Let {x_1}, {x_2}, ldots, {x_n} be n observati | vedclass

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(D) Given observations are ${x_1}, {x_2}, \ldots, {x_n}$ with mean $\bar x$ and variance ${\sigma ^2} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar x)^2$.F

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Statement-$1$ is true,Statement-$2$ is false.

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(D) Given observations are ${x_1}, {x_2}, \ldots, {x_n}$ with mean $\bar x$ and variance ${\sigma ^2} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar x)^2$.For Statement-$2$: The arithmetic mean of $2{x_1}, 2{x_2}, \ldots, 2{x_n}$ is $\frac{1}{n} \sum_{i=1}^{n} (2x_i) = 2 \left( \frac{1}{n} \sum_{i=1}^{n} x_i \right) = 2\bar x$.Since the statement claims the mean is $4\bar x$,Statement-$2$ is false.For Statement-$1$: The variance of $2{x_1}, 2{x_2}, \ldots, 2{x_n}$ is $\frac{1}{n} \sum_{i=1}^{n} (2x_i - 2\bar x)^2 = \frac{1}{n} \sum_{i=1}^{n} 4(x_i - \bar x)^2 = 4 \left( \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar x)^2 \right) = 4{\sigma ^2}$.Thus,Statement-$1$ is true.Therefore,Statement-$1$ is true and Statement-$2$ is false.

Marks	$1-3$	$3-5$	$5-7$	$7-9$
Number of students	$40$	$30$	$20$	$10$

Let ${x_1}, {x_2}, \ldots, {x_n}$ be $n$ observations,and let $\bar x$ be their arithmetic mean and ${\sigma ^2}$ be the variance.
Statement-$1$: The variance of $2{x_1}, 2{x_2}, \ldots, 2{x_n}$ is $4{\sigma ^2}$.
Statement-$2$: The arithmetic mean of $2{x_1}, 2{x_2}, \ldots, 2{x_n}$ is $4\bar x$.

Similar Questions

Find the coefficient of variation of the first $n$ natural numbers.

The variance of the following distribution is:
Marks $1-3$ $3-5$ $5-7$ $7-9$
Number of students $40$ $30$ $20$ $10$

$A$ data consists of $n$ observations $x_1, x_2, ......, x_n$. If $\sum_{i=1}^n (x_i + 1)^2 = 9n$ and $\sum_{i=1}^n (x_i - 1)^2 = 5n$,then the standard deviation of this data is

The mean and standard deviation of $10$ observations are $20$ and $2$ respectively. Later on,it was observed that one observation was recorded as $50$ instead of $40$. Then the correct variance is:

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Let ${x_1}, {x_2}, \ldots, {x_n}$ be $n$ observations,and let $\bar x$ be their arithmetic mean and ${\sigma ^2}$ be the variance.Statement-$1$: The variance of $2{x_1}, 2{x_2}, \ldots, 2{x_n}$ is $4{\sigma ^2}$.Statement-$2$: The arithmetic mean of $2{x_1}, 2{x_2}, \ldots, 2{x_n}$ is $4\bar x$.