If $x_1, x_2, x_3, \ldots, x_n$ are $n$ observations such that $\sum(x_i+2)^2 = 28n$ and $\sum(x_i-2)^2 = 12n$,then the variance is:

The diameters of circles (in mm) drawn in a design are given below:

Diameters	$33-36$	$37-40$	$41-44$	$45-48$	$49-52$
No. of circles	$15$	$17$	$21$	$22$	$25$

Calculate the standard deviation and mean diameter of the circles.
[ Hint : First make the data continuous by making the classes as $32.5-36.5, 36.5-40.5, 40.5-44.5, 44.5-48.5, 48.5-52.5$ and then proceed.] (in $\text{ mm}$)

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

In a discrete data,$\frac{1}{4}$ of the observations are equal to $a$,another $\frac{1}{4}$ of the observations are equal to $-a$. Out of the remaining,half of them are equal to $b$ and the rest are equal to $-b$. If the variance of all the observations is $ab$,then:

Find the variance of the first $n$ natural numbers.

If $\sum_{i=1}^{18} (x_i - 8) = 9$ and $\sum_{i=1}^{18} (x_i - 8)^2 = 45$,find the standard deviation of $x_1, x_2, \dots, x_{18}$.