Explain the relationship between A.M, G.M, H.M with suitable examples?

 

Arithmetic Mean (AM)

Arithmetic Mean is calculated as the simple average of all the observations. The value of this average is obtained by dividing the sum of the observations by their number. The arithmetic mean measures central tendency and is commonly used to find the average of a set of data points. This is simply referred to as mean and is represented by $\bar{X}$.

$$ $A\mathrm{.}M\mathrm{.}=\frac{\mathrm{\Sigma }X}{n}$

Geometric Mean (GM)

Geometric Mean is calculated as the nth root of the product of n values. The geometric mean is useful for calculating the average rates of change, such as compound interest rates or growth rates of investments. The geometric mean tends to reduce the impact of extreme values and is often used in financial and scientific contexts.

$G\mathrm{.}M\mathrm{.}=\sqrt[n]{X_1\times X_2\times X_3\times X_4\times \dots \dots \mathrm{.}{X}_n}$

Harmonic Mean (HM)

Harmonic Mean is defined as the reciprocal of the arithmetic mean of reciprocals. It is particularly used in situations where someone needs to find an average that reflects the ‘rate of work’ or ‘rate of speed.’

$H\mathrm{.}M\mathrm{.}=Rec\frac{\mathrm{\Sigma }RecX}{n}$

Relationship between AM, GM and HM

1. For any set of unequal positive numbers, the relationship between AM, GM, and HM is expressed as: AM > GM > HM.
2. If the values in the set data are equal, then the three averages would also be equal.
3. In case of set of two numbers, the relationship between AM, GM, and HM is expressed as:

$G\mathrm{.}M\mathrm{.}=\sqrt{\bar{X}\times H\mathrm{.}M\mathrm{.}}$

They depict this relationship; let’s take two numbers, a and b:

We know that,

$A\mathrm{.}M\mathrm{.}(\bar{X})=\frac{a+b}{2}$

$G\mathrm{.}M\mathrm{.}=\sqrt{ab}$

$H\mathrm{.}M\mathrm{.}=\frac{2}{\frac{1}{a}+\frac{1}{b}}$

Multiplying AM and HM, we get,

$A\mathrm{.}M\mathrm{.}(\bar{X})\times H\mathrm{.}M\mathrm{.}=\frac{a+b}{2}\times \frac{2}{\frac{1}{a}+\frac{1}{b}}$

$=\frac{ab(a+b)}{a+b}=ab=G\mathrm{.}M{\mathrm{.}}^2$

Thus, we have $G\mathrm{.}M{\mathrm{.}}^2=\bar{X}\times H\mathrm{.}M\mathrm{.}$ or $G\mathrm{.}M\mathrm{.}=\sqrt{\bar{X}\times H\mathrm{.}M\mathrm{.}}$

Example 1:

Find the harmonic mean of two numbers a and b, if the arithmetic mean is 25 and the geometric mean is 10 provided that a>b>0.

Solution:

Given, A.M. = 25 and G.M. = 10

The relationship between AM, GM, and HM is, $G\mathrm{.}M{\mathrm{.}}^2=A\mathrm{.}M\mathrm{.}\times H\mathrm{.}M\mathrm{.}$

$10^2=25\times H\mathrm{.}M\mathrm{.}$

$100=25\times H\mathrm{.}M\mathrm{.}$

$H\mathrm{.}M\mathrm{.}=\frac{100}{25}$

H.M. = 4

Example 2:

Show that $G\mathrm{.}M{\mathrm{.}}^2=A\mathrm{.}M\mathrm{.}\times H\mathrm{.}M\mathrm{.}$, using numbers 16 and 4.

Solution:

Here, $A\mathrm{.}M\mathrm{.}=\frac{16+4}{2}$

$AM=\frac{20}{2}$

AM = 10

$G\mathrm{.}M\mathrm{.}=\sqrt{16\times 4}$

G.M. = 8

$H\mathrm{.}M\mathrm{.}=\frac{2}{\frac{1}{a}+\frac{1}{b}}$

$H\mathrm{.}M\mathrm{.}=\frac{2}{\frac{1}{16}+\frac{1}{4}}$

$H\mathrm{.}M\mathrm{.}=\frac{2}{(\frac{5}{16})}$

$H\mathrm{.}M\mathrm{.}=\frac{32}{5}$

H.M. = 6.4

Verifying relationship, $G\mathrm{.}M{\mathrm{.}}^2=A\mathrm{.}M\mathrm{.}\times H\mathrm{.}M\mathrm{.}$

82 = 10 x 6.4

Hence, 64 = 64.

Summary

There are several measures of central tendency, or averages, each of which is typical in some unique way and has particular characteristics. The commonly used averages are:

Arithmetic Mean (AM): Calculated as the simple average of all observations. It is obtained by dividing the sum of the observations by their number.

$A\mathrm{.}M\mathrm{.}={\mathrm{\Sigma }}_i^n\frac{X_i}{n}$

Geometric Mean (GM): Calculated as the nth root of the product of n values. It reduces the impact of extreme values and is often used in financial and scientific contexts.

$G\mathrm{.}M\mathrm{.}=\sqrt[n]{X_1\times X_2\times X_3\times \mathrm{.}\mathrm{.}\mathrm{.}\times X_n}$

Harmonic Mean (HM): Defined as the reciprocal of the arithmetic mean of reciprocals. It reflects the ‘rate of work’ or ‘rate of speed.’

$H\mathrm{.}M\mathrm{.}=\frac{n}{{\mathrm{\Sigma }}_i^n\frac{1}{X_i}}$

Relationship between AM, GM, and HM:

• For any set of unequal positive numbers: $AM>GM>HM$
• If the values are equal, AM, GM, and HM will also be equal.
• For two numbers, the relationship is: $G\mathrm{.}M{\mathrm{.}}^2=A\mathrm{.}M\mathrm{.}\times H\mathrm{.}M\mathrm{.}$.

Practice Problems

1. Find the arithmetic mean, geometric mean, and harmonic mean of the numbers 5, 10, and 15.
2. Given two numbers a and b, where A.M.=12 and G.M. =8, find HM.
3. Show that $G\mathrm{.}M{\mathrm{.}}^2=A\mathrm{.}M\mathrm{.}\times H\mathrm{.}M\mathrm{.}$ for 9, 25, 35.
4. Find the arithmetic mean, geometric mean, and harmonic mean of the numbers 2, 8, 32, and 128.
5. A sequence of numbers follows a geometric progression with the first term a = 2 and common ratio r = 3. If there are 5 terms in the sequence, calculate the arithmetic mean, geometric mean, and harmonic mean of the sequence.
6. Given a dataset where the values follow a harmonic progression: 1, 1/2, 1/3, 1/4, and 1/5, calculate the arithmetic mean, geometric mean, and harmonic mean.

Relationship between AM, GM and HM – FAQs

What is the primary difference between Arithmetic Mean and Geometric Mean?

The Arithmetic Mean is the sum of all observations divided by their number, providing a simple average. The Geometric Mean is the nth root of the product of n values, reducing the impact of extreme values.

When should the Harmonic Mean be used?

The Harmonic Mean is useful in situations requiring an average rate, such as speeds or rates of work.

Can the Geometric Mean be used for negative numbers?

No, the Geometric Mean is only defined for positive numbers.

What happens to AM, GM, and HM if all values in a dataset are equal?

If all values are equal, then AM, GM, and HM will all be equal to that value.

How is the relationship $AM>GM>HM$ significant?

It shows that for any set of unequal positive numbers, the Arithmetic Mean will always be greater than the Geometric Mean, which in turn will be greater than the Harmonic Mean.