The coefficient of variation of the first n n | vedclass

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(D) The mean of the first $n$ natural numbers is $\bar{x} = \frac{n+1}{2}$.The variance $\sigma^2$ of the first $n$ natural numbers is given by $\sigm

Vedclass · Accepted Answer

$\frac{100}{\sqrt{3}} \sqrt{\frac{n-1}{n+1}}$

Vedclass · Answer

(D) The mean of the first $n$ natural numbers is $\bar{x} = \frac{n+1}{2}$.The variance $\sigma^2$ of the first $n$ natural numbers is given by $\sigma^2 = \frac{n^2-1}{12}$.Thus,the standard deviation is $\sigma = \sqrt{\frac{n^2-1}{12}}$.The coefficient of variation $(CV)$ is defined as $CV = \frac{\sigma}{\bar{x}} 	imes 100$.Substituting the values:$CV = \frac{\sqrt{\frac{n^2-1}{12}}}{\frac{n+1}{2}} 	imes 100 = \frac{\sqrt{\frac{(n-1)(n+1)}{12}}}{\frac{n+1}{2}} 	imes 100$$= \sqrt{\frac{(n-1)(n+1)}{12} 	imes \frac{4}{(n+1)^2}} 	imes 100 = \sqrt{\frac{n-1}{3(n+1)}} 	imes 100$$= \frac{100}{\sqrt{3}} \sqrt{\frac{n-1}{n+1}}$.

$C$.$I$.	$75$-$175$	$175$-$275$	$275$-$375$	$375$-$475$	$475$-$575$	$575$-$675$	$675$-$775$
$f_i$	$3$	$2$	$1$	$0$	$1$	$2$	$3$

	Section $A$	Section $B$
Number of students	$50$	$60$
Average marks in the test	$45$	$45$
Variance of distribution of marks	$64$	$81$

Variate $(x)$	$x_{1}$	$x_{2}$	$x_{3} \ldots x_{15}$
Frequency $(f)$	$f_{1}$	$f_{2}$	$f_{3} \ldots f_{15}$

The coefficient of variation of the first $n$ natural numbers is

Similar Questions

Consider the following frequency distribution:
$C$.$I$. $75$-$175$ $175$-$275$ $275$-$375$ $375$-$475$ $475$-$575$ $575$-$675$ $675$-$775$
$f_i$ $3$ $2$ $1$ $0$ $1$ $2$ $3$
If the variance of this distribution is $60000$,then the coefficient of variation of the distribution is:

All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of $10$ to each of the students. Which of the following statistical measures will not change even after the grace marks were given?

Students of two sections $A$ and $B$ of a class show the following results in a test conducted for $100$ marks. Then

Section $A$ Section $B$

Number of students $50$ $60$

Average marks in the test $45$ $45$

Variance of distribution of marks $64$ $81$

In a series of $2n$ observations,half of them are equal to $a$ and the remaining half are equal to $-a$. If the standard deviation of these observations is $2$,then $|a|$ equals:

For the frequency distribution:

Variate $(x)$ $x_{1}$ $x_{2}$ $x_{3} \ldots x_{15}$

Frequency $(f)$ $f_{1}$ $f_{2}$ $f_{3} \ldots f_{15}$

where $0 < x_{1} < x_{2} < x_{3} < \ldots < x_{15} = 10$ and $\sum_{i=1}^{15} f_{i} > 0$,the standard deviation cannot be:

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