7.2 Mode, Median and Range

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: finding the mode and the median, using the range, and spotting outliers. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Find the mode of a set of data, including data that is not numbers
• Find the median of an odd or an even number of values
• Compare data sets using the median and the range
• Recognise an outlier and explain how it affects each average and the range

Key words

• Mode: The value that occurs most often. Also called the modal value. There can be more than one mode, or none.
• Median: The middle value when the data is written in order of size.
• Range: Largest value − smallest value. It measures spread.
• Outlier: A value that is much bigger or much smaller than the rest of the data.

The mode

The mode is the most common value. It is the only average you can use for data that is not numbers, such as colours or types of pet.

Worked example: shoe sizes

The shoe sizes of seven children are 5, 6, 4, 6, 7, 6, 5. Find the mode.
6 appears three times, 5 twice, 4 and 7 once each.

Answer: Mode = size 6

• red, blue, red, green, blue, red; Mode red
• 3, 8, 3, 5, 8, 1; Mode 3 and 8 (two modes – bimodal)
• 2, 4, 7, 9, 10; Mode no mode (every value appears once)

The median

Write the values in order first. For n values the median is the (n + 1) ÷ 2 th value. With an even number of values, find the number halfway between the two middle values.

Worked example: an odd number of values

Reaction times (hundredths of a second): 14, 9, 11, 17, 12, 8, 15
In order: 8, 9, 11, 12, 14, 15, 17. (7 + 1) ÷ 2 = 4th value.

Answer: Median = 12

Worked example: an even number of values

Marks: 23, 18, 30, 25, 21, 27. In order: 18, 21, 23, 25, 27, 30.
(6 + 1) ÷ 2 = 3.5, so halfway between the 3rd and 4th values: (23 + 25) ÷ 2 = 24.

Answer: Median = 24

Comparing with the median and range

As with the mean, compare an average (here the median) and the spread (the range), and say what each means in context.

Worked example: two shops

Waiting times (minutes) at two shops. Shop X: 3, 5, 4, 6, 4. Shop Y: 1, 9, 2, 8, 5.
Shop X in order: 3, 4, 4, 5, 6. Median 4, range 3.
Shop Y in order: 1, 2, 5, 8, 9. Median 5, range 8.
Shop X has the lower median, so customers usually wait less. Its range is smaller, so waits are more consistent.

Answer: Shop X: shorter and more consistent waiting times

Outliers

An outlier is a value that does not fit the pattern. It may be a mistake, or a real but unusual value. Outliers have a big effect on the mean and the range, but very little effect on the median.

Worked example: hourly pay

Six people are paid £7, £8, £9, £8, £10 and £45 per hour. £45 is an outlier.
With £45: mean = 87 ÷ 6 = £14.50, median = (8 + 9) ÷ 2 = £8.50, range = £38.
Without £45: mean = 42 ÷ 5 = £8.40, median = £8, range = £3.

Answer: The median (£8.50) represents the pay best; the mean is pulled up by £45

Key fact: If there is an outlier, the median is usually a better average than the mean.

Common mistakes

• Finding the “middle” value without putting the data in order first.
• Giving the position of the median (for example “the 4th”) instead of the value.
• Saying the mode is how many times a value appears, instead of the value itself.
• With an even number of values, choosing one of the two middle values instead of the value halfway between them.
• Using the mean to represent data that contains an outlier.

Revision questions

Try these, then check your answers below each one.

Question 1. Find the mode of 4, 9, 6, 9, 2, 6, 9

Answer: 9

Working: 9 appears three times, more than any other value.

Question 2. Find the median of 15, 7, 12, 20, 9

Answer: 12

Working: In order: 7, 9, 12, 15, 20. The middle value is 12.

Question 3. Find the median of 3, 8, 6, 10, 4, 11

Answer: 7

Working: In order: 3, 4, 6, 8, 10, 11. Median = (6 + 8) ÷ 2 = 7.

Related lessons

• Previous lesson: 7.1 Mean and Range
• Next lesson: 7.3 Types of Average
• In this chapter: 7.4 Estimating the Mean
• In this chapter: 7.5 Sampling

Continue in the full course

This lesson is section 7.2 (EGCMF0702) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.