Find the standard deviation of the first n natural numbers.
Find the standard deviation of the first n natural numbers.
$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline x_{i} & 1 & 2 & 3 & 4 & 5 & \ldots & \ldots & n \\ \hline x_{i}^{2} & 1 & 4 & 9 & 16 & 25 & \ldots & \ldots & n^{2} \\ \hline \end{array}$
Now, $\quad \Sigma x_{i}=1+2+3+4+\ldots+n=\frac{n(n+1)}{2}$
and $\Sigma x_{i}^{2}=1^{2}+2^{2}+3^{2}+\ldots+n^{2}=\frac{n(n+1)(2 n+1)}{6}$
$\therefore \quad \alpha=\sqrt{\frac{\Sigma x_{i}^{2}}{n}-\left(\frac{\Sigma x_{i}}{n}\right)^{2}}=\sqrt{\frac{n(n+1)(2 n+1)}{6 n}-\frac{n^{2}(n+1)^{2}}{4 n^{2}}}$
$=\sqrt{\frac{(n+1)(2 n+1)}{6}-\frac{(n+1)^{2}}{4}}=\sqrt{\frac{2\left(2 n^{2}+3 n+1\right)-3\left(n^{2}+2 n+1\right)}{12}}$
$=\sqrt{\frac{4 n^{2}+6 n+2-3 n^{2}-6 n-3}{12}}=\sqrt{\frac{n^{2}-1}{12}}$
Similar Questions
The mean and variance of $20$ observations are found to be $10$ and $4,$ respectively. On rechecking, it was found that an observation $9$ was incorrect and the correct observation was $11$. Then the correct variance is
The mean and variance of $20$ observations are found to be $10$ and $4,$ respectively. On rechecking, it was found that an observation $9$ was incorrect and the correct observation was $11$. Then the correct variance is
- [JEE MAIN 2020]
Find the mean and variance for the data
${x_i}$
$92$
$93$
$97$
$98$
$102$
$104$
$109$
${f_i}$
$3$
$2$
$3$
$2$
$6$
$3$
$3$
Find the mean and variance for the data
| ${x_i}$ | $92$ | $93$ | $97$ | $98$ | $102$ | $104$ | $109$ |
| ${f_i}$ | $3$ | $2$ | $3$ | $2$ | $6$ | $3$ | $3$ |
The frequency distribution:
$\begin{array}{|l|l|l|l|l|l|l|} \hline X & 2 & 3 & 4 & 5 & 6 & 7 \\ f & 4 & 9 & 16 & 14 & 11 & 6 \\ \hline \end{array}$
Find the standard deviation.
The frequency distribution:
$\begin{array}{|l|l|l|l|l|l|l|} \hline X & 2 & 3 & 4 & 5 & 6 & 7 \\ f & 4 & 9 & 16 & 14 & 11 & 6 \\ \hline \end{array}$
Find the standard deviation.
The mean and variance of $7$ observations are $8$ and $16,$ respectively. If five of the observations are $2,4,10,12,14 .$ Find the remaining two observations.
The mean and variance of $7$ observations are $8$ and $16,$ respectively. If five of the observations are $2,4,10,12,14 .$ Find the remaining two observations.
For $(2n+1)$ observations ${x_1},\, - {x_1}$, ${x_2},\, - {x_2},\,.....{x_n},\, - {x_n}$ and $0$ where $x$’s are all distinct. Let $S.D.$ and $M.D.$ denote the standard deviation and median respectively. Then which of the following is always true
For $(2n+1)$ observations ${x_1},\, - {x_1}$, ${x_2},\, - {x_2},\,.....{x_n},\, - {x_n}$ and $0$ where $x$’s are all distinct. Let $S.D.$ and $M.D.$ denote the standard deviation and median respectively. Then which of the following is always true