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Mean, median and mode

The mean is the sum of all the values divided by the number of values. The median is the middle value when the values are in order of size. With an even number of values, it is the mean of the two middle ones. The mode is the value that occurs most often.

Years 4–9 and upper secondary · Updated

Watch Fox work it out on the board

Fox works one step at a time. Press play.

Three averages

AverageHow we calculateIt answers
MeanThe sum of all the values ÷ the number of valuesHow much is it each time if everything is shared out equally?
MedianThe middle value in order of sizeWhere is the middle, so that half are below and half above?
ModeThe value that occurs most oftenWhich value is the most common?

The median step by step

  • Always put the values in order of size first.
  • Odd number of values: the median is the value in the middle.
  • Even number of values: the median is (one middle value + the other middle value) ÷ 2.

Median or mean?

A single extreme value can pull the mean away. The median stays in the middle. That is why the median is used for salaries and house prices.

The mode works even when the values are not numbers, for example favourite colours.

Example

Example 1: all three averages for the same values

Ella read 14, 9, 12, 20, 9, 16 and 11 pages in one week. We want to find all three averages.

  1. The mean: we add all seven values.

    14 + 9 + 12 + 20 + 9 + 16 + 11 = 91

  2. There are seven days, so we divide by seven.

    91 ÷ 7 = 13

  3. The median: we sort from smallest to largest.

    9, 9, 11, 12, 14, 16, 20

  4. The middle one of seven values is the fourth.

    median = 12

  5. The mode: 9 appears twice, all the others once.

    mode = 9

13 · 7 = 91, the same as the sum. The median 12 has three values below it and three above.

Example 2: the median with an even number of values

Adam's journey to school took 12, 8, 15, 10, 9 and 14 minutes. We want to find the median.

  1. We sort the values.

    8, 9, 10, 12, 14, 15

  2. Six values give two in the middle: 10 and 12.

    8, 9, [10, 12], 14, 15

  3. We find the number halfway between them: we add and divide by two.

    (10 + 12) ÷ 2 = 22 ÷ 2 = 11

  4. The median is 11 minutes, although no day took exactly 11.

Three values are below 11 and three are above. 11 is the middle.

Example 3: an extreme value pulls the mean away

Five friends have saved 200, 250, 300, 350 and 2 400 kr. We compare the mean and the median.

  1. The mean: we add all the amounts.

    200 + 250 + 300 + 350 + 2 400 = 3 500

  2. We divide by five.

    3 500 ÷ 5 = 700 kr

  3. The median: five sorted values, so the third.

    median = 300 kr

  4. Four out of five have 350 kr or less. The median describes the group better.

700 · 5 = 3 500, the same as the sum. Without the 2 400 kr, the mean is 1 100 ÷ 4 = 275 kr, close to the median of 300 kr.

Pitfalls

  • Taking the number in the middle of the list without sorting first.

    Sort from smallest to largest. The median is the middle of the sorted values.

  • Dividing the sum by the wrong number, for example forgetting a value that appears twice.

    A value that appears twice counts twice, both in the sum and in the number of values.

  • Choosing one of the two middle values when the number of values is even.

    Add the two middle values and divide by two. The answer may be a decimal.

Practise on your own

0 of 4 correct

  1. Work out the mean of 6, 10, 8 and 4.

  2. What is the mode of 3, 5, 3, 8, 5 and 3?

  3. What is the median of 4, 11, 6 and 9?

  4. Noor has a mean score of 15 points on four tests. How many points does she need on the fifth test for the mean to be 16?

Common questions

Can there be more than one mode?

Yes, if several values are equally the most common. In 2, 2, 5, 5, 7, both 2 and 5 are modes. If all the values occur equally often, there is no mode.

Does the median have to be one of the numbers?

No. With an even number of values it can be, for example, 7.5 when the middle values are 6 and 9.

What is the range?

The difference between the largest and the smallest value. For 9, 9, 11, 12, 14, 16, 20 it is 20 − 9 = 11. It shows how spread out the values are.

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