Find the mean and variance for the first n natural numbers

Solution:

The mean of first n natural numbers is calculated as follows.

Sum of all observations / Number of observations

= [n (n + 1)/2]/n

= (n + 1)/2

Variance,

(σ²) = 1/n ∑ⁿ_{i = 1}(x_i - x)²

= 1/n ∑ⁿ_{i = 1}[x_i - (n + 1)/2]²

= 1/n ∑ⁿ_{i = 1}(x_i)² - 1/n ∑ⁿ_{i = 1}[2 (n + 1)/n]x_i + 1/n ∑ⁿ_{i = 1}[(n + 1)/2]²

= 1/n [n (n + 1)(2n + 1)]/6 - (n + 1)²/2 + (n + 1)²/4

= (n + 1)(2n + 1)/6 - (n + 1)²/4

= (n + 1) [4n + 2 - 3n - 3]/12

= [(n + 1)(n - 1)]/12

= (n² - 1)/12

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NCERT Solutions Class 11 Maths Chapter 15 Exercise 15.2 Question 2

Find the mean and variance for the first n natural numbers

Summary:

The mean of first n natural numbers is (n² - 1)/12