27.25 but the median is only 3.5.
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Mixing up the range with the mode or forgetting to subtract.
The range is a measure of spread: largest value minus smallest value. It is a single number describing spread, not the most frequent value. For 4, 9, 15 the range is 15 − 4 = 11.
Frequently asked questions
What are the mean, median, mode, and range?
The mean is the average (add all values and divide by how many there are). The median is the middle value once the numbers are sorted. The mode is the value that appears most often. The range is the largest value minus the smallest value. The mean, median, and mode describe a typical value; the range describes how spread out the data is.
How do you find the mean of a data set?
Add up all the numbers to get the total, then divide the total by how many numbers there are. For example, the mean of 4, 6, 8, 2 is (4 + 6 + 8 + 2) ÷ 4 = 20 ÷ 4 = 5.
How do you find the median when there is an even number of values?
Sort the numbers, then take the two values in the middle and average them (add them and divide by 2). For 5, 8, 12, 20 the two middle values are 8 and 12, so the median is (8 + 12) ÷ 2 = 10. The result can be a number that is not in the original set.
What is the difference between the mean and the median?
The mean is the average of all values, so every number affects it. The median is only the middle value in order, so it is barely affected by very large or very small numbers (outliers). When a data set has outliers, the median often gives a more typical value than the mean.
Can a data set have more than one mode, or no mode?
Yes. If no value repeats, the data set has no mode. If two or more values tie for the most appearances, the set has multiple modes — two modes is called bimodal. Having no mode or several modes is perfectly valid.
What are measures of central tendency?
Measures of central tendency are single numbers that describe the center or typical value of a data set. The three main ones are the mean, the median, and the mode. The range is not a measure of central tendency — it measures spread instead.
