Table of Contents

Difference Between Arithmetic Mean and Geometric Mean

Relation Between AM, GM, and HM

How to Calculate the Geometric Mean?

Real-life Applications on Geometric Mean

Common Mistakes and How to Avoid Them in Geometric Mean

Solved Examples

FAQs on Geometric Mean

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Last updated on June 4th, 2025

Geometric Mean

Q: When should we use geometric mean and arithmetic mean?

We use geometric mean when we want to deal with the rate of change, growth rates, or ratios.

Q: Can geometric mean be calculated with negative numbers?

No, geometric mean cannot be calculated with negative numbers, because taking the root of negative numbers can lead to complex numbers.

Q: What if the data contains a zero?

If any value in the dataset is zero, then the product becomes zero as well. So the geometric mean would be zero, which would not be correct.

Q: Why is Geometric mean important in finance?

Geometric mean is used to calculate average returns on investments, so it provides an accurate measure of the compounded performance over time.

Q: Can we use percentage values in geometric mean?

Yes, we can use percentage values in geometric mean. They are quite often used when using geometric mean, as geometric mean deals with growth rates. But we first need to convert the percentage to a decimal value.

Q: How can children in Australia use numbers in everyday life to understand Geometric Mean?

Numbers appear everywhere—from counting money to measuring ingredients. Kids in Australia see how Geometric Mean helps solve real problems, making numbers meaningful beyond the classroom.

Q: What are some fun ways kids in Australia can practice Geometric Mean with numbers?

Games like board games, sports scoring, or even cooking help children in Australia use numbers naturally. These activities make practicing Geometric Mean enjoyable and connected to their world.

Q: What role do numbers and Geometric Mean play in helping children in Australia develop problem-solving skills?

Working with numbers through Geometric Mean sharpens reasoning and critical thinking, preparing kids in Australia for challenges inside and outside the classroom.

Q: How can families in Australia create number-rich environments to improve Geometric Mean skills?

Families can include counting chores, measuring recipes, or budgeting allowances, helping children connect numbers and Geometric Mean with everyday activities.

Professor Greenline Explaining Math Concepts

Geometric mean is the average of numbers, which is calculated by multiplying the numbers and taking the appropriate root. In statistics, the measures we use to calculate the central tendency of the whole data set are mean, median, and mode. The mean is a statistical measure that calculates the average of a dataset, providing insight into its overall distribution. In this topic, we are going to learn about geometric mean, the relation, and differences between GM, AM, HM, and its uses in our daily lives.

Geometric Mean for Australian Students

What is Geometric Mean?

Geometric mean is the average of a set of data that is calculated by multiplying the ‘n’ variables and then taking the nth root. The nth root is the total number of values in the dataset. This is why the geometric mean is also defined as the nth root of the product of ‘n’ numbers.

It must be noted that geometric mean is different from arithmetic mean, where the data values are added and then divided by the total number of values. To calculate the geometric mean, we use the formula:

Geometric mean (GM) = √nx1 × x2 × x3 ......× xn

Here n is the total number of terms.

Some Key Takeaways From Geometric Mean

It is the most appropriate for series that show serial correlation (data that relates to previous values).

Geometric mean is an average of a set of values that are calculated using the product of terms.

Difference Between Arithmetic Mean and Geometric Mean

Students sometimes get confused between AM and GM so we are going to discuss the differences between Arithmetic mean and Geometric mean:

Arithmetic Mean	Geometric Mean
The arithmetic mean is the sum of all numbers divided by the total number count in a dataset.	The geometric mean is the nth root of the product of the values in the dataset. Where nth root is the total number of values in a dataset.
The formula we use is: AM = x1 + x2 + x3 ......+ xnn	The formula we use is: GM = nx1 × x2 × x3 ......× xn
We use arithmetic mean when we want to calculate a general average of the values, such as test scores or income	We use geometric mean when we want to calculate population growth or investment returns.
For example, the arithmetic mean of 2, 3, and 5 is: 2 + 3 + 53 = 103 ≈ 3.33	For example, the geometric mean of 2, 3, and 5 is: 32 × 3 × 5 ≈ 3.11

Relation Between AM, GM, and HM

Let us first understand what AM, GM, and HM mean before learning how they are related to one another.

The arithmetic mean (AM) is the average of two or more numbers. It is calculated by adding all the numbers and then dividing the sum by the total count. The formula we use is:

x1 + x2 + x3 ......+ xnn

Geometric mean is when we multiply all the numbers of the dataset and take the nth root. Where nth is the total number of values in the dataset. The formula we use:
GM = nx1 × x2 × x3 ......× xn

Harmonic mean is a type of Pythagorean mean, where we divide the numbers of terms in the data by the sum of all reciprocal terms. The formula we use is n1x1 + 1x2 + 1x3 ..... + 1x3

Now that we have understood the meaning of each of them, let us take a look at how they are related to each other.

For two positive numbers a and b:

Arithmetic mean (AM): AM = a + b2

Geometric Mean (GM): GM = a × b

Harmonic Mean (HM): HM = 21a + 1b = 2aba + b

Now we multiply AM and HM:

AM × HM = (a + b2) × (2aba + b)

(a + b) cancels out in the numerator and denominator we get:

AM × HM = a × b × a × b = (GM)2

So the relation between AM, HM, and GM is: GM2 = AM × HM.

How to Calculate the Geometric Mean?

Now that we know the relationship between geometric mean, arithmetic mean, and harmonic mean. Let us now learn how to calculate the geometric mean using the formula. nx1 × x2 × x3 ......× xn

Here n is the total number of values, x1, x2, and so on are the numbers in the dataset.

Let us use this formula in an example,

Find the geometric mean of two numbers (2 and 8)

The formula is GM = nx1 × x2 × x3 ......× xn

GM = 2 × 8=16=4

Real-life Applications on Geometric Mean

Geometric mean is widely used in our daily lives. Here are some real-world applications of where we use geometric mean.

Finance and investment: Geometric mean can be used to calculate the average growth rate of investments or stock market returns.

Studying population growth: We use geometric mean to calculate the average growth rate in the population, where the growth factors are multiplied.

In sports: Geometric mean is used in sports statistics to average the performance scores were again the results can be multiplicative, such as in races.

Common Mistakes and How to Avoid Them in Geometric Mean

Students can make mistakes when solving geometric mean. So here are a few mistakes that students make and ways to avoid them:

Mistake 1

Not realizing that only positive numbers can be used

Geometric mean can only be calculated using positive numbers. If we calculate using negative numbers or zeros, then the result will be invalid. Students must ensure that all numbers in the dataset are positive.