The mean of a set of $30$ observations is $75$. If each observation is multiplied by a non-zero number $\lambda$ and then each of them is decreased by $25$,their mean remains the same. The value of $\lambda$ is equal to

For a group of $100$ students,the mean $\bar{x}_1$ and the standard deviation $\sigma_1$ of their marks were found to be $40$ and $15$ respectively. Later,it was observed that the scores $40$ and $50$ were misread as $30$ and $60$ respectively. If the mean and the standard deviation with the corrected observations of the scores are $\bar{x}_2$ and $\sigma_2$ respectively,then:

The mean and standard deviation of $20$ observations are found to be $10$ and $2$ respectively. On rechecking,it was found that an observation $8$ was incorrect. Calculate the correct mean and standard deviation if the wrong item is omitted.

One set containing five numbers has mean $8$ and variance $18$,and the second set containing $3$ numbers has mean $8$ and variance $24$. Then the variance of the combined set of numbers is

If the mean of the data:

Class	$5 - 10$	$10 - 15$	$15 - 20$	$20 - 25$	$25 - 30$	$30 - 35$
Frequency	$2$	$k$	$28$	$54$	$k + 1$	$5$

is $21$,then $k$ is one of the roots of the equation:

Let $n \geq 3$. $A$ list of numbers $x_1, x_2, \ldots, x_n$ has mean $\mu$ and standard deviation $\sigma$. $A$ new list of numbers $y_1, y_2, \ldots, y_n$ is made as follows: $y_1 = \frac{x_1+x_2}{2}$,$y_2 = \frac{x_1+x_2}{2}$ and $y_j = x_j$ for $j = 3, 4, \ldots, n$. The mean and the standard deviation of the new list are $\hat{\mu}$ and $\hat{\sigma}$. Which of the following is necessarily true?