Let $X_{1}, X_{2}, \ldots, X_{18}$ be eighteen observations such that $\sum_{i=1}^{18}(X_{i}-\alpha)=36$ and $\sum_{i=1}^{18}(X_{i}-\beta)^{2}=90$,where $\alpha$ and $\beta$ are distinct real numbers. If the standard deviation of these observations is $1$,then the value of $|\alpha-\beta|$ is ...... .

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

If each of the observations $x_{1}, x_{2}, \ldots, x_{n}$ is increased by $a$,where $a$ is a positive or negative number,show that the variance remains unchanged.

The mean of $5$ observations is $7$. If four of these observations are $6, 7, 8, 10$ and one is missing,then the variance of all the five observations is:

The mean and standard deviation of some data for the time taken to complete a test are calculated with the following results:
Number of observations $= 25$,mean $= 18.2 \text{ seconds}$,standard deviation $= 3.25 \text{ s}$.
Further,another set of $15$ observations $x_{1}, x_{2}, \ldots, x_{15}$,also in seconds,is now available and we have $\sum_{i=1}^{15} x_{i} = 279$ and $\sum_{i=1}^{15} x_{i}^{2} = 5524$. Calculate the standard deviation based on all $40$ observations.

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 $A$.