The arithmetic mean, or "average" is a measure of the "middle" or "typical" value of a data set. It is the sum of a collection of numbers divided by the number of numbers in that collection. While it is often referred to simply as "mean" or "average," the term "arithmetic mean" is preferred in some contexts because it helps distinguish it from other means, such as the geometric mean and the harmonic mean.

The arithmetic mean is used frequently not only in mathematics and statistics but also in fields such as economics, sociology, and history. For example, per capita income is the arithmetic mean income of a nation's population.

Suppose we have a data set containing the values $a_1, \dots, a_n$. The arithmetic mean $A$ is defined via the expression:

$\displaystyle A = \frac{1}{n} \sum_{i=1}^n a_i = \frac{1}{n}(a_1 + \cdots + a_n)$

This is simply a mathematical way of writing "the mean equals the sum of all of the values in the set, divided by the number of values in the set."

Example 1

To see how this applies to an actual set of numbers, consider the following set: $\{3,5,10\}$.

In order to find the average, we must first find the sum of the numbers:

$3 + 5 + 10 = 15$

Next, divide their sum by 3, the number of values in the set:

$\dfrac{15}{3} = 5$

Therefore, the average of the set of numbers $\{3,5,10\}$ is 5.

Example 2

Find the average of the following set of numbers: $\{12, 25, 34, 17, 8, 42\}$

We need to add the values together and then divide that sum by the total number of values, which is 6.

$\dfrac{12 + 25 + 34 + 17 + 8 + 42}{6} = \dfrac{136}{6} = 23$

The average of this set is 23. 

Example 3

Consider the following real-life situation. A small company has 8 employees. Two of those employees are paid $35 per hour, two of them are paid $27 per hour, and four are paid $25 per hour. What is the average hourly wage of these 8 employees?

We need to add together each of the hourly salaries and then divide by 8, the number of employees: 

$\dfrac{35 + 35 + 27 + 27 + 25 + 25 + 25 + 25}{8}= \dfrac{224}{8} = 28$

Therefore, employees of this company are paid an average hourly wage of $28.