$A$ teacher wanted to analyze the performance of two sections of students in a mathematics test of $100$ marks. Looking at their performances,she found that a few students got under $20$ marks and a few got $70$ marks or above. So she decided to group them into intervals of varying sizes as follows: $0-20, 20-30, ..., 60-70, 70-100$. Then she formed the following table:

Marks	Number of students
$0-20$	$7$
$20-30$	$10$
$30-40$	$10$
$40-50$	$20$
$50-60$	$20$
$60-70$	$15$
$70-100$	$8$
Total	$90$

$A$ histogram for this table was prepared by a student as shown in Fig. Carefully examine this graphical representation. Do you think that it correctly represents the data?

(N/A) No,the graph is giving us a misleading picture. In a histogram,the areas of the rectangles are proportional to the frequencies. This condition is satisfied when the widths of all class intervals are equal. However,in this case,the widths of the rectangles are varying,so the histogram does not give a correct representation. For example,it shows a greater frequency in the interval $70-100$ than in $60-70$,which is incorrect.
To correct this,we must modify the lengths of the rectangles so that their areas become proportional to the frequencies. The steps are:
$1$. Select the minimum class size,which is $10$ in this case.
$2$. Modify the lengths of the rectangles to be proportionate to the class size of $10$. For a class size of $20$,the length is $7$. For a class size of $10$,the length becomes $\frac{7}{20} \times 10 = 3.5$.
Following this method,we get the modified table:

Marks	Frequency	Width	Length of rectangle
$0-20$	$7$	$20$	$\frac{7}{20} \times 10 = 3.5$
$20-30$	$10$	$10$	$\frac{10}{10} \times 10 = 10$
$30-40$	$10$	$10$	$\frac{10}{10} \times 10 = 10$
$40-50$	$20$	$10$	$\frac{20}{10} \times 10 = 20$
$50-60$	$20$	$10$	$\frac{20}{10} \times 10 = 20$
$60-70$	$15$	$10$	$\frac{15}{10} \times 10 = 15$
$70-100$	$8$	$30$	$\frac{8}{30} \times 10 = 2.67$

These lengths represent the "proportion of students per $10$ marks interval". Plotting these values gives the correct histogram.