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
Number of numbers = 4
Harmonic mean = =
= = = 7.5
Product of numbers = 5 x 10 x 15 x 6 = 4500
G.M = = 8.2
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
Number of numbers = 5
Root mean square = =
= ≈ 9.3
(b) weighted arithmetic mean =
= = 80