The mean of $100$ items is $49$. It was discovered that three items which should have been $60, 70, 80$ were wrongly read as $40, 20, 50$ respectively. The correct mean is

If the coefficients of variation of two distributions are $40$ and $20$ and their variances are $144$ and $64$ respectively,then the mean of their arithmetic means is
$(A)$ $40$
$(B)$ $12$
$(C)$ $30$
$(D)$ $35$

The mean of five observations is $4$ and their variance is also $4$. If three of the five observations are $1, 3, 4$,then the product of the other two is:

If the mean and the standard deviation of the data $3, 5, 7, a, b$ are $5$ and $2$ respectively,then $a$ and $b$ are the roots of the equation:

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

There are $60$ students in a class. The following is the frequency distribution of the marks obtained by the students in a test:
$\begin{array}{|l|l|l|l|l|l|l|} \hline \text{Marks} & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \text{Frequency} & x-2 & x & x^2 & (x+1)^2 & 2x & x+1 \\ \hline \end{array}$
where $x$ is a positive integer. Determine the mean and standard deviation of the marks.