Mean

The mean of a set of numbers is what we usually refer to as the ‘average’.

We calculate it by adding up all the numbers in the set, and then dividing that total by the number of values in the set:

Calculating mean

1. Add up all the numbers in the set to get the total.
2. Count how many numbers there are in the set to get the number of values.
3. Divide the total by the number of values to get the mean.

Notation for mean

We can represent the mean of a set of numbers x using the symbol \overline{x} (x with a bar on top):

\text{mean:}\quad\overline{x}

Formula

We can use summations to write the mean for any set x:

\overline{x} = \frac{\sum x}{n}

Where:

• \sum x means “the sum of all the values in the set x”
• n is the number of values in the set x

Examples

Find the mean of \{2, 3, 5, 7, 11\}

• \text{total} = 2 + 3 + 5 + 7 + 11 = 28
• \text{number of values} = 5
• \text{mean} = \frac{\text{total}}{\text{number of values}}
• \text{mean} = \frac{28}{5}
• Answer: \overline{x} = \frac{28}{5}

Find the mean of \{1, 4, 9, 16, 25\}

• \text{total} = 1 + 4 + 9 + 16 + 25 = 55
• \text{number of values} = 5
• \text{mean} = \frac{\text{total}}{\text{number of values}}
• \text{mean} = \frac{55}{5}
• Answer: \overline{x} = 11

Find the mean of \{3, 1, 4, 1, 5, 9\}

• \text{total} = 3 + 1 + 4 + 1 + 5 + 9 = 23
• \text{number of values} = 6
• \text{mean} = \frac{\text{total}}{\text{number of values}}
• \text{mean} = \frac{23}{6}
• Answer: \overline{x} = \frac{23}{6}

Find the mean of \{1, 2, 3, 4, 5, 6, 7, 8, 9\}

• \text{total} = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45
• \text{number of values} = 9
• \text{mean} = \frac{\text{total}}{\text{number of values}}
• \text{mean} = \frac{45}{9}
• Answer: \overline{x} = 5