7.3 Types of Average

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: choosing an average, averages from frequency tables, and the modal class. 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

• Choose the most suitable average for a set of data and explain why
• Find the mode, median, mean and range from a frequency table
• Find the modal class of grouped data
• Find the class that contains the median for grouped data

Key words

• Frequency: How many times a value occurs.
• Frequency table: A table showing each value (or group) and its frequency.
• Grouped data: Data sorted into classes, such as 10 < t ≤ 20, when there are many different values.
• Modal class: The class with the highest frequency.
• Typical value: A value that represents the data well. A good average is representative of the data.

Choosing the right average

• Mode; Good points Works for data that is not numbers. Shows the most popular value.; Weak points There may be no mode, or several. Can be far from the centre.
• Median; Good points Not affected by outliers. Easy to find for ordered data.; Weak points Does not use every value.
• Mean; Good points Uses every value, so it reflects all the data.; Weak points Pulled up or down by outliers.

Worked example: which average?

A shoe shop wants to know which size to order most of. Which average should it use?
It needs the most popular size, not a calculated value (the mean might be 6.38, which is not a size).

Answer: The mode

Mode and range from a frequency table

A frequency table is a short way to write lots of repeated values. The table below means four 0s, seven 1s, five 2s and so on.

• Frequency (matches); 0 4; 1 7; 2 5; 3 3; 4 1

Worked example: mode and range

The highest frequency is 7, for 1 goal, so the mode is 1 goal (not 7!).
Range = largest value − smallest value = 4 − 0 = 4 goals.

Answer: Mode = 1 goal, range = 4 goals

Mean and median from a frequency table

For the mean, add a column for value × frequency. Divide its total by the total frequency.

• 0; Frequency f 4; x × f 0
• 1; Frequency f 7; x × f 7
• 2; Frequency f 5; x × f 10
• 3; Frequency f 3; x × f 9
• 4; Frequency f 1; x × f 4
• Total; Frequency f 20; x × f 30

Worked example: the mean

Mean = 30 ÷ 20 = 1.5

Answer: 1.5 goals per match

Worked example: the median

20 matches, so the median is halfway between the 10th and 11th values.
Running total of frequencies: 4 (up to 0 goals), 11 (up to 1 goal). The 10th and 11th values are both 1.

Answer: Median = 1 goal

Grouped data: the modal class and the median class

When data is grouped you cannot see the exact values, so you give a class instead of a single value.

• Frequency; 0 < t ≤ 10 6; 10 < t ≤ 20 11; 20 < t ≤ 30 8; 30 < t ≤ 40 5

Worked example: modal class and median class

Modal class: highest frequency is 11, so 10 < t ≤ 20.
Median: 30 pupils, so between the 15th and 16th values. Running totals: 6, then 6 + 11 = 17. The 15th and 16th are both in the second class.

Answer: Modal class 10 < t ≤ 20; median is in the class 10 < t ≤ 20

Key fact: With grouped data you cannot find the exact range or median. You can only say which class the median is in.

Common mistakes

• Giving the highest frequency (7) as the mode instead of the value with that frequency (1 goal).
• Adding the frequencies and dividing by the number of rows to find the mean.
• Finding the “middle” of the values column (0 to 4) instead of the middle of all 20 values.
• For grouped data, giving a single number for the modal class instead of the whole class.

Revision questions

Try these, then check your answers below each one.

Question 1. A café wants to know which flavour of ice cream is most popular. Which average should it use? (Choose from: Mode, Median, Mean.)

Answer: Mode

Working: Flavours are not numbers, so only the mode can be used.

Question 2. Using the same table (0 goals: 3, 1: 6, 2: 8, 3: 2, 4: 1), find the median number of goals.

Answer: 2

Working: Running totals: 3, 9, 17. The 10th and 11th values are both 2 goals.

Question 3. The times taken by 30 pupils to solve a puzzle are grouped: 0 < t ≤ 5: 8 pupils; 5 < t ≤ 10: 9; 10 < t ≤ 15: 10; 15 < t ≤ 20: 3. What is the modal class? (Choose from: 0 < t ≤ 5, 5 < t ≤ 10, 10 < t ≤ 15, 15 < t ≤ 20.)

Answer: 10 < t ≤ 15

Working: The highest frequency is 10, for 10 < t ≤ 15.

Related lessons

• Previous lesson: 7.2 Mode, Median and Range
• Next lesson: 7.4 Estimating the Mean
• In this chapter: 7.1 Mean and Range
• In this chapter: 7.5 Sampling

Continue in the full course

This lesson is section 7.3 (EGCMF0703) 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.