The incomes of the parents of 100 students in a class in a certain university are tabulated below.

Income (in thousand ₹)	No. of students
0 - 8	8
8 - 16	35
16 - 24	35
24 - 32	14
32 - 40	8

(i) Draw a cumulative frequency curve to estimate the median income.

(ii) If 15% of the students are given freeships on the basis of the income of their parents, find the annual income of parents, below which the freeships will be awarded.

(iii) Calculate the Arithmetic mean.

(i) Cumulative frequency distribution table :

Here, n = 100, which is even.

Median = $\dfrac{n}{2} = \dfrac{100}{2}$ = 50th term.

Steps of construction of ogive :

1. Take 2 cm = 8 thousand on x-axis.

2. Take 1 cm = 10 students on y-axis.

3. Plot the point (0, 0) as ogive starts from x-axis representing the lower limit of first class.

4. Plot the points (8, 8), (16, 43), (24, 78), (32, 92) and (40, 100).

5. Join the points by a free hand curve.

6. Draw a line parallel to x-axis from point G (no. of students) = 50, touching the graph at point H. From point H draw a line parallel to y-axis touching x-axis at point I.

From graph, I = 17.6 (thousands)

Hence, median = 17.6 thousands.

(ii) 15% of 100 students = $\dfrac{15}{100} \times 100$ = 15.

Draw a line parallel to x-axis from point J (no. of students) = 15, touching the graph at point K. From point K draw a line parallel to y-axis touching x-axis at point L.

From graph,

L = 9.6 thousands.

Hence, the annual income of parents, below which the freeships will be awarded is 9.6 thousands.

(iii)

Mean = $\dfrac{Σfx}{Σf} = \dfrac{1832}{100}$ = 18.32 (thousands)

Hence, mean = 18.32 (thousands).