Find the standard deviation for the following data:

${x_i}$	$3$	$8$	$13$	$18$	$23$
${f_i}$	$7$	$10$	$15$	$10$	$6$

Let in a series of $2n$ observations,half of them are equal to $a$ and the remaining half are equal to $-a$. Also,by adding a constant $b$ to each of these observations,the mean and standard deviation of the new set become $5$ and $20$,respectively. Then the value of $a^{2} + b^{2}$ is equal to ....... .

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

Variance of first $n$ natural numbers is $\qquad$ .

Let $a, b, c, d$ and $e$ be the observations with mean $m$ and standard deviation $S$. The standard deviation of the observations $a+k, b+k, c+k, d+k$ and $e+k$ is

The variance of the observations $2, 3, 5, 7, 11, 13, 17, 22$ is