Arithmetic mean, Geometric Mean and Harmonic Mean

Arithmetic mean, geometric mean, and harmonic mean are all measures of central tendency. If all the observations of the series are the constant ‘K’, the mean will also be ‘K’. The same property applies for all the means be it arithmetic mean, geometric mean or harmonic mean. If the deviation in the series is taken from the mean, the sum of deviations from the mean will be zero. If we change the origin, the same change will consequently be seen in the mean as well. Similarly, if we change the scale, the same scale change will be seen eventually in the mean. All the concepts discussed here will be studied in detail below while studying the meaning and calculation of all the means and their relationship with each other, i.e., the relationship between the arithmetic mean, geometric mean and harmonic mean.

Arithmetic Mean (A.M.)

Arithmetic Mean of Numbers:

The arithmetic mean of numbers primarily means average. If we are given two numbers, say a and b, then the arithmetic mean of those two numbers will be (a + b)/2. Therefore, we can define the arithmetic mean of numbers as the sum of observations divided by the no. of observations.

 ‘N’ Arithmetic Mean between Two Numbers:

If A is one arithmetic mean between two numbers a and b, then a, A and b are in A.P. (arithmetic progression),

If A1 and A2 are two means between a and b then a, A1, A2, b are in A. P. (arithmetic progression).

In simple words, we can understand it as follows:

If three terms are in arithmetic progression, then the middle term is called the arithmetic mean of the other two. We can interpret this mathematically as, if three terms p,q and r are in arithmetic progression, then q = (p+r)/2 is the arithmetic mean of p and r.

Similarly, for n numbers, If a1,a2,a3,………………………..an are the given n numbers, then the arithmetic mean (A) of these numbers will be:

A= 1/n (a1+ a2+ a3+………………………..+ an)

In general, if n arithmetic means, A1, A2, A3,………………………….An, are inserted between a and b,

Then d = (b – a) / (n+1)

And

Ak = a + kd 

Geometric Mean (G.M.)

Geometric Mean of Numbers:

The geometric mean of two numbers (x1 and x2) is the product of two numbers raised to the power ½     or  (x1*x2)1/2

Similarly, the geometric mean of three numbers (x1,x2and x3) is the product of three numbers raised to the power ⅓  or (x1*x2*x3)1/3

The same pattern will follow for n numbers. Also, remember that geometric mean is defined for positive numbers.

‘N’ Geometric Mean between Two Numbers a and b:

If G1 is one geometric mean between a and b, then a, G1, b will form a Geometric Progression (G.P.).

If G1, G2 is one geometric mean between a and b, then a, G1, G2, b will form a Geometric Progression (G.P.).

The same pattern will follow for n geometric means inserted between two numbers a and b.

In general, if n geometric means G1, G2, G3, ……………………………,Gn, are inserted between two numbers a and b,

Then r = (b/a) 1/n+1

and Gk = ark

Harmonic Mean (H.M.):

Harmonic Mean of Numbers:

The harmonic mean of given numbers will be equal to the number of observations divided by the sum of reciprocal of the given numbers.

n / (1/x1 + 1/x2 + 1/x3+…………………………..+ 1/xn )

‘n’ Harmonic Mean between Two Numbers a and b:

H1 is called one Harmonic Mean between two numbers ‘a’ and ‘b’ if a, H1, b form a Harmonic Progression (H.P.).

Similarly, H1 and H2 are called two harmonic means between two numbers ‘a’ and ‘b’ if a, H1, H2, b form a Harmonic Progression (H.P.).

Relationship between AM,GM and HM

Let a and b are two unequal numbers  and are positive then there

AM ≥GM ≥HM

Let’s see an example for better clarity:

Example:

Find the minimum value of 4x+4/4x

Solution:

The given expression is = 4x+4/4x

The first term of the expression = 4x

The second term of the expression = 4/4x

Note that all the terms in the given expression should be positive. We can see that both the terms are positive here.

 4x and 4/4x > 0

Product of terms = 4x ( 4/4x) = 4 which is a constant.

Now, applying AM>=GM on both the terms, 4x and 4/4x

AM =( 4x+4/4x)/2

GM = √ 4x ( 4/4x)

( 4x+4/4x)/2 >= √ 4x ( 4/4x)

4x+4/4x >= 2*2

4x+4/4x >= 4

Hence, Minimum Value of 4x+4/4x = 4

Conclusion:

In this article, we discussed in detail the relationship between arithmetic mean, geometric mean and harmonic mean. First, we discussed the definition of arithmetic mean, geometric mean and harmonic mean. We learnt formulas for Am, GM and HM and derivation of all the formulas in detail as well. We also learnt further that the arithmetic mean is always greater than or equal to the geometric mean, and the same relationship holds true for geometric mean and harmonic mean as well. We also solved some practical problems on the AM, GM and HM for a better understanding of the concepts learnt here. We hope this study material will be helpful to clear all your basic concepts on the relationship between Arithmetic mean, Geometric mean and Harmonic mean.