Skip to content
Get started
Mathematics 05809.1

Averages and representing data

Mean, median, mode, range, and choosing the right average.

Learning objectives

What you need to be able to do

Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.

  • 9.1.1Calculate and interpret mean, median, mode and range, including from frequency tables.

6 minute read

Averages and representing data

The three averages

  • Mean — add all values, divide by how many. Uses every value, but is distorted by extreme values.
  • Median — the middle value when the data is in order. Unaffected by extremes.
  • Mode — the most frequent value. The only average usable for non-numerical data such as favourite colour.
  • Range — largest minus smallest. A measure of spread, not an average.

Choosing which average

This is a favourite exam question:

  • Use the median when there are outliers, such as one very high salary in a small company.
  • Use the mode for categorical data.
  • Use the mean when data is fairly symmetric and every value should count.

Median position

For n values in order, the median is at position (n + 1)/2. With an even number of values, take the mean of the middle two. Forgetting to order the data first is the single most common error in the topic.

Mean from a frequency table

Multiply each value by its frequency, add those products, then divide by the total frequency: mean = Σ(fx) / Σf Dividing by the number of rows rather than the total frequency is a frequent mistake.

Grouped data

With grouped data you cannot know exact values, so use the midpoint of each class as x. The result is an estimate of the mean — and questions often award a mark for saying so. The modal class is the group with the highest frequency, not the highest midpoint.

Think of it like this

The mean is a see-saw balance point, so one very heavy value at the far end tips it a long way. The median just counts along to the middle, which is why it shrugs off a millionaire in a room of ordinary earners.

Worked examples

Method, step by step

Find the median of 7, 3, 9, 4, 12, 6.

  1. 1Put the values in order: 3, 4, 6, 7, 9, 12.
  2. 2There are 6 values, an even number, so the median is the mean of the 3rd and 4th.
  3. 3Those are 6 and 7, so the median is (6 + 7)/2 = 6.5.

6.5

The salaries in a small firm are £18k, £20k, £21k, £22k and £150k. Which average best represents a typical salary, and why?

  1. 1The mean is (18+20+21+22+150)/5 = £46.2k, which is higher than four of the five salaries.
  2. 2The £150k value is an outlier distorting the mean.
  3. 3The median is the middle value when ordered: £21k, which is typical of most employees.

The median (£21k), because the single very high salary is an outlier that pulls the mean up to a figure no ordinary employee earns.

Common misconceptions

  • Finding the median without sorting the data. The middle of an unordered list is meaningless.
  • Dividing by the number of rows in a frequency table instead of the total frequency.
  • Calling the range an average. It measures spread.

In the exam

  • For grouped data, always write "estimate of the mean" — the word estimate is often worth a mark because midpoints are assumptions.
  • When asked which average is "most appropriate", name it **and** justify with reference to outliers or data type.