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What is the comparison between the arithmetic, geometric, and harmonic means?

  • Last Updated : 19 Jan, 2022

Algebra is the branch of mathematics that deals with numbers and symbols and provides the rules to manipulate these symbols and numbers. An algebraic expression is formed of numbers, variables, and mathematical operators. Example: 6y+2 is an algebraic expression. Comparing numerical values is an important aspect of mathematics, and algebra helps us deal with this. Mean is another entity that is used for finding relationships between Numbers/ Variables. There are three kinds of mean available: Arithmetic Mean(AM), Geometric Mean(GM), Harmonic Mean(HM) in algebraic mathematics which helps us in calculating Mean. Now, we will discuss the three means in mathematics: Arithmetic Mean(AM), Geometric Mean(GM), Harmonic Mean(HM), and compare three of them on the basis of magnitude.

Arithmetic Mean(AM)

In mathematics, arithmetic Mean(AM) evaluates the total sum of the numbers upon the total count of numbers. Arithmetic Mean(AM) in simple terms is basically the average of all the given numbers whose Arithmetic Mean needs to be calculated. Consider two variables R and S then the arithmetic mean(AM) of these variables would be given by the formula:

Let the total number of entities under consideration, here the two variables are entities so, assign entities(N) the value of 2.

AM = (R + S)/N

= (R + S)/2

Geometric Mean(GM)

In mathematics, geometric mean(GM) evaluates to the finding the nth root of the product of all the numbers whose geometric mean(GM) needs to be calculated. Consider two variables R and S then the geometric mean(GM) of these variables would be given by the formula:

Let the total number of entities under consideration, here the two variables are entities so, assign entities(N) the value of 2.

GM = (R + S)1/N

= (R + S)1/2

Harmonic Mean(HM) 

In mathematics, Harmonic Mean(HM) evaluates the total count of numbers upon the sum of reciprocal of the given numbers whose Harmonic Mean(HM) needs to be calculated. Consider two variables R and S then the Harmonic Mean(HM) of these variables would be given by the formula:

Let the total number of entities under consideration, here the two variables are entities so, assign entities(N) the value of 2.

AM = N/(1/R + 1/S)

= 2/(1/R + 1/S)

= 2/(R + S/RS)

= 2RS/(R + S)

What is the comparison between the arithmetic, geometric, and harmonic means?

Solution:

Comparing Arithmetic Mean(AM), Geometric Mean(GM), Harmonic Mean(HM) on the basis of magnitude. So consider two numbers 4 and 5 replacing these variables in the above formulas.

Hence the arithmetic mean(AM) of these numbers would be given by the formula:

Let the total number of entities under consideration, here the two variables are entities so, assign entities(N) the value of 2.

AM = (4 + 5)/2

= (4 + 5)/2

= 4.5

The Geometric Mean(GM) of these numbers would be given by the formula:

Let the total number of entities under consideration, here the two variables are entities so, assign entities(N) the value of 2.

GM = (4 + 5)1/2

= (4 + 5)1/2

= 3

The Harmonic Mean(HM) of these numbers would be given by the formula:

Let the total number of entities under consideration, here the two variables are entities so, assign entities(N) the value of 2.

AM = N/(1/4 + 1/5)

= 2/(1/4 + 1/5)

= 2/(4 + 5/(4 * 5))

= (2 * 20)/ 9

= 40/9

= 4.44

Comparing Arithmetic Mean(AM), Geometric Mean(GM), Harmonic Mean(HM) of the two numbers 4 and 5.

We have,

Arithmetic Mean(AM) = 4.5

Geometric Mean(GM) = 3

Harmonic Mean(HM) = 4.44

We see that arithmetic mean is the largest in magnitude, followed by Harmonic Mean and then by Geometric Mean.

So, 

Arithmetic Mean > Harmonic Mean > Geometric Mean

or

Geometric Mean < Harmonic Mean < Arithmetic Mean

Similar Questions

Question 1: Name the three popular comparing ‘Mean’ in Mathematics. 

Solution:

The three popular ‘Mean’ are Arithmetic Mean, Geometric Mean, Harmonic Mean

Question 2: Arrange in Descending Order: Arithmetic Mean, Geometric Mean, Harmonic Mean.

Solution:

Descending Order of Arrangement: Arithmetic Mean, Harmonic Mean, Geometric Mean.

Question 3: Which among the given options is the Greatest in Magnitude: Arithmetic Mean, Geometric Mean, Harmonic Mean.

Solution:

Arithmetic Mean is the greatest in magnitude.

Question 4: Which among the given options is the Smallest in Magnitude: Arithmetic Mean, Geometric Mean, Harmonic Mean.

Solution:

Geometric Mean is the smallest in magnitude.

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