Consider the following statements:
$(1)$ Mode can be computed from a histogram.
$(2)$ Median is not independent of change of scale.
$(3)$ Variance is independent of change of origin and scale.
Which of these is/are correct?

Consider the following frequency distribution:

Value	$4$	$5$	$8$	$9$	$6$	$12$	$11$
Frequency	$5$	$f_1$	$f_2$	$2$	$1$	$1$	$3$

Suppose that the sum of the frequencies is $19$ and the median of this frequency distribution is $6$. For the given frequency distribution,let $\alpha$ denote the mean deviation about the mean,$\beta$ denote the mean deviation about the median,and $\sigma^2$ denote the variance. Match each entry in List-$I$ to the correct entry in List-$II$ and choose the correct option.

List-$I$	List-$II$
$(P) \ 7f_1+9f_2$ is equal to	$(1) \ 146$
$(Q) \ 19\alpha$ is equal to	$(2) \ 47$
$(R) \ 19\beta$ is equal to	$(3) \ 48$
$(S) \ 19\sigma^2$ is equal to	$(4) \ 145$
	$(5) \ 55$

If the mean and variance of six observations $7, 10, 11, 15, a, b$ are $10$ and $\frac{20}{3}$ respectively,then the value of $|a-b|$ is equal to:

The average marks of boys in a class is $40$ and that of girls is $45$. The average marks of both boys and girls combined is $42$. Then the percentage of boys in the class is (in $\%$)

One set containing five numbers has mean $8$ and variance $18$,and the second set containing $3$ numbers has mean $8$ and variance $24$. Then the variance of the combined set of numbers is

$A$ set of four observations has mean $1$ and variance $13$. Another set of six observations has mean $2$ and variance $1$. Then,the variance of all these $10$ observations is equal to: