# Finding the Mean

We can find the mean of a set of numbers.

Imagine a teacher wanted to find the class's average test score in mathematics.

# How to Find the Mean

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

#### An Example Question

What is the mean of the test scores?

Step 1
Add up the numbers.
7 + 10 + 8 + 6 + 4 = 35
Step 2
Divide the answer (35) by how many numbers there are. In our example, there are 5 students, so there are 5 test scores.
35 ÷ 5 = 7
The mean of the test scores is 7.

# A Real Example of How to Find the Mean

The slider below gives another example of finding the mean.

# A Formula to Find the Mean

The formula for finding the mean is shown below:

In this formula,
• is the symbol for the mean. It is said "x bar".

• The Σ symbol means "sum of". It sums what comes after it: xi.

• xi is each value, where i = 1, 2... n. n is how many numbers there are. In our example of the test scores:
x1 = 7, x2 = 10, x3 = 8, x4 = 6, x5 = 4
• Below the Σ is i = 1. This means start summing from i = 1. Above the Σ is n. This means stop summing when i = n.

Taken together, the Σxi means:
Σxi = x1 + x2 + ... + xn
In our example, n = 5
Σxi = x1 + x2 + x3 + x4 + x5
Σxi = 7 + 10 + 8 + 6 + 4
Σxi = 35
• This is all written above a line, with n under it. This means divide by n. In our example, n = 5:
x̄ = Σxi / n = 35 ÷ 5 = 7
The mean of the test scores is 7.

# How to Find the Mean from a Frequency Table

Sometimes data is presented in frequency tables.

A frequency table representing the test scores is shown below:

It is possible to find that the mean test score is 7.

Read more about finding the mean from a frequency table.

##### Interactive Widget
Here is an interactive widget to help you learn about the mean.
##### Interactive Test
show

Here's a second test on finding the mean.
Here's a third test on finding the mean.

# The Mean Is the Most Common Type of Average

The mean is the most common type of average.

People often use the word average to mean the mean, even though there are other types of average.

# What's in a Name?

"Mean" comes from the Old French "meien", which comes from the Latin "medianus", meaning "middle".

# Other Types of Average

The most commonly commonly used types of average, other than the mean, are:

# The Sample Mean and the Population Mean

Imagine we wanted to find out the mean income in a country. One way to do this would be to find the incomes of everybody in the country and find the mean. This would be the population mean, because you find data for the whole population.

The trouble with this approach is that you would have to find data on millions of people, which would be time-consuming, costly and perhaps not possible.

Instead, you would take a sample from the population. This would be a subset of the population, a group selected from the populaion but much smaller.

You could then find the mean income of this sample. This would be the sample mean.

You would use the sample mean to estimate the population mean.

# Outliers

Sometimes in a group of numbers, a few of them are much larger...

3, 5, 2, 7, 150

...or much smaller than the rest:

1, 502, 847, 564, 980

These relatively large (150) or small (1) numbers are called outliers.

The mean is very affected by outliers, unlike the median. For example, consider the numbers:
1, 2, 3
The mean is 2 and the median is 2.

If instead the numbers have a large outlier:
1, 2, 300
The mean is now 101 while the median is still 2.

If you read about the average salary in a country, ask if it is the mean or median average?

The mean average may dragged up by some very rich people, while the median won't be.

# Sum Over Count

The mean is the sum of the number divided by a count of the numbers.