Standard deviation of first n even natural numbers
Answers
Given : first n even natural numbers
To Find : standard deviation
Solution:
first n even natural numbers
2 , 4 , 6 , ... _______ , 2n
= 2( 1 , 2 , 3 , ... ____ , n)
∑x² = 2²(1² + 2² + 3² +... ____ + n²)
∑n² = n(n + 1)(2n + 1)/6
∑x² / n =2² n(n + 1)(2n +1 )/6n
= 2(n + 1)(2n + 1)/3
∑x = 2( 1 + 2 + 3 + ... ____ + n )
∑n = n(n + 1)/2
∑x/ n =2n (n + 1)/2n = n + 1
SD = √( ∑x² / n - (∑x/ n)²
=√ ( 2(n + 1)(2n + 1)/3 - (n + 1)² )
= √ (4n² + 6n + 2 - 3n² - 6n - 3)/3
= √ ( n² - 1)/3
Standard deviation of first n even natural numbers = √ ( n² - 1)/3
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