The mean of $25$ observations is $36$. If the mean of the first $13$ observations is $32$ and the mean of the last $13$ observations is $40$,find the $13^{\text{th}}$ observation.

Heights (in $cm$) of $30$ girls of Class $IX$ are given below:
$140, 140, 160, 139, 153, 153, 146, 150, 148, 150, 152$
$146, 154, 150, 160, 148, 150, 148, 140, 148, 153, 138$
$152, 150, 148, 138, 152, 140, 146, 148$
Prepare a frequency distribution table for this data.

In a diagnostic test in mathematics given to students,the following marks (out of $100$) are recorded:
$46, 52, 48, 11, 41, 62, 54, 53, 96, 40, 98, 44$
Which 'average' will be a good representative of the above data and why?

Is it correct to say that in a histogram,the area of each rectangle is proportional to the class size of the corresponding class interval? If not,correct the statement.

The mean of the data: $2, 8, 6, 5, 4, 5, 6, 3, 6, 4, 9, 1, 5, 6, 5$ is given to be $5$. Based on this information,is it correct to say that the mean of the data: $10, 12, 10, 2, 18, 8, 12, 6, 12, 10, 8, 10, 12, 16, 4$ is $10$? Give reason.

If $\bar{x}$ represents the mean of $n$ observations $x_{1}, x_{2}, \ldots, x_{n},$ then the value of $\sum_{i=1}^{n}(x_{i}-\bar{x})$ is