Statistics Questions

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

In a series of $2n$ observations, half of them equal $a$ and remaining half equal $-a.$ If the standard deviation of the observations is $2,$ then $|a|$ equals

Suppose a population $A$ has $100$ observations $101,102,\dots ,200$, and another population $B$ has $100$ observations $151,152,\dots ,250$. If $V_A$ and $V_B$ represent the variances of the two populations, respectively, then $\frac{V_A}{V_B}$ is:

The average marks of boys in a class is $52$ and that of girls is $42$. The average marks of boys and girls combined is $50$. The percentage of boys in the class is:

If in a frequency distribution, the mean and median are $21$ and $22$ respectively, then its mode is approximately:

Let $x_1,x_2,\dots ,x_n$ be $n$ observations such that $\sum x_i^2=400$ and $\sum x_i=80$. Then a possible value of $n$ among the following is:

The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the set is increased by 2, then the median of the new set:

In an experiment with 15 observations on $x$, the following results were available: $\sum x^2=2830$, $\sum x=170$. One observation that was 20 was found to be wrong and was replaced by the correct value 30. Then the corrected variance is:

The mean of the numbers $a,b,8,5,10$ is $6$ and the variance is $6.80$. Then which one of the following gives possible values of $a$ and $b$?

If the mean deviation of numbers $1,1+d,1+2d,\dots ,1+100d$ from their mean is $255$, then $d$ is equal to:

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

For two data sets, each of size $5$, the variances are given to be $4$ and $5$ and the corresponding means are given to be $2$ and $4$, respectively. The variance of the combined data set is:

If the mean deviation about the median of the numbers $a,2a,\dots ,50a$ is $50$, then $|a|$ equals:

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