Mean - GCSE Maths Definition

Definition

The mean is an average that is used to summarise a set of data values. The mean is calculated by dividing the sum of all the data values by the number of data values.

Explanation

The mean, also known as the mean average or arithmetic mean, is one of three types of average you need to know for GCSE Mathematics.

If working with raw data, then the mean is found in two stages. The first stage is to add up all the data values. The second stage is to then divide this total by the number of data values.

One way of thinking about the mean is that it shares everything equally - so if five people contributed different amounts of money to a central pot, then that pot was shared between those five, they would all get the same amount. This would mean some of them would get back more than they put in, some would get less.

The mean average is what is usually referred to in the news and media when phrases such as “the average” or “on average” are used.

You may be required to find the mean of a set of data listed as a set of values, find the mean of values presented in a frequency table, or estimate the mean from a grouped frequency table.

Example 1 -  a list of data values (raw data)

Question: The same product is purchased in six different shops. The prices paid were £2, £3.50, £2, £2.50, £3 and £2.84. Find the mean price of the item. 

Solution:

The data values are the prices paid, so first add these up

2 + 3.50 + 2 + 2.50 + 3 + 2.84 = 15.84

Then, divide by the number of values; there are 6 of them (6 shops)

15.84 ÷ 6 = 2.64

So the mean price of the item across the six shops is £2.64.

Example 2 -  data in a frequency table

Question:

The table below shows the number of pets owned by 30 people.

Find the mean number of pets per person.

Solution:

We need to think carefully about what the data is showing us - 5 people had no pets, 3 had one pet (each) and so on. We can add each number of pets up by multiplication, then find the total number of pets as shown below

Mean number of pets = 55 ÷ 30 = 1.833 333 … = 1.83 pets per person  (2 d.p.)

Example 3 - estimating the mean from grouped data

Question:

25 athletes took part in a fun run. The times for completion, in minutes, are summarised below.

Find an estimate for the mean time to complete the race.

Solution:

As the data is grouped, we do not have the raw data. So to estimate the mean, we assume each time would be the middle value for its group, then find the mean using the same approach as for frequency tables, using the midpoints as the data values.

Estimate of mean time = 235 ÷ 20 = 11.75 minutes

Common mistakes (and how to avoid them)

Mistake 1: Using a calculator incorrectly

Calculators follow the rules of order of operations (BIDMAS). A common mistake is to not ensure the calculator does the adding step first.

For example, typing in 8 + 4 + 9 ÷ 3 will give the result 15. This is not the mean! The calculator has correctly worked out 9 ÷ 3 first, then worked out 8 + 4 + 3 to get 15. The mean would require all the adding stages to be done before the division; 8 + 4 + 9 = 21; then 21 ÷ 3 = 7. So the mean is 7, not 15. 

How to avoid this: Always use brackets or the fraction facility when calculating the mean (of a list of values) on a calculator. For the above, we could type in (8 + 4 + 9) ÷ 3, or, using the fraction facility, type in .

For frequency tables and grouped data, you may be able to use a TABLE mode (or similar) instead. Consider if your answer for the mean seems right - if most of the values are low (like for the number of pets example) then a mean of 102.4 would not be sensible!

Mistake 2: Finding the mean of the frequencies in a table

When data is presented in a table, a common mistake is to find the mean of the values in the frequency column. The frequency column tells us how many of each data value there are, and are not part of the actual (raw) data.

Typically, the data values are in the left-hand column of a table, and the frequency values are in the right-hand column.

In Example 2 above, the table tells us the value “0” appears 5 times in the data and the value “1”  appears 3 times; if we were to write these as a list of the data values (i.e. the raw data) it would be 0, 0, 0, 0, 0, 1, 1, 1 and so on.

How to avoid this: Avoid learning how to find the mean from frequency tables (and grouped data) as a trick. Immerse yourself in the question context - think carefully about what a frequency table shows. Again also consider if your final answer seems reasonable compared to the other data values.

Frequently asked questions

What are the other types of average?

Median - the value that is in the middle position from a list of ordered data values.

Mode - the value (or item/object) that occurs most frequently (most popular). 

Why can we only estimate the mean from grouped data?

When data is presented in a grouped frequency table, we do not have the original, recorded data values (raw data), so we cannot find the exact mean; we simply do not have the accurate information required. However, by assuming that the data is evenly spread out across each group, we can estimate that each data value is the middle value of its group and then treat the table as a frequency table.

Related GCSE Mathematics glossaries

• Arithmetic mean

• Average

• Mean

• Median

• Mode

• Modal class