(B) Let the original observations be $x_i$ with mean $\bar{x} = \frac{1}{n} \sum x_i$ and variance $\sigma^2 = \frac{1}{n} \sum (x_i - \bar{x})^2$.For

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

(B) Let the original observations be $x_i$ with mean $\bar{x} = \frac{1}{n} \sum x_i$ and variance $\sigma^2 = \frac{1}{n} \sum (x_i - \bar{x})^2$.For the new observations $y_i = 2x_i$,the new mean $\bar{y}$ is:$\bar{y} = \frac{1}{n} \sum (2x_i) = 2 \left( \frac{1}{n} \sum x_i \right) = 2\bar{x}$.Thus,Statement $-2$ is false because the mean is $2\bar{x}$,not $4\bar{x}$.The new variance $\sigma_y^2$ is:$\sigma_y^2 = \frac{1}{n} \sum (y_i - \bar{y})^2 = \frac{1}{n} \sum (2x_i - 2\bar{x})^2 = \frac{1}{n} \sum 4(x_i - \bar{x})^2 = 4 \sigma^2$.Thus,Statement $-1$ is true.

Class 11 Mathematics Statistics Variance and Standard Deviation

Difficult

Let $x_1, x_2, x_3, \dots, x_n$ be $n$ observations,$\bar{x}$ be their arithmetic mean,and $\sigma^2$ be their variance.
Statement $-1$: The variance of observations $2x_1, 2x_2, 2x_3, \dots, 2x_n$ is $4\sigma^2$.
Statement $-2$: The arithmetic mean of $2x_1, 2x_2, 2x_3, \dots, 2x_n$ 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,Statement $-2$ is true.
D
Statement $-1$ is true,Statement $-2$ is true,and Statement $-2$ is the correct explanation for Statement $-1$.

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Statistics

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Variance and Standard Deviation

The raw data $x_1, x_2, \ldots, x_{n}$ is an $A$.$P$. with common difference $d$ and first term $0$. If $\bar{x}$ and $\sigma^2$ are the mean and variance of $x_{i}, i=1, 2, \ldots, n$,then $\sigma^2$ is:

MediumMHT CET 2023

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The mean and standard deviation of six observations are $8$ and $4,$ respectively. If each observation is multiplied by $3,$ find the new mean and new standard deviation of the resulting observations.

Medium

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Let in a series of $2n$ observations,half of them are equal to $a$ and the remaining half are equal to $-a$. Also,by adding a constant $b$ to each of these observations,the mean and standard deviation of the new set become $5$ and $20$,respectively. Then the value of $a^{2} + b^{2}$ is equal to ....... .

DifficultJEE MAIN 2021

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The mean of four observations is $3$. If the sum of the squares of these observations is $48$,then their standard deviation is

MediumAP EAMCET 2014

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The standard deviation of the following distribution is:
Class interval $0-10$ $10-20$ $20-30$ $30-40$
Frequency $1$ $3$ $4$ $2$

MediumAP EAMCET 2017

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Class interval	$0-10$	$10-20$	$20-30$	$30-40$
Frequency	$1$	$3$	$4$	$2$

Let $x_1, x_2, x_3, \dots, x_n$ be $n$ observations,$\bar{x}$ be their arithmetic mean,and $\sigma^2$ be their variance.Statement $-1$: The variance of observations $2x_1, 2x_2, 2x_3, \dots, 2x_n$ is $4\sigma^2$.Statement $-2$: The arithmetic mean of $2x_1, 2x_2, 2x_3, \dots, 2x_n$ is $4\bar{x}$.

Similar Questions

The raw data $x_1, x_2, \ldots, x_{n}$ is an $A$.$P$. with common difference $d$ and first term $0$. If $\bar{x}$ and $\sigma^2$ are the mean and variance of $x_{i}, i=1, 2, \ldots, n$,then $\sigma^2$ is:

The mean and standard deviation of six observations are $8$ and $4,$ respectively. If each observation is multiplied by $3,$ find the new mean and new standard deviation of the resulting observations.

The mean of four observations is $3$. If the sum of the squares of these observations is $48$,then their standard deviation is

The standard deviation of the following distribution is:Class interval$0-10$$10-20$$20-30$$30-40$Frequency$1$$3$$4$$2$

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Let $x_1, x_2, x_3, \dots, x_n$ be $n$ observations,$\bar{x}$ be their arithmetic mean,and $\sigma^2$ be their variance.
Statement $-1$: The variance of observations $2x_1, 2x_2, 2x_3, \dots, 2x_n$ is $4\sigma^2$.
Statement $-2$: The arithmetic mean of $2x_1, 2x_2, 2x_3, \dots, 2x_n$ is $4\bar{x}$.

The standard deviation of the following distribution is:
Class interval $0-10$ $10-20$ $20-30$ $30-40$
Frequency $1$ $3$ $4$ $2$