# Mean

## What Is the Mean?

The mean is an average of a set of numbers.

The mean is found by adding all the numbers together and dividing by how many numbers there are.

### Dictionary Definition

The Oxford English Dictionary defines the mean as "the average of a set of numerical values, as calculated by adding them together and dividing by the number of terms in the set."

## Interactive Widget

Use this interactive widget to see the mean, median, and mode averages of your data set.

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## Finding the Mean

Imagine we wanted to find the mean of the numbers given below: To find the mean, add the numbers (1, 2, 3, 4 and 5) together and divide by how many numbers there are (5):

Mean = (1 + 2 + 3 + 4 + 5) ÷ 5

Mean = 15 ÷ 5

Mean = 3

## Understanding the Mean

An average is a single value that is typical for a set of values. It is a value that can represent, or stand in, for all the values in a set.

In the example above, we found that the mean of 1, 2, 3, 4, 5 is 3.

3 is a typical of these numbers, it is somewhere near the middle of them. It is also representative of the numbers.

Imagine that the numbers in our example represent the amount of money that 5 friends have: The mean amount of money is \$3. It is the amount each would have if the friends added all their money together and shared it equally among the 5 of them: This is a useful way of understanding the mean. It is the total of all the items shared out equally amongst each item.

## A Formula for the Mean

Imagine we are finding the mean of n numbers, where n is a number. In the example below, there are 5 numbers, so n = 5: We label each number in a set xi, where i = 1 for the 1st number, 2 for the 2nd number, up to n for the nth number. (In our example, n = 5, so x5 is the final number). This is shown below: The formula for finding the mean is shown below: In this formula,

• is the symbol for the mean. It is said "x bar".

• Σxi means "sum of xi" from i = 1 (below the Σ) to n (above the Σ).

Σxi = x1 + x2 + ... + xn

In our example, n = 5

Σxi = x1 + x2 + x3 + x4 + x5

Σxi = 1 + 2 + 3 + 4 + 5

Σxi = 15

• This is all written above a line, with n under it. This means divide by n.

## The Sample Mean and the Population Mean

We often wish to take the average of a lot of numbers. For example, imagine you wished to find the average income in a country.

To find the mean, we would first have to find the income of everybody who lives in the country.

The mean is found by adding all the numbers together and dividing by how many numbers there are. We would call everybody's income the population. The population is all the numbers in a set for which we are trying to find the average.

The population mean would be the average of everybody's income. It is denoted μ.

However, finding every number in the population is very time-consuming and may not be possible.

Instead, we choose a sample from the population.

We would find the incomes from a subset of the population. We would find a much smaller number of people (perhaps several hundred) who are representative of the total population and find their incomes. We would then find the average of these numbers, finding the sample mean, which is denoted .

We use the sample mean to estimate the population mean.