The Lesson

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.

Question

What is the mean of the test scores below?

Step-by-Step:

1

Add up the numbers.
7 + 10 + 8 + 6 + 4 = 35

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

Answer:

The mean of the test scores is 7.

Lesson Slides

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

Top Tip

Sum Over Count

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

Note

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.

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