Question

Find the mean and the standard deviation of the first n natural Numbers hence find the standard deviation of the first 200 natural numbers

Answer 1

Answer:

200 is the mean of this question

Answer 2

The value of standard deviation of first n natural numbers S.D. = 57.73

Step-by-step explanation:

Mean for first n natural numbers is given by

$u = \frac{1 + 2 + 3 ----- + n}{n}$

⇒ $u = \frac{n + 1}{2}$ ------ (1)

This is the mean of first n natural numbers.

Now variance of the first n natural numbers is given by

$\sigma^{2}$ = $\frac{n^{2} - 1 }{12}$

Standard deviation S.D. = $\sqrt{\sigma^{2}}$ ------- (2)

This is the standard deviation for n natural numbers.

Now, Standard deviation of first 200 natural numbers S.D. = $\sqrt{\sigma_{200} ^{2}}$

⇒ Variance of 200 numbers is given by  $\sigma_{200} ^{2}$ = $\frac{200^{2} - 1 }{12}$

⇒ Variance ( $\sigma_{200} ^{2}$ ) = 3333.25

Now , S.D. = $\sqrt{variance}$

⇒ S.D. = $\sqrt{3333.25}$

⇒ S.D. = 57.73

This is the value of standard deviation of first n natural numbers.