Statement $1$ : The variance of first $n$ odd natural numbers is $\frac{{{n^2} - 1}}{3}$
Statement $2$ : The sum of first $n$ odd natural number is $n^2$ and the sum of square of first $n$ odd natural numbers is $\frac{{n\left( {4{n^2} + 1} \right)}}{3}$

  • [AIEEE 2012]
  • A

    Statement $1$ is true, Statement $2$ is false.

  • B

    Statement $1$ is true, Statement $2$ is true;
    Statement $2$ is not a correct explanation for Statement $1$.

  • C

    Statement $1$ is false, Statement $2$ is true.

  • D

    Statement $1$ is true, Statement $2$ is true,
    Statement $2$ is a correct explanation for Statement $1$.

Similar Questions

The mean and the standard deviation (s.d.) of $10$ observations are $20$ and $2$ resepectively. Each of these $10$ observations is multiplied by $\mathrm{p}$ and then reduced by $\mathrm{q}$, where $\mathrm{p} \neq 0$ and $\mathrm{q} \neq 0 .$ If the new mean and new s.d. become half of their original values, then $q$ is equal to

  • [JEE MAIN 2020]

Find the variance of the following data: $6,8,10,12,14,16,18,20,22,24$

Find the mean and variance for the data

${x_i}$ $6$ $10$ $14$ $18$ $24$ $28$ $30$
${f_i}$ $2$ $4$ $7$ $12$ $8$ $4$ $3$

 

Mean and standard deviation of 100 observations were found to be 40 and 10 , respectively. If at the time of calculation two observations were wrongly taken as 30 and 70 in place of 3 and 27 respectively, find the correct standard deviation.

If the mean and variance of the frequency distribution

$x_i$ $2$ $4$ $6$ $8$ $10$ $12$ $14$ $16$
$f_i$ $4$ $4$ $\alpha$ $15$ $8$ $\beta$ $4$ $5$

are $9$ and $15.08$ respectively, then the value of $\alpha^2+\beta^2-\alpha \beta$ is $............$.

  • [JEE MAIN 2023]