Geometric Mean Questions

1.0Geometric Mean Definition

The geometric mean (GM) of a set of n positive numbers is the n^th root of their product. It provides a measure of central tendency that is particularly useful when comparing quantities that multiply together over time, such as growth rates, interest rates, or ratios. The geometric mean is defined as follows:

For a set of numbers $a_1,a_2,\dots ,a_n$ , the geometric mean is given by:

$GM=\sqrt[n]{a_1\cdot a_2\cdots a_n}$

Alternatively, this can be expressed using logarithms, which is often more convenient for large datasets:

$\ln (GM)=\frac{1}{n}{\sum }_{i=1}^n\ln \left(a_i\right)$

$GM=e^{\frac{1}{n}{\sum }_{i=1}^n\ln \left(a_i\right)}$

$\ln (GM)=\frac{1}{n}{\sum }_{i=1}^n\ln \left(a_i\right)$

$GM=e^{\frac{1}{n}{\sum }_{i=1}^n\ln \left(a_i\right)}$

2.0Key Properties of the Geometric Mean

1. Applicability: The geometric mean is only defined for positive numbers.
2. Multiplicative Nature: It is particularly suited for datasets that involve multiplication or exponential growth.
3. Relationship with Arithmetic Mean: The geometric mean of a dataset is always less than or equal to the arithmetic mean of the same dataset, with equality holding only when all the elements of the dataset are equal.

Example Calculation

Consider the numbers 4, 8, and 16. The geometric mean is calculated as follows:

$GM=\sqrt[3]{4\cdot 8\cdot 16}=\sqrt[3]{512}=8$

This result shows that the geometric mean provides a central value that represents the data in a multiplicative sense.

3.0How to Calculate the Geometric Mean of 3 Numbers

The geometric mean (GM) of three numbers a, b, and c is calculated using the following formula:

$GM=\sqrt[3]{a\cdot b\cdot c}$

This involves taking the product of the three numbers and then finding the cube root of the result.

Step-by-Step Example

Let's calculate the geometric mean of the numbers 4, 8, and 16.

1. Multiply the numbers together:

4×8×16=512

2. Find the cube root of the product:

$GM=\sqrt[3]{512}$

3. Calculate the cube root:

GM = 8 

So, the geometric mean of 4, 8, and 16 is 8.

Additional Example

Calculate the geometric mean of 3, 5, and 7.

1. Multiply the numbers together:

3×5×7=105

2. Find the cube root of the product:

$GM=\sqrt[3]{105}$

3. Calculate the cube root:

$GM\approx 4.72$

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

4.0What is Meant by Geometric Mean?

In mathematics, the geometric mean (GM) is a type of mean that represents the central tendency of a given set of data. The geometric mean is determined by taking the n^th root of the product of the data values, where ‘n’ is the total number of values in the dataset.

Note: The arithmetic mean differs from the geometric mean.

5.0Geometric Mean Formula

To find the geometric mean (GM) of a dataset with values $a_1,a_2,a_3,\dots ,a_n$ , use the following formula:

Geometric Mean $=\sqrt[n]{a_1\times a_2\times \cdots \times a_n}$

Alternatively, the geometric mean can be expressed as:

$\text{ Geometric Mean }={\left(a_1\times a_2\times \cdots \times a_n\right)}^{1/n}$

These formulas provide a way to calculate the central tendency of a set of values by taking the n^th root of their product, where n represents the total number of values in the dataset.

6.0Geometric Mean Solved Questions

Example 1: Find the geometric mean of the numbers 3, 7, 10, 4, and 15.

Solution: Given data values: 3, 7, 10, 4, and 15

To find the geometric mean, we use the formula:

$\text{ Geometric Mean }={\left(a_1\times a_2\times \cdots \times a_n\right)}^{1/n}$

Substituting the values into the formula, we get:

$\text{ Geometric Mean, }\mathrm{GM}=(3\times 7\times 10\times 4\times 15{)}^{1/5}$

Calculating the product:

$3\times 7\times 10\times 4\times 15=12600$

Finding the 5th root of 12600:

$\mathrm{GM}=(12600{)}^{1/5}\approx 6.608$

Therefore, the geometric mean of 3, 7, 10, 4, and 15 is approximately 6.608 (rounded to two decimal places).

Example 2: Find the geometric mean of the numbers 4, 8, 15, 16, and 23.

Solution: Given data values: 4, 8, 15, 16, and 23

$\text{ Geometric Mean, }\mathrm{GM}=(4\times 8\times 15\times 16\times 23{)}^{1/5}$

Calculating the product:

$4\times 8\times 15\times 16\times 23=176640$

Finding the 5th root of 176640:

$\mathrm{GM}=(176640{)}^{1/5}\approx 11.205$

Therefore, the geometric mean of 4, 8, 15, 16, and 23 is approximately 11.205.

Example 3: Find the geometric mean of the numbers 1, 3, 9, 27, and 81.

Solution: Given data values: 1, 3, 9, 27, and 81

$\text{ Geometric Mean, }\mathrm{GM}=(1\times 3\times 9\times 27\times 81{)}^{1/5}$

Calculating the product:

$1\times 3\times 9\times 27\times 81=59049$

Finding the 5th root of 59049:

$\mathrm{GM}=(59049{)}^{1/5}=9$

Therefore, the geometric mean of 1, 3, 9, 27, and 81 is 9.

Example 4: Calculate the geometric mean of 5, 25, 125, and 625.

Solution: Given data values: 5, 25, 125, and 625

$\text{ Geometric Mean, GM }=(5\times 25\times 125\times 625{)}^{1/4}$

Calculating the product:

$5\times 25\times 125\times 625=9765625$

Finding the 4th root of 9765625:

$\mathrm{GM}=(9765625{)}^{1/4}=55.901$

Therefore, the geometric mean of 5, 25, 125, and 625 is 55.901.

Example 5: Determine the geometric mean of 2, 5, 11, and 13.

Solution:

Given data values: 2, 5, 11, and 13

$\text{ Geometric Mean, GM }=(2\times 5\times 11\times 13{)}^{1/4}$

Calculating the product:

$2\times 5\times 11\times 13=1430$

Finding the 4th root of 1430:

$\mathrm{GM}=(1430{)}^{1/4}\approx 6.149$

Therefore, the geometric mean of 2, 5, 11, and 13 is approximately 6.149.

Example 6: Find the geometric mean of 10, 50, 100, and 500.

Solution: Given data values: 10, 50, 100, and 500

$\text{ Geometric Mean, }\mathrm{GM}=(10\times 50\times 100\times 500{)}^{1/4}$

Calculating the product:

$10\times 50\times 100\times 500=25000000$

Finding the 4th root of 25000000:

$\mathrm{GM}=(25000000{)}^{1/4}\approx 70.710$

Therefore, the geometric mean of 10, 50, 100, and 500 is approximately 70.710.

7.0Geometric Mean Practice Question

1. Find the geometric mean of the numbers 5, 10, 15, 20, and 25.
2. Calculate the geometric mean of the numbers 7, 14, 21, and 28.
3. Determine the geometric mean of 3, 6, 9, and 12.
4. Find the geometric mean of the numbers 2, 4, 8, and 16.
5. Calculate the geometric mean of 1, 4, 16, and 64.

Solutions:

1. 13.025
2. 15.4936
3. 6.640
4. 5.656
5. 8

8.0Sample Question on Geometric Mean

Q. What is the formula for the Geometric Mean?

Ans: The formula for the geometric mean of a set of values ${\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3,\dots ,{\mathrm{a}}_{\mathrm{n}}$ is:

$\text{ Geometric Mean }={\left(a_1\times a_2\times \cdots \times a_n\right)}^{1/n}$