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?

Definition of 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 n^th 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

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:

$$\begin{gathered} \frac{left}{altitude}=\frac{altitude}{right} \\ \Rightarrow \frac{a}{x}=\frac{x}{b} \\ \Rightarrow ab=x^2 \\ \Rightarrow x=\sqrt{ab} \end{gathered}$$

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 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.

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.