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Watch this video lesson to learn how different the geometric mean is from the arithmetic mean, or average, that we are all so familiar with. Learn in what circumstance the geometric mean is preferred. Also, learn the formula and how to use it.

Geometric Mean

The geometric mean is the nth root when you multiply n numbers together. It is not the same as the arithmetic mean, or average, that we know. For the arithmetic mean, we add our numbers together and divide by how many numbers we have. The geometric mean uses multiplication and roots. For example, for the product of two numbers, we would take the square root. For the product of three numbers, we take the third root.

Uses

When would we use the geometric mean as opposed to the arithmetic mean? We would use the geometric mean when we want to figure out the average rate of growth if the growth rate is determined by multiplication. For example, the average percentage amount of growth in a bank account per year uses the geometric mean since the growth each year is determined by multiplying the amount in the bank account by the percentage growth.

The arithmetic mean is used when the growth is determined by addition. For example, the average amount it costs to feed people at a party uses the arithmetic mean because the growth in cost is determined by addition. The geometric mean does have one limitation - it cannot be used for negative numbers; all the numbers have to be positive. You can, however, find the geometric mean for as many numbers as you want.

Formula

The formula to find the geometric mean is this:

It tells us to first multiply all the numbers we have together and then to take the nth root of that product. If we had two numbers, we take the square root; if we have four numbers, we take the fourth root. Most calculators have a button next to the square root button that lets you choose the root. This button lets you calculate the third root, fourth root, fifth root, etc. - whichever root you need. Now some calculators will have you choose the root before pushing the button, while others have you choose the root after pushing the button. Get familiar with your calculator first so you know how yours will calculate your roots.

Using the Formula

Let's look at some examples to see how we can use this formula. Say we want to find the geometric mean of 3, 4, and 9. What does the formula tells us to do first? It tells us to multiply our numbers together first. So let's do that: 3 * 4 * 9 = 108. Then the formula tells us to take the third root since we have three numbers. Doing that, we get 4.7622 for our answer.

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Now, what if we complicated things a little bit and we wanted to get the geometric mean of the numbers 3, 4, 3, 2, 8, and 10? What do we do first? We do the same as before by multiplying all our numbers together. We don't stress out even if the numbers get really big. Multiplying everything together, we get 3 * 4 * 3 * 2 * 8 * 10 = 5760.

Do you see how we multiplied the 3 twice? That is because our data has the number 3 twice. Remember, whatever numbers the data shows, that is what you multiply together. Even if the numbers repeat, you still multiply them together. Okay, so we've multiplied all our numbers together - now what? Now we figure out which root we need. Let's see - how many numbers do we have? We have six numbers, so that means we will be taking the sixth root. Okay, so the sixth root of 5760 is 4.234, and that is our answer.

Lesson Summary

To review, the geometric mean is the nth root when you multiply n numbers together. It is used to calculate the average rate of growth when the growth is determined by multiplication as in the case of annual percentage growth of a bank account. The formula for geometric mean is found from its definition. You first multiply your numbers together and then you take the nth root where n is the number of numbers you multiplied together. So if you are multiplying three numbers together, you will take the third root. If you are multiplying eight numbers together, then you will take the eighth root.

Learning Outcomes

Following this lesson, you should have the ability to:

Define geometric mean

Explain when you would want to use the geometric mean instead of the arithmetic mean

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