What is the weighted mean of the first $n$ natural numbers when the weights are equal to the corresponding natural numbers?

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:

Statement $(I)$: The range of the ungrouped data does not change even if certain intermediate observations are removed.
Statement $(II)$: The value of the mean deviation of an ungrouped data about the median is always less than or equal to the value of the mean deviation computed about any other measure of central tendency.
Statement $(III)$: For a grouped data,range is approximated as the difference between the lower limit of the largest class and the upper limit of the smallest class.

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:

Find the least positive value of $k$,if the range of $15, 14, k, 25, 30, 35$ is $23$.

Consider the given data with frequency distribution:
$x_{i} = \{3, 8, 11, 10, 5, 4\}$
$f_{i} = \{5, 2, 3, 2, 4, 4\}$
Match each entry in List-$I$ to the correct entries in List-$II$.

List-$I$	List-$II$
$(P)$ The mean of the above data is	$(1) 2.5$
$(Q)$ The median of the above data is	$(2) 5$
$(R)$ The mean deviation about the mean of the above data is	$(3) 6$
$(S)$ The mean deviation about the median of the above data is	$(4) 2.7$
	$(5) 2.4$

The correct option is :