Relationship Between Arithmetic Mean and Geometric Mean: The most commonly used measure of central tendency is the mean. The different mean types are arithmetic mean, geometric mean, weighted arithmetic mean, and harmonic mean. In general, arithmetic mean is denoted as mean or AM, geometric mean as GM, and harmonic mean as HM. The mean for any set is the average of the set of values present in that set. It is used to calculate the rate of cell growth by division in biology, solve linear transformations, and calculate growth rate and risk factors in finance.

In this article, let us learn in detail about the relationship between arithmetic mean and geometric mean.

What are the Types of Mean?

1. Arithmetic Mean

2. Geometric Mean

3. Harmonic Mean

How are the Types of Mean Different From Each Other?

Type of Mean	Arithmetic Mean	Geometric Mean	Harmonic Mean
Definition	Ratio of sum of values of observations to the number of observations	\({n^{th}}\) root of the product of \(n\) values of observations	Reciprocal of the arithmetic mea of the reciprocals of values of observations
Formula	\(AM = \overline X = \frac{{\sum x }}{n}\)	\(GM = \sqrt[x]{{\left({{x_1}} \right)\left({{x_2}} \right) \ldots \left({{x_n}} \right)}}\)	\(HM = \frac{n}{{\sum \left({\frac{1}{x}} \right)}}\)
Advantages	Repeated samples result in similar means. Hence, arithmetic mean best resists fluctuation between samples	Determines correct average when dealing with ratios and percentages	Does not give much weightage to karge items
Disadvantages	Highly affected by the presence of abnormally high or low values	Cannot be calculated when the value of any observations is zero or negative	Gives high weightage to small items

How is Geometric Mean Different From Arithmetic Mean?

The geometric mean differs from the arithmetic mean or average in how it is calculated, as it considers the compounding that occurs across periods. Hence, investors and finance people prefer the geometric mean, as it is more accurate than the arithmetic mean.

Diagrammatic Representation of AM and GM

We have learnt that,
Arithmetic mean, \(AM = \frac{{a + b}}{2}\)
Geometric mean, \(GM = \sqrt[2]{{ab}}\)
The diagrammatic representation of the arithmetic and geometric means of two positive integers \(a\) and \(b\) is as shown below.

Let us now learn the various theorems that state the relationship between the arithmetic mean and the geometric mean of a given data.

How are AM and GM related?

Theorem 1:
If AM and GM are the arithmetic mean and the geometric mean of two positive integers \(a\) and \(b,\) respectively, then, \(AM > GM.\)
Proof:
Given:
Arithmetic mean, \(AM = \frac{{a + b}}{2}\)
Geometric mean, \(GM = \sqrt[2]{{ab}}\)
\( \Rightarrow AM – GM = \frac{{a + b}}{2} – \sqrt {ab} \)
\(AM – GM = \frac{{a + b – 2\sqrt {ab} }}{2}\)
\(AM – GM = \frac{{{{\left({\sqrt a – \sqrt b } \right)}^2}}}{2}\)
We know that, \(\frac{{{{\left({\sqrt a – \sqrt b } \right)}^2}}}{2} > 0\)
\(\therefore AM – GM > 0\)
\(AM > GM\)
Hence proved that the arithmetic mean of two positive numbers is always greater than their GM.
This is also called the arithmetic mean – geometric mean (AM-GM) inequality.

Theorem 2:
If \(A\) and \(G\) are the arithmetic mean and the geometric mean of two positive integers \(a\) and \(b,\) respectively, then the quadratic equation having \(a\) and \(b\) as its roots is \({x^2} – 2Ax + {G^2} = 0.\)
Proof:
Given:
Arithmetic mean, \(A = \frac{{a + b}}{2}\)
Geometric mean, \(G = \sqrt[2]{{ab}}\)
Substituting the values of \(A\) and \(G\) in the quadratic equation, we get,
\({x^2} – 2\left({\frac{{a + b}}{2}} \right)x + {\left({\sqrt {ab} } \right)^2} = 0\)
\({x^2} – \left({a + b}\right)x + ab = 0\)
\({x^2} – ax – bx + ab = 0\)
\(x\left({x – a} \right) – b\left({x – a} \right) = 0\)
\(\left({x – a} \right)\left({x – b} \right) = 0\)
\(x = a,x = b\)
The roots of the quadratic equations are \(a\) and \(b.\)
Hence proved.

Theorem 3:
If \(A\) and \(G\) are the arithmetic mean and the geometric mean of two positive integers \(a\) and \(b,\) respectively, then the numbers are given by \(A \pm \sqrt {{A^2} – {G^2}} .\)
Proof:
For the quadratic equation \({x^2} – 2Ax + {G^2} = 0,\) the value of \(x\) is calculated using the formula,
\(x = \frac{{ – b \pm \sqrt {{b^2} – 4ac} }}{{2a}}\)
Here,
\(a = 1\)
\(b = – 2A\)
\(c = {G^2}\)
\(\therefore x = \frac{{2A \pm \sqrt {{{\left({2A} \right)}^2} – 4\left( 1 \right) \times {G^2}} }}{{2 \times 1}}\)
\(x = \frac{{2A \pm \sqrt {4{A^2} – 4{G^2}} }}{2}\)
\(x = \frac{{2A \pm 2\sqrt {{A^2} – {G^2}} }}{2}\)
\( \Rightarrow x = A \pm \sqrt {{A^2} – {G^2}} \)
Hence proved.

How are AM, GM and HM Related?

The geometric representation of arithmetic, geometric and harmonic means is as shown below.

Now, let’s derive the algebraic relation of the three types of means.
Let \(A,G\) and \(H\) are the arithmetic, geometric and harmonic means of two positive integers \(a\) and \(b,\) respectively.
We know that,
Arithmetic mean, \(A = \frac{{a + b}}{2}\)
Geometric mean, \(G = \sqrt[2]{{ab}}\)
Harmonic mean, \(H = \frac{2}{{\frac{1}{a} + \frac{1}{b}}} = \frac{2}{{\frac{{a + b}}{{ab}}}} = \frac{{2ab}}{{a + b}}\)
Substituting the values of \(A\) and \(G,\) we get,
\(H = \frac{{2\left({{G^2}} \right)}}{{2A}} = \frac{{{G^2}}}{A}\)
\( \Rightarrow {G^2} = H \times A\)
\(\therefore G = \sqrt{H \times A} \)
Hence, this is the relation between the arithmetic mean, geometric mean and harmonic mean of a given data.

What is the Use of AM, GM and HM?

The different types of means have several applications in fields like statistics, mathematics, photography, biology, etc. A few of them are listed below:

1. Arithmetic Mean: The arithmetic mean income of a country’s population is the per capita income of that country.
2. Geometric Mean: Comparison of review ratings of different products is achieved using a geometric mean.
3. Harmonic Mean: The length of the perpendicular or the height \(\left( h \right),\) in a right triangle, \({h^2}\) is half the harmonic mean of \({a^2}\) and \({b^2}.\)

Solved Problems – Relationship Between Arithmetic Mean and Geometric Mean

Q.1. Find the two numbers if their geometric and arithmetic means are 7 and 25, respectively.
Ans: Let the two numbers be \(c\) and \(d.\)
\(\therefore AM = \frac{{c + d}}{2} = 25\)
\( \Rightarrow d = 50 – c\)
And, \(GM = \sqrt {cd} = 7\)
Substituting the value of \(d,\) we get,
\(\sqrt {c\left({50 – c} \right)} = 7\)
\(\sqrt {50c – {c^2}} = 7\)
\(50c – {c^2} – 49 = 0\)
\(c\left({c – 49} \right) – 1\left({c – 49} \right) = 0\)
\(\left({c – 49}\right)\left({c – 1}\right) = 0\)
\( \Rightarrow c = 49,\) or \(c = 1\)
\(\therefore d = 1,\) or \(d = 49\)
The two numbers are \(49\) and \(1.\)

Q.2. Find the harmonic mean of two positive numbers whose arithmetic mean is 16 and geometric mean is 8.
Ans: Using the relation, \({G^2} = H \times A\)
We get, \({8^2} = H \times 16\)
\(H = \frac{{64}}{{16}} = 4\)
The harmonic mean of the data is \(4.\)

Q.3. If the arithmetic mean of two numbers is 5 times the geometric mean, then find the value of \(\frac{{p – q}}{{p + q}}.\)
Ans: Given: \(AM = 5 \times GM\)
Let the two numbers be \(p\) and \(q.\)
\(\therefore \frac{{p + q}}{2} = 5 \times \sqrt {pq} \)
\(p + q = \sqrt {100\,pq} \)
Squaring on both sides,
\({\left({p + q} \right)^2} = 100\, pq\)
\({p^2} + {q^2} + 2\, pq – 100\, pq = 0\)
\({p^2} + {q^2} – 98\, pq = 0\)
\({p^2} + {q^2} – 2\,pq – 96\,pq = 0\)
\({\left({p – q} \right)^2} = 96\, pq\)
\( \Rightarrow p – q = \sqrt {96\,pq} \)
\(\therefore \frac{{p – q}}{{p + q}} = \sqrt {\frac{{96\,pq}}{{100\,pq}}} = \frac{{4\sqrt 6 }}{{10}}\)

Q.4. Find two numbers whose difference is 12, and their arithmetic mean is 2 more than their geometric mean.
Ans: Let the two numbers be \(u\) and \(v.\)
Given:
\(AM = GM + 2\)
\(\frac{{u + v}}{2} = \sqrt {uv} + 2\)
\(u + v = 2\sqrt {uv} + 4\)
\(u + v – 2\sqrt {uv} + 4\)
\({\left({\sqrt u – \sqrt v } \right)^2} = 4\)
\(\sqrt u – \sqrt v = \pm 2\,\,\,\,\,\,…\left( 1 \right)\)
Also given, \(u – v = 12\)
\(\left({\sqrt u + \sqrt v } \right)\left({\sqrt u – \sqrt v }\right) = 12\)
Substituting \(\left( 1 \right),\) we get,
\(\left({\sqrt u + \sqrt v } \right)\left({ \pm 2} \right) = 12\)
\(\sqrt u + \sqrt v = \pm 6\,\,…\left( 2 \right)\)
Solving \(\left( 1 \right)\) and \(\left( 2 \right),\) we get,
\(u = 16\) and \(v = 4\)

Q.5.If the AM of two numbers is 34 and their GM is 16, find the numbers.
Ans: Let the two numbers be \(x\) and \(y.\)
\(\therefore AM = \frac{{x + y}}{2} = 34\)
\( \Rightarrow y = \left({34 \times 2} \right) – x\)
And \(GM = \sqrt {xy} = 16\) Substituting the value of \(y,\) we get,
\(\sqrt {x\left({68 – x}\right)} = 16\)
\(\sqrt {68x – {x^2}} = 16\)
\(68x – {x^2} – 256 = 0\)
Solving for roots of the equation, we get,
\(x = 64\) and \(y = 4\)
The two numbers are \(64\) and \(4.\)

Summary

This article explains the differences between arithmetic mean, geometric mean, and harmonic mean. It also states and proves the various ways in which the arithmetic mean and the geometric mean of data are related to each other. The problems explain the steps involved to calculate one or two of the unknown values of the lot – arithmetic mean, geometric mean, harmonic mean, and the numbers in the data set. It also illustrates the geometric representation of the relationship of the three types of means.

Learn All the Concepts on Arithmetic Mean

Frequently Asked Questions (FAQs)

Q.1. What is the relationship between arithmetic mean and geometric mean?
Ans: The relation between the different types of means – arithmetic, geometric, and harmonic are shown below.

Note that \(AM > GM.\)

Q.2. Why is geometric mean less than arithmetic?
Ans: Geometric mean is always lesser than the arithmetic mean because it takes into consideration the compounding that occurs. This is the reason geometric mean is preferred for financial calculations over arithmetic mean.

Q.3. What is the difference between arithmetic mean and geometric mean?
Ans: While the arithmetic mean is the ratio of the sum of values to the number of observations, geometric mean is the nth root of the product of values of n observations.

Q.4. How do you find the arithmetic mean and geometric mean?
Ans: The formulas to find the arithmetic mean and geometric mean are as follows.

Arithmetic Mean	Geometric Mean
\(AM = \overline X = \frac{{\sum x }}{n}\)	\(GM = \sqrt[x]{{\left({{x_1}} \right)\left({{x_2}} \right) \ldots \left({{x_n}} \right)}}\)

Q.5. Which is better, arithmetic or geometric mean?
Ans: Arithmetic mean of data is useful and accurate when the data set is not skewed, and the values are independent of each other. On the other hand, the geometric mean of data is effective when the data set is volatile. While the arithmetic mean is used in simple, daily calculations, the geometric mean is used for financial analysis.

Q.6. Is harmonic mean greater than the arithmetic mean?
Ans: Harmonic mean is always lesser than the arithmetic and geometric mean of the given data set.

Q.7. Which mean is the most affected by extreme values?
Ans: Although arithmetic mean is the commonly used measure of central tendency, it is also the most affected in the presence of extreme values in data.

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