The Lesson

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."

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,
  • x̄ 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 x̄. We use the sample mean to estimate the population mean.

Note

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".

The Arithmetic Mean

The mean we have mean discussing on this page is more specifically called the arithmetic mean. It is the result of adding up the numbers and dividing. There are other types of mean. For example, the geometric mean is the result of multiplying the numbers together and finding the nth root (where n is the number of numbers).

The Mean and Outliers

The mean has a weakness.

Consider the following numbers:
1, 2, 3, 4, 5
The mean is 3.

Let's replace the 5 with 50:
1, 2, 3, 4, 50
The mean is now 12.

We call the 50 an outlier, because it is so much larger than the other numbers. (An outlier called also be much smaller, like 0.05).

The mean has increased significantly because of the large number, but it is now less typical of the numbers. Most of the numbers are very small (less than 5), but the presence of one large number has made the mean larger than is typical. This mean is not wrong, but might be misleading.

For example, imagine you wanted to find the average income in a country. The presence of a few billionaires would drag the mean upwards, so that the mean might be higher than a typical middle-class income. To get around this problem, you might use the median (or middle) number. The median of our five numbers is 3, regardless of whether the fifth number is 5 or 50. The median is less sensitive to outliers.

Therefore, the median income might be more typical of middle-income citizen.