The mean of the following frequency distribution is $50$ and $\Sigma f = 120$. Find the missing frequencies $f_1$ and $f_2$.

Class	$0-20$	$20-40$	$40-60$	$60-80$	$80-100$
$f$	$17$	$f_1$	$32$	$f_2$	$19$

$A$ set of four observations has mean $1$ and variance $13$. Another set of six observations has mean $2$ and variance $1$. Then, the variance of all these $10$ observations is equal to:

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.

Consider the frequency distribution of the given numbers. If the mean is known to be $3$,then the value of $f$ is:

Value	$1$	$2$	$3$	$4$
Frequency	$5$	$4$	$6$	$f$

Based on the following statements,choose the correct option.
Statement-$I:$ The variance of the first $n$ even natural numbers is $\frac{n^2-1}{4}$.
Statement-$II:$ The difference between the variance of the first $20$ even natural numbers and their arithmetic mean is $112$.

$A$ factory is operating in two shifts,day and night,with $70$ and $30$ workers respectively. If the per day mean wage of the day shift workers is $Rs. 54$ and the per day mean wage of all the workers is $Rs. 60$,then the per day mean wage of the night shift workers (in $Rs.$) is: