The mean of $n$ items is $\bar x$. If the first term is increased by $1$,the second by $2$,and so on,then the new mean is:

If the variance of the frequency distribution is $3$,then $\alpha$ is ......

$X_i$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
Frequency $f_i$	$3$	$6$	$16$	$\alpha$	$9$	$5$	$6$

The mean and standard deviation of $20$ observations were calculated as $10$ and $2.5$ respectively. It was found that by mistake one data value was taken as $25$ instead of $35$. If $\alpha$ and $\sqrt{\beta}$ are the mean and standard deviation respectively for the correct data,then $(\alpha, \beta)$ is:

If for some $x \in R^{+} \cup \{0\}$,the frequency distribution of the marks obtained by $20$ students in a test is given by the table below,then find the mean of the marks.

Marks:	$2$	$3$	$5$	$7$
Frequency:	$(x+1)^2$	$2x-5$	$x^2-3x$	$x$

The frequency distribution of the daily household expenditure of some families in a residential area is given below. If the mode of the distribution is $140$,what is the value of $b$?

Daily Expenditure (	$0-50$	$50-100$	$100-150$	$150-200$	$200-250$
Number of Families $(f)$	$24$	$33$	$37$	$b$	$25$

The mean and standard deviation of $100$ observations were calculated as $40$ and $5.1$,respectively,by a student who took by mistake $50$ instead of $40$ for one observation. What are the correct mean and standard deviation?