The mean of $100$ observations is $45$. It was later found that two observations $19$ and $31$ were incorrectly recorded as $91$ and $13$. The correct mean is...

The average age of a group of men and women is $30$ years. If the average age of men is $32$ years and that of women is $27$ years,what is the percentage of women in the group?

Let $a$ and $b$ be two real numbers. If the arithmetic mean and the variance of $a, b, 8, 5$ and $10$ are respectively $6$ and $6.8$,then an ordered pair $(a, b) =$

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

Let $x_1, x_2, \ldots, x_{11}$ be the observations satisfying $\sum_{i=1}^{11}(x_i-4)=22$ and $\sum_{i=1}^{11}(x_i-4)^2=154$. If the mean and variance of the observations are $\alpha$ and $\beta$,then the quadratic equation having the roots $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$ is

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.