Relation between A.M., G.M. and H.M.

In Maths, when we learn about sequences, we also come across the relation between AM, GM and HM. These three are basically the average or mean of the respective series. AM stands for Arithmetic Mean, GM stands for Geometric Mean and HM stands for Harmonic Mean. AM, GM and HM are the mean of Arithmetic Progression (AP), Geometric Progression (GP) and Harmonic Progression (HP). Before learning about the relationship between them, one should know a brief of all the three means along with their formulas.

AM, GM, HM Formulas

Before we relate the three means in Statistics, which are Arithmetic Mean, Geometric Mean and Harmonic Mean, let us understand them better.

Arithmetic Mean

Arithmetic mean represents a number that is achieved by dividing the sum of the values of a set by the number of values in the set. If a1a2a3….,an, is a number of group of values or the Arithmetic Progression, then;

AM=(a1+a2+a3+….,+an)/n

Geometric Mean

The Geometric Mean for a given number of values containing n observations is the nth root of the product of the values.

GM = √(a1a2a3….an)

Or

GM = (a1a2a3….an)1/n

Harmonic Mean

HM is defined as the reciprocal of the arithmetic mean of the given data values. It is represented as:

HM = n/(1/a1,1/a2,1/a3,….,1/an)

What is the Relation between AM, GM and HM?

The relationship between AM, GM and HM is given by:

AM x HM = GM2

Now let us understand how this relation is derived;

First, consider a, AM, b is an Arithmetic Progression.

Now the common difference of Arithmetic Progression will be;

AM – a = b – AM

a + b = 2 AM …………..(1)

Secondly, let a, GM, b is a Geometric Progression. Then, the common ratio of this GP is;

GM/a = b/GM

ab = GM2……………(2)

Third, is the case of harmonic progression, a, HM, b, where the reciprocals of each term will form an arithmetic progression, such as:

1/a, 1/HM, 1/b is an AP.

Now the common difference of the above AP is;

1/HM – 1/a = 1/b – 1/HM

2/HM = 1/b – 1/a

2/HM = (a+b)/ab ………….(3)

Put eq. 1 and eq.2 in eq. 3 to get;

2/HM = 2AM/GM2

GM2=AM x HM

Hence, this is the relation between Arithmetic, Geometric and Harmonic mean.