Mean, median, mode and range: how to find each one

Averages and spread

The mean, median and mode are three kinds of average. Each one gives a single number that stands for a typical value in the data. The range is different: it is not an average but a measure of spread, showing how far apart the values are.

A memory aid helps. The median is the middle, like the median strip down the middle of a highway. The mode is the most common value: mode and most both start with mo.

How to find the mean

The mean is what each value would be if the total were shared out equally. For 4, 8, 6, 2, the total is 20 and there are 4 numbers, so the mean is 20 ÷ 4 = 5. If four friends had 4, 8, 6 and 2 stickers and shared them out evenly, each would have 5.

The mean does not have to be a whole number, or one of the values. The mean of 3, 4 and 6 is 13 ÷ 3 = 4⅓, or about 4.33.

Working backward is a favorite test question. If the mean of 5 numbers is 8, their total must be 5 × 8 = 40. If four of them are 6, 11, 7 and 10, which add up to 34, the missing number is 40 − 34 = 6.

How to find the median

Order the data first. With an odd number of values, the median is the one in the middle: for 2, 5, 6, 9, 12, it is 6. A quick way to find it is to cross off one value from each end until only one is left.

With an even number of values, two numbers share the middle. The median is halfway between them, which is their mean. For 4, 6, 10, 15, the middle two are 6 and 10, so the median is (6 + 10) ÷ 2 = 8.

How to find the mode and range

• The mode is the value that appears most often. In 3, 7, 7, 2, 9, 7, 4, the mode is 7.
• A data set can have two modes: 2, 2, 5, 8, 8 has modes of 2 and 8.
• If every value appears only once, as in 1, 4, 6, 9, there is no mode.
• The range is the largest value minus the smallest. For 12, 5, 20, 8, it is 20 − 5 = 15.
• Take care with negative numbers. Temperatures of −4, 3 and 7 °C have a range of 7 − (−4) = 11 °C.

Which average is best?

The mean uses every value, so one very large or very small value can drag it a long way. The numbers 3, 4, 4, 5 and 34 have a mean of 50 ÷ 5 = 10 but a median of 4. The median gives a much better picture of a typical value here, because the one unusual value, 34, pulls the mean up.

Use the mean when the values are fairly even, the median when there are unusual values, and the mode for the most popular choice, such as the most common shoe size or the favorite fruit in a class survey.

How it is taught in school

In the US, Common Core makes the mean and median the main measures of center in grade 6. Students learn that a measure of center sums up a data set with one number (6.SP.A.3), find the median and mean, and choose between them depending on the shape of the data (6.SP.B.5.c, 6.SP.B.5.d). Mode and range are not named in the standards, but most textbooks teach them alongside. The idea of sharing a total out equally starts with line plots in grade 5 (5.MD.B.2).

In England, Year 6 pupils calculate and interpret the mean as an average. The median, mode and range are part of Key Stage 3 (Years 7 to 9), where pupils use averages and the range to describe and compare data.

How parents can help

• Use real numbers: the minutes your child read each night this week, the family's shoe sizes or the points scored in a game.
• Show the mean as sharing out. Stack blocks in uneven towers, then move blocks until every tower is the same height.
• Ask your child to order the data before anything else. It makes the median, mode and range much easier to spot.
• Ask which average would be fairest in a real situation, such as a set of test scores with one very low score.
• Keep sessions short. Print a fresh worksheet, let your child mark it with the answer key, and go through the worked steps for any they missed.