When I say the word "mean," you probably automatically think of your math teacher. But "mean" isn't just an adjective used to describe your math teacher. It's also a word used to describe an average. For example, if you have test grades of 72, 98, and 87, you'll want to average those to find out what your grade is. That's called averaging, or taking the arithmetic mean.

But did you know that there are multiple ways of averaging a set of data points? It's true! That's why we put the word "arithmetic" in front of the word "mean," because there are multiple types of averages, or means. The three we will look at in this unit are the arithmetic mean (the one you're used to using), the geometric mean, and the harmonic mean. On this page we'll talk about the arithmetic mean.

The arithmetic mean is easy to calculate. If you have two numbers, you add them up and divide them by two. If you have three numbers, you add them up and divide by three. If you have n numbers, you add them up and divide by n.

A fancy formula for the arithmetic mean is:

A =

Let's find your average test scores given above:

A = = 85.67

Congratulations, you've got a B!

Sometimes students will want to figure out what they need to get on a test to get a certain grade. That's easy to do using arithmetic means.

Example
My test grades are 86, 98, and 104. I have one more test to take, and I want my average to be at least 80. What's the lowest grade I could get?

Solution
Never mind the fact that students should never be asking what's the minimum that's okay, it's still easy to figure out. We want the average to be 80, so we plug that in for A in our formula:

80 =

Solving this gives x = 32.

Example
I measured the temperature every hour from 8:00 until 12:00, and the average was 48 degrees. Then my dog chewed on the paper I wrote the data on, and now I can't read my 12:00 value. Fortunately I can read all the others: 45, 45, 47, 48. What was the temperature at 12:00?

Solution
48 =

x = 48 degrees.

When to Use
So when do you use the arithmetic mean (instead of the harmonic or geometric)? Well, most of the time you'll use the arithmetic mean. Basically, you use the arithmetic mean in all the circumstances when you wouldn't use the harmonic or geometric mean. I know that doesn't seem to be very helpful, but until we study the other two means, that's the best answer I can give you!

1.

Find the arithmetic mean of the numbers 32, 34, 36, and 38

2.

Find the arithmetic mean of the numbers 100 and 149

3.

Two numbers have an arithmetic mean of 13. One of the numbers is 18. What is the other?

4.

Three numbers have an arithmetic mean of 30. Two of the numbers are 24 and 26. What is the other number?

5.

Three numbers have an average of 55. One of the numbers is n less than the average, and one is n more than the average. What is the third number?

6.

Three numbers have an arithmetic mean of 45. One of the numbers is 42. What is the arithmetic mean of the other two numbers?

7.

My current test average is 88. If I get a 95 on my next test, I'll have an average of 89. How many tests will I have in all?