Averages are used in everyday life to give us information about a set of numerical data, give an overview of the values seen and tell us the most common outcome. Range measures the spread of the data.

A set of numbers can be difficult to interpret and it can be a challenge to gain an overview of what they show.

Finding different averages of a set of data gives us a tool to describe the results.
The main averages, which can also be referred to as **measures of central tendency**, are the mean, mode and median.

Robert is preparing for his Mathematics GCSE exams. Each paper is marked out of 100. He attempts 10 tests and gets the following scores:

63, 86, 64, 67, 71, 42, 79, 64, 80, 64.

We can use these values to calculate the mean, median and mode to find out more information about his scores.

**The mean**

The mean uses all the values in the data. To calculate the mean:

- Add all the numbers up
- Divide by how many values there are:

\[\frac {63~+~86~+~64~+~67~+~71~+~42~+~79~+~64~+~80~+~64} {10}~=~\frac {680} {10}~=~68\]

Mean = 68

**The median**

The median is the middle value in the sorted set of data. To calculate the median:

- List the values in order from smallest to largest
- Cross values off from each end to identify the middle value

If there are two numbers in the middle you must calculate the mean of these two values. This means we add them up and divide by 2.

- Order: 42, 63, 64, 64, 64, 67, 71, 79, 80, 86
- Middle: 64, 67
- \[\frac {64~+~67} {2}~=~\frac {131} {2}~=~65.5\]

Median = 65.5

**The mode**

The mode is the most common value that appears in the data.

The mode is the only average where there can be more than one. If there are two modes we say it is **bimodal**; if there are more than two modes it is **multimodal**. If all the values appear the same number of times, we can say **there is no mode**.

It is often useful to use the ordered set of numbers; 42, 63, **64**, **64**, **64**, 67, 71, 79, 80, 86.

Mode = 64

The value 64 appears three times. All the rest appear only once.

- Question
Emily is also practising for her Mathematics GCSE exams. Her practice paper results are:

61, 73, 82, 90, 61, 67, 76, 40, 80, 62.

Calculate the mean, median and mode of her scores.

Mean = 69.2

\[\frac {61~+~73~+~82~+~90~+~61~+~67~+~76~+~40~+~80~+~62} {10}~=~\frac {692} {10}~=~69.2\]

Median = 70

67 and 73 are in the middle: \(\frac {67~+~73} {2}~=~\frac {140} {2}~=~70\)

Mode = 61

The value 61 appears twice. All other numbers appear only once.