Maths
What are the mean, median and mode?
Quick answer
Mean, median and mode are three kinds of average. The mean shares the total out equally, the median is the middle value in order, and the mode is the most common value.
In one sentence
Mean, median and mode are three kinds of average. The mean shares the total out equally, the median is the middle value when the numbers are in order, and the mode is the most common value.
Explained
Five friends score 2, 3, 3, 6 and 11 goals in a season. What is a typical score? There are three good answers.
football- football
- football
How to find each one
Mean. Add all the values and divide by how many there are. 2 + 3 + 3 + 6 + 11 = 25, and there are 5 friends, so the mean is 25 ÷ 5 = 5. When people say "average" without saying which, they usually mean the mean.
Median. Put the values in order and pick the middle one. In 2, 3, 3, 6, 11 the middle value is 3. With an even number of values, the median is halfway between the two middle ones: for 4, 7, 9, 12 it is (7 + 9) ÷ 2 = 8.
Mode. The value that appears most often. Here 3 appears twice, so the mode is 3. A set can have no mode (all different) or more than one. The mode is the only average that works for non-numbers too, like the most popular ice cream flavor, and it is easy to read from a tally chart.
Range. Not an average, but useful: the biggest value minus the smallest, 11 − 2 = 9. It shows how spread out the data is.
Memory trick: mode sounds like "most", and the median is in the middle, like a highway's median strip (the central reservation in the UK).
Which average is best?
The mean uses every value, but one extreme value pulls it a long way. Here, the 11 drags the mean up to 5, even though three of the five friends scored 3 or fewer. The median ignores extremes, which is why house prices and incomes are often reported as medians.
Examples
| Data | Mean | Median | Mode | Range |
|---|---|---|---|---|
| 5, 5, 6, 8, 11 | 35 ÷ 5 = 7 | 6 | 5 | 6 |
| 20, 22, 22, 25, 96 | 185 ÷ 5 = 37 | 22 | 22 | 76 |
In the last row, the 96 makes the mean bigger than four of the five numbers; the median, 22, is more typical.
Common mistakes
- Finding the median without sorting. Always put the numbers in order first.
- Dividing by the wrong number for the mean. Count the values, including any zeros.
- Giving how many times the mode appears instead of the value itself. In 4, 4, 4, 7 the mode is 4, not 3.
Quick quiz
Mean, median and mode
Check what you know with 5 quick questions.
Question 1 of 5
What is the mean of 4, 6 and 11?
Practise it
Try it
For questions 1 and 2, use 8, 3, 11, 3, 5.
- Find the mean and the median.
- Find the mode and the range.
- Which average describes 10, 11, 12, 12, 100 best, and why?
Answers: 1) Mean = 30 ÷ 5 = 6. In order, 3, 3, 5, 8, 11, so the median is 5. 2) Mode = 3; range = 11 − 3 = 8. 3) The median, 12: the extreme value 100 pulls the mean up to 29.
Mean, median and mode practice
Practise it step by step in a full course: Grade 8 Maths
For grown-ups
Related topics
Bar charts
A bar chart shows data using bars of different lengths. The longer the bar, the bigger the number. In the US it is usually called a bar graph.
Probability
Probability measures how likely something is to happen, on a scale from 0 (impossible) to 1 (certain). A fair coin landing heads has a probability of 1/2.
Division
Division is sharing into equal groups, or finding how many groups fit. The sign is ÷, so 12 ÷ 3 = 4. The answer is called the quotient.
Pictograms
A pictogram shows data using small pictures or symbols. A key tells you what each picture is worth, such as one apple standing for 2 apples. In the US it is often called a picture graph.
Cumulative frequency
Cumulative frequency is a running total of frequencies. Add each frequency to the total so far; the last value equals the total, and a cumulative frequency graph gives the median and quartiles.
Frequency polygons
A frequency polygon is a line graph of a grouped frequency table. Plot each class's frequency above the class midpoint and join the points with straight lines.