Let $x_1, x_2, x_3, x_4, .......... , x_n$ be $n$ observations and let $\bar x$ be their arithmetic mean and $\sigma ^2$ be their variance.
Statement $-1$ : Variance of observations $2x_1, 2x_2, 2x_3, ......, 2x_n$ is $4\sigma ^2$ .
Statement $-2$ : Arithmetic mean of $2x _1, 2x_2, 2x_3, ......, 2x_n$ is $4\bar x$ .
Let $x_1, x_2, x_3, x_4, .......... , x_n$ be $n$ observations and let $\bar x$ be their arithmetic mean and $\sigma ^2$ be their variance.
Statement $-1$ : Variance of observations $2x_1, 2x_2, 2x_3, ......, 2x_n$ is $4\sigma ^2$ .
Statement $-2$ : Arithmetic mean of $2x _1, 2x_2, 2x_3, ......, 2x_n$ is $4\bar x$ .
- A
Statement $-1$ is true, statement $-2$ is true and statement $-2$ is $NOT$ the correct explanation for statement $-1$
- B
Statement $-1$ is true, statement $-2$ is false
- C
Statement $-1$ is false, stateemnt $-2$ is true
- D
Statement $-1$ is true, statement $-2$ is true and statement $-2$ is correct explanation for statement $-1$
Similar Questions
For the frequency distribution :
Variate $( x )$
$x _{1}$
$x _{1}$
$x _{3} \ldots \ldots x _{15}$
Frequency $(f)$
$f _{1}$
$f _{1}$
$f _{3} \ldots f _{15}$
where $0< x _{1}< x _{2}< x _{3}<\ldots .< x _{15}=10$ and
$\sum \limits_{i=1}^{15} f_{i}>0,$ the standard deviation cannot be
For the frequency distribution :
| Variate $( x )$ | $x _{1}$ | $x _{1}$ | $x _{3} \ldots \ldots x _{15}$ |
| Frequency $(f)$ | $f _{1}$ | $f _{1}$ | $f _{3} \ldots f _{15}$ |
where $0< x _{1}< x _{2}< x _{3}<\ldots .< x _{15}=10$ and
$\sum \limits_{i=1}^{15} f_{i}>0,$ the standard deviation cannot be
- [JEE MAIN 2020]
Calculate mean, variance and standard deviation for the following distribution.
Classes
$30-40$
$40-50$
$50-60$
$60-70$
$70-80$
$80-90$
$90-100$
${f_i}$
$3$
$7$
$12$
$15$
$8$
$3$
$2$
Calculate mean, variance and standard deviation for the following distribution.
| Classes | $30-40$ | $40-50$ | $50-60$ | $60-70$ | $70-80$ | $80-90$ | $90-100$ |
| ${f_i}$ | $3$ | $7$ | $12$ | $15$ | $8$ | $3$ | $2$ |
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
If both the means and the standard deviation of $50$ observations $x_1, x_2, ………, x_{50}$ are equal to $16$ , then the mean of $(x_1 - 4)^2, (x_2 - 4)^2, …., (x_{50} - 4)^2$ is
If both the means and the standard deviation of $50$ observations $x_1, x_2, ………, x_{50}$ are equal to $16$ , then the mean of $(x_1 - 4)^2, (x_2 - 4)^2, …., (x_{50} - 4)^2$ is
- [JEE MAIN 2019]
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 $B$
$10$
$20$
$30$
$25$
$43$
$15$
$7$
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 $B$ | $10$ | $20$ | $30$ | $25$ | $43$ | $15$ | $7$ |