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**Arithmetic mean**(A.M) or average is equal to the ratio of sum of all numbers in the given data to the number of numbers in the data.

For an arithmetic sequence,

nth term, Tn = a + (n-1)d

Sum of n terms, Sn = a + l or 2a + (n-1)d

Here n stands for number of terms, a is the first term, l is the last term and d is the common difference.

**Geometric mean(G.M)** is defined as the nth root of product of numbers. n stands for number of numbers.

For a Geometric sequence,

nth term, Tn = a

Sum of n terms, Sn = if r >1

Sn = if r < 1

**Harmonic mean(H.M)** is defined as the ratio of number of numbers to the sum of reciporocal of numbers in the given data.

Always A.M > G.M > H.M and

A.M = G.M = H.M when all the elements in the given data are equal.

**Root mean Square(R.M.S)** is the root of the ratio of the sum of squares of numbers to the number of numbers in the given data.

**Weighted Arithmetic Mean** is the ratio of the sum of product of numbers with their respective weights to the sum of weights in the given data.

**Example 1:** Find the Harmonic mean, Geometric mean and Arithmetic mean of numbers 5, 10, 15, 6

Harmonic mean:

Number of numbers = 4

Harmonic mean = =

= = = 7.5

Geometric mean:

Product of numbers = 5 x 10 x 15 x 6 = 4500

G.M = = 8.2

Arithmetic mean:

Sum of all numbers = 5 + 10 + 15 + 6 = 36

A.M = = = 9

**Example 2:** (a) Find the Root mean square of the data 12, 14, 2, 5, 8

(b) A student secured 90 in math, 72 in physics, 80 in chemistry, 62 in Biology and the weights are 5, 4, 2 and 1 respectively. Find the weighted arithmetic mean.

(a) Sum of squares of all numbers = + + + +

= 144 + 196 + 4 + 25 + 64

= 433

Number of numbers = 5

Root mean square = =

= ≈ 9.3

(b) weighted arithmetic mean =

= = 80