Let $X = \{x \in \mathbb{N} : 1 \le x \le 19\}$ and for some $a, b \in \mathbb{R}$,$Y = \{ax + b : x \in X\}$. If the mean and variance of the elements of $Y$ are $30$ and $750$,respectively,then the sum of all possible values of $b$ is

In a data set with $15$ observations $x_1, x_2, x_3, \ldots, x_{15}$,we are given $\sum_{i=1}^{15} x_i^2 = 3600$ and $\sum_{i=1}^{15} x_i = 175$. If the value of one observation $20$ was found to be incorrect and was replaced by its correct value $40$,then the corrected variance of the data is:

The variance of the variates $112, 116, 120, 125, 132$ about their $A.M.$ is

The mean of two samples of size $200$ and $300$ were found to be $25$ and $10$ respectively. Their standard deviations $(S.D.)$ are $3$ and $4$ respectively. Then,the variance of the combined sample of size $500$ is:

The frequency distribution:
$\begin{array}{|l|l|l|l|l|l|l|} \hline X & A & 2 A & 3 A & 4 A & 5 A & 6 A \\ \hline f & 2 & 1 & 1 & 1 & 1 & 1 \\ \hline \end{array}$
where $A$ is a positive integer,has a variance of $160$. Determine the value of $A$.

The mean and standard deviation of marks obtained by $50$ students of a class in three subjects,Mathematics,Physics and Chemistry are given below:

Subject	Mathematics,Physics,Chemistry
Mean	$42, 32, 40.9$
Standard deviation	$12, 15, 20$

Which of the three subjects shows the highest variability in marks and which shows the lowest?