If the mean and standard deviation of $5$ observations $x_1, x_2, x_3, x_4, x_5$ are $10$ and $3$,respectively,then the variance of $6$ observations $x_1, x_2, x_3, x_4, x_5$ and $-50$ is equal to: (in $.5$)

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

If the mean of $100$ observations is $50$ and their standard deviation is $5$,then the sum of squares of all observations is

If each observation of a distribution with variance $\sigma^2$ is multiplied by $\lambda$,find the standard deviation of the new observations.

All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of $10$ to each of the students. Which of the following statistical measures will not change even after the grace marks were given?

Consider three observations $a, b,$ and $c$ such that $b = a + c$. If the standard deviation of $a + 2, b + 2, c + 2$ is $d$,then which of the following holds true?