Given a list of numbers, it is easy to determine the arithmetic mean, or average. The average is simply the sum of the numbers in a given problem, divided by the number of numbers added together. For example, if four number are added together their sum is divided by four to find the average or arithmetic mean.

Average or arithmetic mean is sometimes confused with two other concepts: mode and median. The mode is the most frequent value in a set of numbers, while the median is the number in the middle of the range of a given set.

### Uses and Applications

It's important to know how to calculate the mean or average of a set of numbers. Among other things, this will allow you to calculate your grade point average. However, you'll need to calculate the mean for several other situations, too.

The concept of an average allows statisticians, demographers, economists, biologists, and other researchers to better understand the most common situations. For example, by determining the average income of an American family and comparing it to the average cost of a home, it's possible to better understand the magnitude of economic challenges facing most American families. Similarly, by looking at the average temperature in a particular area at a particular time of year, it's possible to predict the probable weather and make a wide range of decisions appropriately.

### Issues and Pitfalls

While averages can be very useful tools, they can also be misleading for a variety of reasons. In particular, averages can obscure the information contained in data sets. Here are a few examples of how averages can be misleading:

- John's grades include a 4.5 in math, a 4.0 in science, a 2.0 in English and a 2.5 in History. After averaging his scores, his advisor decided that John is a straight "B" student. In fact, however, John is quite talented in math and science and needs remediation in English and history.
- Ten people were in a room. One woman in the room was pregnant. Based on the average, therefore, everyone in the room was .1% pregnant. This is, of course, a false and ridiculous finding!

### The Calculation

In general, you calculate the mean or average of a set of numbers by adding them all up and dividing by how many numbers you have. This can be defined as follows:

For a set of numbers, {x1, x_{2}, x_{3}, ... x_{j}} the mean or average is the sum of all "x" divided by "j".

### Worked Examples

Let's start with an easy example. Calculate the mean of the following set of numbers:

1, 2, 3, 4, 5

To do this, add up the numbers and divide by how many numbers you have (5 of them, in this case).

mean = (1 + 2 + 3 + 4 + 5)/5

mean = 15/5

mean = 3

Here's another example of calculating the mean.

Calculate the mean of the following set of numbers:

25, 28, 31, 35, 43, 48

How many numbers are there? 6. So, add together all of the numbers and divide the total by 6 in order to get the mean.

mean = (25 + 28 + 31 + 35 + 43 + 48)/6

mean = 210/6

mean = 35