Relation Between A.M. and G.M.

Introduction

When learning about sequences in maths, it is obvious to come across the relation between arithmetic mean and geometric mean. This is an important topic while studying progressions. 

Before understanding the relationship between them, it’s important to understand these three methods and their formulas.

Arithmetic Mean

The arithmetic mean is a number obtained by dividing the sum of a set’s values by the total number of values in the set. If a₁, a₂, a₃,….,a_n, is a number of group of values or the Arithmetic Progression, then;

AM=(a₁+a₂+a₃+….,+a_n)/n

Geometric Mean

GM is a unique average that is calculated by multiplying the series together and then taking the n^th root of the result where n is the number of terms. GM = ⁿ√ (a₁ × a₂ × … × a_n)

Example: What is the Geometric Mean of 1, 4, 16, 64 and 256?

Solution:

Given

a₁ = 1

a₂ = 4

a₃ =16

a₄ = 64

a₅ = 256

First, we will multiply the given numbers

 1 × 4 × 16 × 64 × 256 = 1048576

Then take the 5th root (that is, the n^th root where n = 5)

⁵√1048576 = 16

Geometric Mean = ⁵√ (1 × 4 × 16 × 64 × 256) = 16

What Is the Relation Between Arithmetic Mean and Geometric Mean?

The geometric mean is calculated differently from the arithmetic mean or average because it takes into account the compounding that occurs over time. As a result, investors and finance professionals favour the geometric mean over the arithmetic mean since it is more accurate.

The statement that the value of AM is greater than the value of GM and HM explains the relation between AM and GM and HM. For the same set of data points, the arithmetic mean is higher than the geometric mean, which is higher than the harmonic mean. The following expression can be used to represent the relation between HM, Arithmetic Mean, and Geometric Mean.

AM >= GM >= HM [The equality holds if and only if all terms of series are same]

Here, the arithmetic mean is abbreviated as AM, the geometric mean is abbreviated as GM, and the harmonic mean is abbreviated as HM.

To comprehend this, we must first learn how to locate AM, GM, and HM. The arithmetic mean (AM), geometric mean (GM), and harmonic mean (H) formulas are as follows for any two numbers a, b. The arithmetic mean, commonly known as the average of the provided numbers, is equal to the sum of the two numbers divided by two for two numbers a, b.

The square root of the product of the two numbers a, b is equal to the geometric mean of the two numbers. Furthermore, if n data points are present, the geometric mean is equal to the nth root of the product of the n numbers.

GM = ⁿ√a₁a₂…..a_n

The arithmetic mean of these two numbers, 1/a, 1/b, is equal to (a + b)/2ab, and the harmonic mean of the two numbers a and b  is the inverse of this.

HM=2ab/(a+b)

Formula for Relation Between AM, GM and HM

The formula explaining the relationship between AM, GM, and HM is the product of arithmetic and harmonic means equals the square of the geometric mean. This can be expressed in the form of the following expression.

AM × HM = GM²

As a result, the geometric mean’s square equals the product of the arithmetic and harmonic means.

What is the AM, GM and HM Relationship?

AM, GM, and HM have the following relationship:

GM² = AM x HM

Let us now look at how this relationship is derived;

To begin, examine the Arithmetic Progression a, AM, b.

The following are the most prevalent differences in Arithmetic Progression:

AM – a = b – AM

a + b = 2 AM …………… (1)

Second, assume that a, GM, b are all Geometric Progressions. Then, this GP’s common ratio is:

GM/a = b/GM

ab = GM²…………… (2)

HM=2ab/(a+b)………………(3)

Thus it is clear that,

GM² = AM x HM

AM, GM, HM in Statistics:

In key calculations, AM, GM, HM in Statistics plays a critical part.

• One of the measures of the central tendency of a grouped or ungrouped set of data is the mean, which is relatively simple and easy to compute.

• Stock indices are calculated using the geometric mean. The portfolio’s annual returns are also calculated using the geometric mean. The geometric mean is often used to analyse biological processes like cell division and bacterial growth, among others.
• In finance, the harmonic mean is used to calculate the price-earnings ratio and other average multiples. It’s also employed in the Fibonacci sequence calculation.

Conclusion 

We’ve covered the fundamentals of AM, GM, and even HM, as well as the relation between the arithmetic mean (AM) and geometric mean (GM) and harmonic mean and other key topics. 

It also explains and proves the different relationships between the arithmetic mean and the geometric mean of data. The questions demonstrate how to determine the AM, GM, and HM and not to forget the numbers in the set of data. Also, the student may easily calculate one of two of the unknown values of the lot. It also shows the geometric representation of the three forms of meanings’ relationship.