If mean and variance of five observations are frac{24}{5} and frac{194}{25} respectively and mean of

English

Gujarati

Hindi

13.Statistics

hard

If the mean and variance of five observations are $\frac{24}{5}$ and $\frac{194}{25}$ respectively and the mean of first four observations is $\frac{7}{2}$, then the variance of the first four observations in equal to

$\frac{4}{5}$

$\frac{77}{12}$

$\frac{5}{4}$

$\frac{105}{4}$

(JEE MAIN-2024)

Solution

$\bar{X}=\frac{24}{5} ; \sigma^2=\frac{194}{25}$

Let first four observation be $\mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3, \mathrm{x}_4$

Here, $\frac{x_1+x_2+x_3+x_4+x_5}{5}=\frac{24}{5}$.

Also, $\frac{\mathrm{x}_1+\mathrm{x}_2+\mathrm{x}_3+\mathrm{x}_4}{4}=\frac{7}{2}$

$\Rightarrow \mathrm{x}_1+\mathrm{x}_2+\mathrm{x}_3+\mathrm{x}_4=14$

Now from eqn $-1$

$\mathrm{x}_5$=$10$

Now, $\sigma^2=\frac{194}{25}$

$ \frac{\mathrm{x}_1^2+\mathrm{x}_2^2+\mathrm{x}_3^2+\mathrm{x}_4^2+\mathrm{x}_5^2}{5}-\frac{576}{25}=\frac{194}{25} $

$ \Rightarrow \mathrm{x}_1^2+\mathrm{x}_2^2+\mathrm{x}_3^2+\mathrm{x}_4^2=54$

Now, variance of first $4$ observations

Var $=\frac{\sum_{i=1}^4 x_i^2}{4}-\left(\frac{\sum_{i=1}^4 x_i}{4}\right)^2$

$ =\frac{54}{4}-\frac{49}{4}=\frac{5}{4}$

Standard 11

Mathematics

Find the mean and variance for the following frequency distribution.

Classes $0-30$ $30-60$ $60-90$ $90-120$ $120-150$ $50-180$ $180-210$

$f_i$ $2$ $3$ $5$ $10$ $3$ $5$ $2$

hard

From the data given below state which group is more variable, $A$ or $B$ ?

Marks $10-20$ $20-30$ $30-40$ $40-50$ $50-60$ $60-70$ $70-80$

Group $A$ $9$ $17$ $32$ $33$ $40$ $10$ $9$

Group $B$ $10$ $20$ $30$ $25$ $43$ $15$ $7$

hard

For a frequency distribution standard deviation is computed by applying the formula

normal

Let $n \geq 3$. A list of numbers $x_1, x, \ldots, x_n$ has mean $\mu$ and standard deviation $\sigma$. A new list of numbers $y_1, y_2, \ldots, y_n$ is made as follows $y_1=\frac{x_1+x_2}{2}, y_2=\frac{x_1+x_2}{2}$ and $y_j=x_j$ for $j=3,4, \ldots, n$.

The mean and the standard deviation of the new list are $\hat{\mu}$ and $\hat{\sigma}$. Then, which of the following is necessarily true?

normal

(KVPY-2014)

Find the mean and variance for the data

${x_i}$ $6$ $10$ $14$ $18$ $24$ $28$ $30$

${f_i}$ $2$ $4$ $7$ $12$ $8$ $4$ $3$

medium

Classes	$0-30$	$30-60$	$60-90$	$90-120$	$120-150$	$50-180$	$180-210$
$f_i$	$2$	$3$	$5$	$10$	$3$	$5$	$2$

Marks	$10-20$	$20-30$	$30-40$	$40-50$	$50-60$	$60-70$	$70-80$
Group $A$	$9$	$17$	$32$	$33$	$40$	$10$	$9$
Group $B$	$10$	$20$	$30$	$25$	$43$	$15$	$7$

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

If the mean and variance of five observations are $\frac{24}{5}$ and $\frac{194}{25}$ respectively and the mean of first four observations is $\frac{7}{2}$, then the variance of the first four observations in equal to

Solution

Similar Questions

Find the mean and variance for the following frequency distribution. Classes $0-30$ $30-60$ $60-90$ $90-120$ $120-150$ $50-180$ $180-210$ $f_i$ $2$ $3$ $5$ $10$ $3$ $5$ $2$

From the data given below state which group is more variable, $A$ or $B$ ? Marks $10-20$ $20-30$ $30-40$ $40-50$ $50-60$ $60-70$ $70-80$ Group $A$ $9$ $17$ $32$ $33$ $40$ $10$ $9$ Group $B$ $10$ $20$ $30$ $25$ $43$ $15$ $7$

For a frequency distribution standard deviation is computed by applying the formula

Find the mean and variance for the data ${x_i}$ $6$ $10$ $14$ $18$ $24$ $28$ $30$ ${f_i}$ $2$ $4$ $7$ $12$ $8$ $4$ $3$

Start a Free Trial Now

Find the mean and variance for the following frequency distribution.

Classes $0-30$ $30-60$ $60-90$ $90-120$ $120-150$ $50-180$ $180-210$

$f_i$ $2$ $3$ $5$ $10$ $3$ $5$ $2$

From the data given below state which group is more variable, $A$ or $B$ ?

Marks $10-20$ $20-30$ $30-40$ $40-50$ $50-60$ $60-70$ $70-80$

Group $A$ $9$ $17$ $32$ $33$ $40$ $10$ $9$

Group $B$ $10$ $20$ $30$ $25$ $43$ $15$ $7$

Find the mean and variance for the data

${x_i}$ $6$ $10$ $14$ $18$ $24$ $28$ $30$

${f_i}$ $2$ $4$ $7$ $12$ $8$ $4$ $3$