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# Geometric Mean

By this point, you have probably heard the term 'mean' used many times in math. But what do we actually mean? (pardon the pun). There are different types of mean, and one that you have probably come across is called the arithmetic mean. This is where we take a set of numbers, add them up and divide this number by however many numbers you have to find an "average" of the numbers. For example, you might wish to know the mean score in a class test to help determine whether you performed on average, below average or above average. In this article, however, we will be looking at a different type of mean called the geometric mean. So, what do we mean by geometric mean

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## Definition of Geometric Mean

Geometric mean is defined as the average rate of return of set of values which is calculated using the products of its terms.

Suppose we have a set of n numbers. The geometric mean is where we multiply together the set of numbers and then take the positive nth root. So, if we have two numbers we would multiply them and then take the positive square root, if we had three numbers we would multiply them and then take the positive cube root, if we had four numbers we would multiply them and then take the positive fourth root and so on.

## Geometric Mean Formula

### Geometric Mean Definition

For the set of n numbers, ${x}_{1},{x}_{2},...,{x}_{n}$, the formula for the geometric mean is given by the following:

${\left(\underset{i=1}{\overset{n}{\prod {x}_{i}}}\right)}^{\frac{1}{n}}=\sqrt[n]{{x}_{1}{x}_{2}...{x}_{n}}$

### Geometric Mean Examples

Suppose we have the set of two numbers 9 and 4. To find the geometric mean, we would first multiply together 9 and 4 to get 36 and since we have two numbers we would take the square root of 36 to get six. Mathematically, we can write ${\left(9×4\right)}^{\frac{1}{2}}={36}^{\frac{1}{2}}=\sqrt{36}=6$. Thus, the geometric mean is 6.

Suppose we have the set of numbers 4, 8 and 16. To calculate the geometric mean we first multiply together 4, 8 and 16 to obtain 512. Since there are three numbers we then take the cube root. Mathematically, we can write ${\left(4×8×16\right)}^{\frac{1}{3}}={\left(512\right)}^{\frac{1}{3}}=8$. Thus 8 is the geometric mean of our numbers.

Suppose we have the set of numbers 1, 2, 3, 4 and 5. To find the geometric mean we first multiply together 1, 2, 3, 4, and 5 to obtain 120. Since we have five numbers, we take the fifth root of 120 which is 2.61 to 2 decimal places. Mathematically, we can write ${\left(1×2×3×4×5\right)}^{\frac{1}{5}}=\left(120{\right)}^{\frac{1}{5}}=2.61$. Thus, the geometric mean is 2.61.

## The Geometric Mean in a Triangle

Calculating the geometric mean can be particularly useful in geometry. Consider the below triangle ABCD:

Geometric Mean Triangle, Jordan Madge- StudySmarter Originals

The altitude of a triangle is a line drawn from the particular vertex of a triangle which forms a perpendicular line to the base of the triangle. So in this triangle, the altitude is the line AC. We also have the left side of BD, which is BC, as well as the right side of BD, which is CD.

Now, notice that if we "pull apart" the above triangle, we get two smaller triangles. We also notice that if we rotate the triangle on the left, we simply have a smaller version of the triangle on the right. This is shown in the diagram below.

Explaining the geometric mean, Jordan Madge- StudySmarter Originals

Now, notice that the two triangles BAC and ADC are mathematically similar, so we can use ratios to find missing lengths. Naming the left side a, the right side b, and the altitude x, we have the following:

$\frac{left}{altitude}=\frac{altitude}{right}\phantom{\rule{0ex}{0ex}}⇒\frac{a}{x}=\frac{x}{b}\phantom{\rule{0ex}{0ex}}⇒ab={x}^{2}\phantom{\rule{0ex}{0ex}}⇒x=\sqrt{ab}$

Therefore, the altitude, x can be calculated by finding the geometric mean of a and b. This is known as the geometric means theorem for triangles.

Geometric mean example, Jordan Madge- StudySmarter Originals

In the triangle ABCD, BC=6 cm, CD=19 cm and AC= x cm as shown above. Find the value of the altitude x.

Solution:

Using results from the geometric means theorem for triangles, we find that id="2619448" role="math" $x=\sqrt{6×19}=\sqrt{114}=10.7$cm (1. d.p)

Geometric mean example, Jordan Madge- StudySmarter Originals

In the triangle ABCD, BC=4 cm, CD=9 cm and AC= x cm as shown above. Find the value of the altitude x.

Solution:

Using results from the geometric means theorem for triangles, we obtain that id="2619458" role="math" $x=\sqrt{9×4}=\sqrt{36}=6$cm

## Geometric vs Arithmetic mean

When we refer to the mean of a set of numbers, we usually are referring to the arithmetic mean. The arithmetic mean is when we take the sum of the set of numbers and then divide it by how many numbers we have.

### Arithmetic Mean Formula

The formula for the arithmetic mean is given by the following:

$A=\frac{1}{n}\underset{i=1}{\overset{n}{\sum {a}_{i}}}$

Here, A is defined as the value of the arithmetic mean, n is how many values there are in the set, and ${a}_{i}$ are the numbers in the set.

Find both the arithmetic and geometric mean of the numbers 3, 5 and 7.

Solution:

To obtain the arithmetic mean, we would first add together 3, 5 and 7 to obtain 15. Then, since we have three numbers in our set, we would divide 15 by 3 to get 5. Mathematically, we can write:

$A=\frac{1}{3}\left(3+5+7\right)=5$

To obtain the geometric mean, we would first multiply together 3, 5 and 7 to get 105 and then take the cube root of 105 (since we have three numbers in our set). The cube root of 105 is 4.72 to 2. d.p and thus the geometric mean of the numbers is 4.72. Mathematically, we can write:

$G={\left(3×5×7\right)}^{\frac{1}{3}}=\left(105{\right)}^{\frac{1}{3}}=4.72$

Notice that the arithmetic mean of 5 is quite close to the geometric mean of 4.72. We will now explore the different reasons we may use the geometric mean as opposed to the arithmetic mean.

### Geometric and Arithmetic Mean Differences

There are several key differences between both the geometric and arithmetic mean. The first, most obvious difference is the fact that they are calculated using two different formulae. In the previous example, we obtained an arithmetic mean of 5 and a geometric mean of 4.72. It is important to note that the geometric mean is always less than or equal to the arithmetic mean. For example, if we take the singleton set $\left\{2\right\}$,since there is only one number in this set, the geometric mean is 2 and the arithmetic mean is also 2.

Moreover, the arithmetic mean can be used for both positive and negative numbers. However, this is not the case for the geometric mean; the geometric mean can only be used for a set of positive numbers. This is due to the fact that error may arise in eventualities such as taking the square root of negative numbers.

Further, we use the geometric and arithmetic mean for different reasons. The arithmetic mean has a plethora of everyday uses, however, the geometric mean is more commonly used when there is some sort of correlation between the set of numbers. For example, in finance, the geometric mean is used when calculating interest rates. The arithmetic mean may be useful when finding the average temperature over a week.

There is actually a third type of mean called the harmonic mean. The harmonic mean is calculated by squaring the geometric mean and dividing it by the arithmetic mean. This type of mean is commonly used in machine learning.

## Geometric Mean - Key takeaways

• The geometric mean is where we multiply together the set of numbers and then take the positive nth root.
• It can be represented by the formula ${\left(\underset{i=1}{\overset{n}{\prod {x}_{i}}}\right)}^{\frac{1}{n}}=\sqrt[n]{{x}_{1}{x}_{2}...{x}_{n}}$.
• Calculating the geometric mean can be particularly useful in geometry.
• The altitude of a triangle is a line drawn from the particular vertex of a triangle which forms a perpendicular line to the base of the triangle.
• The geometric mean theorem for triangles can be used to calculate the altitude of a triangle.
• The geometric mean is always less than or equal to the arithmetic mean.
• The arithmetic mean is represented by the formula $A=\frac{1}{n}\underset{i=1}{\overset{n}{\sum {a}_{i}}}$.
• The geometric mean is more commonly used when there is some sort of correlation between the set of numbers. For example, when calculating interest rates.

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##### Frequently Asked Questions about Geometric Mean

When to use geometric mean?

The geometric mean is commonly used when there is some sort of correlation between the set of numbers. For example, in finance, the geometric mean is used when calculating interest rates

How to find the geometric mean?

Suppose we have a set of n numbers. The geometric mean is where we multiply together the set of numbers and then take the positive nth root.

How to find the geometric mean of two numbers?

Multiply together the two numbers and take the square root.

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