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(A) Let the observations be $x_1, x_2, \dots, x_n$. The variance is given by $\sigma^2 = \frac{1}{n} \sum (x_i - \bar{x})^2$. If we change the origin

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Origin only

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(A) Let the observations be $x_1, x_2, \dots, x_n$. The variance is given by $\sigma^2 = \frac{1}{n} \sum (x_i - \bar{x})^2$. If we change the origin by a constant $a$,the new observations are $y_i = x_i + a$. The mean becomes $\bar{y} = \bar{x} + a$. The new variance is $\frac{1}{n} \sum (y_i - \bar{y})^2 = \frac{1}{n} \sum (x_i + a - (\bar{x} + a))^2 = \frac{1}{n} \sum (x_i - \bar{x})^2 = \sigma^2$. Thus,the variance is independent of the change of origin. If we change the scale by a constant $b$,the new observations are $y_i = b x_i$. The new variance becomes $b^2 \sigma^2$. Therefore,the variance is independent of the change of origin only.

	Physics	Chemistry	Mathematics	Biology
Mean	$20$	$25$	$23$	$27$
$S$.$D$.	$3$	$2$	$4$	$5$

Class	$30-40$	$40-50$	$50-60$	$60-70$	$70-80$	$80-90$	$90-100$
$f_i$	$3$	$7$	$12$	$15$	$8$	$3$	$2$

Marks of $A$	$30, 20, 40$
Marks of $B$	$70, 0, 5$

The variance is independent of the change of which of the following?

Similar Questions

The following table shows the information about marks obtained in Physics,Chemistry,Mathematics,and Biology by $100$ students in a class. Which subject shows the highest variability in marks?
Physics Chemistry Mathematics Biology
Mean $20$ $25$ $23$ $27$
$S$.$D$. $3$ $2$ $4$ $5$

Calculate the mean,variance,and standard deviation for the following distribution:

Class $30-40$ $40-50$ $50-60$ $60-70$ $70-80$ $80-90$ $90-100$

$f_i$ $3$ $7$ $12$ $15$ $8$ $3$ $2$

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

Statement-$1$: The variance of the first $n$ even natural numbers is $\frac{n^2 - 1}{4}$.
Statement-$2$: The sum of the first $n$ natural numbers is $\frac{n(n + 1)}{2}$ and the sum of the squares of the first $n$ natural numbers is $\frac{n(n + 1)(2n + 1)}{6}$.

The marks obtained by students $A$ and $B$ in $3$ examinations are given below:

Marks of $A$ $30, 20, 40$

Marks of $B$ $70, 0, 5$

The ratio of the coefficient of variation of marks of $A$ and the coefficient of variation of marks of $B$ is:

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The variance is independent of the change of which of the following?

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The following table shows the information about marks obtained in Physics,Chemistry,Mathematics,and Biology by $100$ students in a class. Which subject shows the highest variability in marks?PhysicsChemistryMathematicsBiologyMean$20$$25$$23$$27$$S$.$D$.$3$$2$$4$$5$

Calculate the mean,variance,and standard deviation for the following distribution: Class $30-40$ $40-50$ $50-60$ $60-70$ $70-80$ $80-90$ $90-100$ $f_i$ $3$ $7$ $12$ $15$ $8$ $3$ $2$

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

Statement-$1$: The variance of the first $n$ even natural numbers is $\frac{n^2 - 1}{4}$.Statement-$2$: The sum of the first $n$ natural numbers is $\frac{n(n + 1)}{2}$ and the sum of the squares of the first $n$ natural numbers is $\frac{n(n + 1)(2n + 1)}{6}$.

The marks obtained by students $A$ and $B$ in $3$ examinations are given below: Marks of $A$ $30, 20, 40$ Marks of $B$ $70, 0, 5$ The ratio of the coefficient of variation of marks of $A$ and the coefficient of variation of marks of $B$ is: