Mean and variance for data {xᵢ} 92 93 97 98 102 104 109 {fᵢ} 3 2 3 2 6 3 3

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13.Statistics

hard

Find the mean and variance for the data

${x_i}$ $92$ $93$ $97$ $98$ $102$ $104$ $109$

${f_i}$ $3$ $2$ $3$ $2$ $6$ $3$ $3$

$29.09$

Solution

The data is obtained in tabular form as follows.

${x_i}$	${f_i}$	${f_i}{x_i}$	${{x_i} – \bar x}$	${\left( {{x_i} – \bar x} \right)^2}$	${f_i}{\left( {{x_i} – \bar x} \right)^2}$
$92$	$3$	$276$	$-8$	$64$	$192$
$93$	$2$	$186$	$-7$	$49$	$98$
$97$	$3$	$291$	$-3$	$9$	$27$
$98$	$2$	$196$	$-2$	$4$	$8$
$102$	$6$	$612$	$2$	$4$	$24$
$104$	$3$	$312$	$4$	$16$	$48$
$109$	$3$	$327$	$9$	$81$	$243$
	$22$	$2200$			$640$

Here, $N = 22,\sum\limits_{i = 1}^7 {{f_i}{x_i}} = 2200$

$\therefore \bar x = \frac{1}{n}\sum\limits_{i = 1}^7 {{f_i}{x_i}} = \frac{1}{{22}} \times 2200 = 100$

Variance $\left( {{\sigma ^2}} \right) = \frac{1}{N}\sum\limits_{i = 1}^7 {{f_i}{{\left( {{x_i} – \bar x} \right)}^2} = } \frac{1}{{22}} \times 640 = 29.09$

Standard 11

Mathematics

If the standard deviation of the numbers $-1, 0, 1, k$ is $\sqrt 5$ where $k > 0,$ then $k$ is equal to

hard

(JEE MAIN-2019)

Let $\mu$ be the mean and $\sigma$ be the standard deviation of the distribution

$X_i$ $0$ $1$ $2$ $3$ $4$ $5$

$f_i$ $k+2$ $2k$ $K^{2}-1$ $K^{2}-1$ $K^{2}-1$ $k-3$

where $\sum f_i=62$. if $[x]$ denotes the greatest integer $\leq x$, then $\left[\mu^2+\sigma^2\right]$ is equal $………$.

hard

(JEE MAIN-2023)

The variance of first $50$ even natural numbers is

normal

The mean and standard deviation of $15$ observations were found to be $12$ and $3$ respectively. On rechecking it was found that an observation was read as $10$ in place of $12$ . If $\mu$ and $\sigma^2$ denote the mean and variance of the correct observations respectively, then $15\left(\mu+\mu^2+\sigma^2\right)$ is equal to$……………….$

hard

(JEE MAIN-2024)

If the variance of the following frequency distribution is $50$ then $x$ is equal to:

Class $10-20$ $20-30$ $30-40$

Frequency $2$ $x$ $2$

medium

(JEE MAIN-2020)

$X_i$	$0$	$1$	$2$	$3$	$4$	$5$
$f_i$	$k+2$	$2k$	$K^{2}-1$	$K^{2}-1$	$K^{2}-1$	$k-3$

Find the mean and variance for the data ${x_i}$ $92$ $93$ $97$ $98$ $102$ $104$ $109$ ${f_i}$ $3$ $2$ $3$ $2$ $6$ $3$ $3$

Solution

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Find the mean and variance for the data

${x_i}$ $92$ $93$ $97$ $98$ $102$ $104$ $109$

${f_i}$ $3$ $2$ $3$ $2$ $6$ $3$ $3$

Let $\mu$ be the mean and $\sigma$ be the standard deviation of the distribution

$X_i$ $0$ $1$ $2$ $3$ $4$ $5$

$f_i$ $k+2$ $2k$ $K^{2}-1$ $K^{2}-1$ $K^{2}-1$ $k-3$

where $\sum f_i=62$. if $[x]$ denotes the greatest integer $\leq x$, then $\left[\mu^2+\sigma^2\right]$ is equal $………$.

If the variance of the following frequency distribution is $50$ then $x$ is equal to:

Class $10-20$ $20-30$ $30-40$

Frequency $2$ $x$ $2$