Let r be the range and {S^2} = frac{1}{{n - 1 | vedclass

Name: PDF Generator App | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(a) We have $\mathop {r = \max |{x_i} - {x_j}|}\limits_{\,\,\,\,\,\,\,\,\,\,\,\,\,i\, 
e j} $
and ${S^2} = \frac{1}{{n - 1}}\sum\limits_{i = 1}^n {{

Vedclass · Accepted Answer

$S \le r\sqrt {\frac{n}{{n - 1}}} $

Vedclass · Answer

(a) We have $\mathop {r = \max |{x_i} - {x_j}|}\limits_{\,\,\,\,\,\,\,\,\,\,\,\,\,i\, 
e j} $
and ${S^2} = \frac{1}{{n - 1}}\sum\limits_{i = 1}^n {{{({x_i} - \bar x)}^2}} $
Now, ${({x_i} - \bar x)^2} = {\left( {{x_i} - \frac{{{x_1} + {x_2} + ..... + {x_n}}}{n}} ight)^2}$

$ = \frac{1}{{{n^2}}}[({x_i} - {x_1}) + ({x_i} - {x_2}) + .... + ({x_i} - {x_i} - 1)$

$ + ({x_i} - {x_i} + 1) + .......$$ + ({x_i} - {x_n})] \le \frac{1}{{{n^2}}}{[(n - 1)r]^2}$,

==> ${({x_i} - \bar x)^2} \le {r^2} \Rightarrow \sum\limits_{i = 1}^n {{{({x_i} - \bar x)}^2} \le n{r^2}} $

==> $\frac{1}{{n - 1}}\sum\limits_{i = 1}^n {{{({x_i} - \bar x)}^2} \le \frac{{n{r^2}}}{{(n - 1)}}} $==> ${S^2} \le \frac{{n{r^2}}}{{(n - 1)}}$

==> $S \le r\sqrt {\frac{n}{{n - 1}}} $.

$X$	$c$	$2c$	$3c$	$4c$	$5c$	$6c$
$f$	$2$	$1$	$1$	$1$	$1$	$1$

Let $r$ be the range and ${S^2} = \frac{1}{{n - 1}}\sum\limits_{i = 1}^n {{{({x_i} - \bar x)}^2}} $ be the $S.D.$ of a set of observations ${x_1},\,{x_2},\,.....{x_n}$, then

Similar Questions

The mean and variance of $5$ observations are $5$ and $8$ respectively. If $3$ observations are $1,3,5$, then the sum of cubes of the remaining two observations is

If the variance of the frequency distribution is $160$ , then the value of $\mathrm{c} \in \mathrm{N}$ is

$X$ $c$ $2c$ $3c$ $4c$ $5c$ $6c$

$f$ $2$ $1$ $1$ $1$ $1$ $1$

The mean and variance of $10$ observations were calculated as $15$ and $15$ respectively by a student who took by mistake $25$ instead of $15$ for one observation. Then, the correct standard deviation is$.....$

For a given distribution of marks mean is $35.16$ and its standard deviation is $19.76$. The co-efficient of variation is..

One set containing five numbers has mean $8$ and variance $18$ and the second set containing $3$ numbers has mean $8$ and variance $24$. Then the variance of the combined set of numbers is