Geometric Mean

In mathematics and statistics, summarizing a data set can be effectively achieved using measures of central tendency. The most significant measures include the mean, median, mode, and range. Among these, the mean provides a comprehensive overview of the data. The mean represents the average of the numbers in the data set and comes in different types: Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM). In this article, we will delve into the definition, formula, properties, and applications of the geometric mean, and explore its relationship with the arithmetic and harmonic means, accompanied by solved examples for better understanding.

1.0Geometric Mean Definition

In mathematics, the Geometric Mean (GM) is a measure of central tendency that signifies the average value of a set of numbers by considering the product of their values. Essentially, it is calculated by multiplying all the numbers and then taking the nth root of the total product, where n is the total number of values in the set. For example, for a given set of two numbers, such as 1 and 3, the geometric mean is calculated as follows: $\sqrt{3\times 1}=\sqrt{3}=1.732$ .

In other words, the geometric mean is defined as the nth root of the product of n numbers. It differs from the arithmetic mean, which involves summing the data values and then dividing by the total number of values. In contrast, the geometric mean involves multiplying the data values together and then taking the nth root, where n is the total number of values. For instance, if we have two data values, we take the square root; if we have three data values, we take the cube root; if we have four data values, we take the fourth root, and so on.

2.0Geometric Mean Formula

The formula to calculate the Geometric Mean(GM) is as follows:

The Geometric Mean (G.M) of a set of n observations is calculated by taking the nth root of the product of these values.

If we have observations x₁, x₂, …., x_n , then the G.M is defined as:

$\text{ G.M }=\sqrt[n]{x_1\cdot x_2\cdot \dots \cdot x_n}\text{ or }\mathrm{GM}={\left(x_1\cdot x_2\cdot \dots \cdot x_n\right)}^{\frac{1}{n}}$

This can also be expressed as:

$\text{ G.M }={\left(x_1\cdot x_2\cdot \dots \cdot x_n\right)}^{\frac{1}{n}}$

Therefore, the Geometric Mean is:

$\text{ G.M }=\sqrt[n]{x_1\cdot x_2\cdot \dots \cdot x_n}$

Where $\mathrm{n}=f_1+f_2+\dots +f_n$ , representing the sum of frequencies for each observation.

For any grouped data, the Geometric Mean can be written as

$\text{ G.M }={\left({\prod }_{i=1}^n{x}_i^{f_i}\right)}^{{\sum }^1{f}_i}$

$GM=$$\text{ Antilog }\frac{\sum \log x_i}{n}$

3.0Difference Between Arithmetic Mean (AM) and Geometric Mean (GM)

4.0Relationship Between Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM)

The Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM) are all measures of central tendency, each useful in different contexts. They are related through various mathematical inequalities and properties, often summarized by the following relationship:

$AM\ge GM\ge HM$

This relationship holds for any set of positive numbers and can be demonstrated through inequalities. Here's a detailed look at each mean and their relationships:

Arithmetic Mean (AM)

• Definition: The sum of all data values divided by the total number of values.
• Formula: $\mathrm{AM}=\frac{x_1+x_2+\dots +x_n}{n}$

Geometric Mean (GM)

• Definition: The nth root of the product of all data values.
• Formula: $\mathrm{GM}=\sqrt[n]{x_1\cdot x_2\cdot \dots \cdot x_n}$

Harmonic Mean (HM)

• Definition: The reciprocal of the arithmetic mean of the reciprocals of the data values.
• Formula: $\mathrm{HM}=\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\dots +\frac{1}{x_n}}$

5.0Relationship and Inequalities

Arithmetic Mean-Geometric Mean Inequality (AM-GM Inequality)

For any set of positive numbers $x_1,x_2,\dots ,x_n$ ,

$\frac{x_1+x_2+\dots +x_n}{n}\ge \sqrt[n]{x_1\cdot x_2\cdot \dots \cdot x_n}$

or

$\mathrm{AM}\ge \mathrm{GM}$

Geometric Mean-Harmonic Mean Inequality (GM-HM Inequality)

For any set of positive numbers $x_1,X_2,\dots ,x_n$ ,

$\sqrt[n]{x_1\cdot x_2\cdot \dots \cdot x_n}\ge \frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\dots +\frac{1}{x_n}}$

or

$\mathrm{GM}\ge \mathrm{HM}$

Combined Inequality

Combining these inequalities, we get the comprehensive relationship:

$\mathrm{AM}\ge \mathrm{GM}\ge \mathrm{HM}$

Example to Illustrate the Relationship

Consider three numbers: 2, 3, and 6.

1. Arithmetic Mean (AM):

$\mathrm{AM}=\frac{2+3+6}{3}=\frac{11}{3}=3.67$

2. Geometric Mean (GM):

$\mathrm{GM}=\sqrt[3]{2\cdot 3\cdot 6}=\sqrt[3]{36}=\sqrt[3]{36}\approx 3.30$

3. Harmonic Mean (HM):

$\mathrm{HM}=\frac{3}{\frac{1}{2}+\frac{1}{3}+\frac{1}{6}}=\frac{3}{\frac{3+2+1}{6}}=\frac{3}{1}=3.00$

Thus,

$\mathrm{AM}=3.67\ge \mathrm{GM}\approx 3.30\ge \mathrm{HM}=3.00$

Understanding these relationships helps in choosing the appropriate measure of central tendency for different types of data and mathematical problems.

6.0Important Properties of Geometric Mean

Some of the important properties of the Geometric Mean (G.M) are:

1. The G.M for a given data set is always less than or equal to the arithmetic mean (A.M) of the data set.
2. If each value in the data set is replaced by the G.M, the product of the values remains unchanged.
3. The ratio of corresponding values in two data sets is equal to the ratio of their geometric means.
4. The product of corresponding values in two data sets is equal to the product of their geometric means.

7.0Geometric Mean Solved Examples 

Example for Ungrouped Data

Example 1: Consider a set of numbers: 4, 1, and 16. We want to find the geometric mean.

Solution:

1. Multiply the numbers:  4 × 1 × 16 = 64. 

2. Take the nth root (since there are 3 numbers, we take the cube root):

$\mathrm{GM}=\sqrt[3]{64}=4$

So, the geometric mean of 4, 1, and 16 is 4.

Example 2: Consider the numbers: 5, 2, 8, and 10. We want to find the geometric mean.

Solution:

1. Multiply the numbers:  5×2×8×10 = 800

2. Take the nth root (since there are 4 numbers, we take the fourth root):

$\mathrm{GM}=\sqrt[4]{800}\approx 4.64$

So, the geometric mean of 5, 2, 8, and 10 is approximately 4.64.

Example 3: Consider the set of numbers: 7, 3, and 5. We want to find the geometric mean.

Solution:

1. Multiply the numbers: 7×3×5=105 

2. Take the nth root (since there are 3 numbers, we take the cube root):

$\mathrm{GM}=\sqrt[3]{105}\approx 4.72$

So, the geometric mean of 7, 3, and 5 is approximately 4.72.

Example 4: Consider the set of numbers: 9, 4, 12, and 6. We want to find the geometric mean.

Solution:

1. Multiply the numbers: 9×4×12×6=2592 

2. Take the nth root (since there are 4 numbers, we take the fourth root):

$\mathrm{GM}=\sqrt[4]{2592}\approx 6.25$

So, the geometric mean of 9, 4, 12, and 6 is approximately 6.25.

Example for Grouped Data:

Example 5: Consider the following grouped data:

Solution: We want to find the geometric mean for this grouped data.

1. Multiply each value by its frequency:

$2^3=8$

$3^2=9$

$6^1=6$

2. Multiply these results together:

$8\times 9\times 6=432$

3. Take the nth root (since the total frequency $\sum f_i=3+2+1=6$ ):

$\mathrm{GM}=\sqrt[6]{432}\approx 2.72$

So, the geometric mean for this grouped data is approximately 2.72.

Example 6: Consider the following grouped data:

We want to find the geometric mean for this grouped data.

1. Multiply each value by its frequency:

$\begin{array}{rl}& 4^3=64\\ & 5^2=25\\ & 8^1=8\end{array}$

Multiply these results together:

$64\times 25\times 8=12800$

2. Take the nth root (since the total frequency $\sum f_i=3+2+1=6$ ):

$\mathrm{GM}=\sqrt[6]{12800}\approx 4.57$

So, the geometric mean for this grouped data is approximately 4.57.

Example 7: Consider the following grouped data:

We want to find the geometric mean for this grouped data.

1. Multiply each value by its frequency:

$\begin{array}{rl}& 10^4=10000\\ & 15^3=3375\\ & 20^2=400\end{array}$

2. Multiply these results together:

$10000\times 3375\times 400=13500000000$

3. Take the nth root (since the total frequency $\sum f_i=4+3+2=9$):

$\mathrm{GM}=\sqrt[9]{13500000000}\approx 13.75$

So, the geometric mean for this grouped data is approximately 13.75.

8.0Geometric Mean Practice Questions

Ungrouped Data

1. Find the geometric mean of the numbers 5, 25, and 125.
2. Calculate the geometric mean for the numbers 1, 2, 4, and 8.
3. Determine the geometric mean for the set of numbers: 7, 14, 28.
4. What is the geometric mean of the numbers 3, 6, 9, 12, and 15?
5. Compute the geometric mean of 2, 5, and 10.

Grouped Data

6. The following table shows the frequencies of certain values. Calculate the geometric mean.

7. Given the grouped data, find the geometric mean.

Solutions

Ungrouped Data

1. $\mathrm{GM}=\sqrt[3]{5\times 25\times 125}=\sqrt[3]{15625}=25$
2. $\mathrm{GM}=\sqrt[4]{1\times 2\times 4\times 8}=\sqrt[4]{64}=2.83$
3. $\mathrm{GM}=\sqrt[3]{7\times 14\times 28}=\sqrt[3]{2744}\approx 14$
4.  $\mathrm{GM}=\sqrt[5]{3\times 6\times 9\times 12\times 15}=\sqrt[5]{29160}\approx 7.58$
5. $\mathrm{GM}=\sqrt[3]{2\times 5\times 10}=\sqrt[3]{100}\approx 4.64$

Grouped Data

6. $\mathrm{GM}=\sqrt[9]{2^4\times 5^3\times 8^2}=\sqrt[9]{2^4\times 5^3\times 64}=\sqrt[9]{102400}\approx 4.64$
7. $\mathrm{GM}=\sqrt[9]{3^3\times 6^4\times 9^2}=\sqrt[9]{27\times 1296\times 81}=\sqrt[9]{28224}\approx 4.55$

9.0Sample Questions on Geometric Mean

Q1. How do you calculate the geometric mean for ungrouped data?

Ans: For ungrouped data, the geometric mean is determined by multiplying all the values together and then extracting the nth root, where n is the number of values. The formula is:

$\mathrm{GM}=\sqrt[n]{x_1\cdot x_2\cdot \dots \cdot x_n}$

Q2. What is the geometric mean formula for ungrouped data?

Ans: For ungrouped data, the geometric mean is calculated as:

$\mathrm{GM}={\left(x_1\cdot x_2\cdot \dots \cdot x_n\right)}^{\frac{1}{n}}$