Using a graph paper, draw an ogive for the following distribution which shows a record of the weight in kilograms of 200 students.

Weight (in kg)	No. of students
40 - 45	5
45 - 50	17
50 - 55	22
55 - 60	45
60 - 65	51
65 - 70	31
70 - 75	20
75 - 80	9

Use your ogive to estimate the following :

(i) the percentage of students weighing 55 kg or more

(ii) the weight above which the heaviest 30% of the students fall

(iii) the number of students who are (a) under weight and (b) Over-weight, if 55.70 kg is considered as standard weight.

Cumulative frequency distribution table :

Here, n = 200, which is even.

Steps of construction:

1. Take 2 cm along x-axis = 5 kg

2. Take 2 cm along y-axis = 20 units.

3. Since, scale on x-axis starts at 40, a break (kink) is shown near the origin on x-axis to indicate that the graph is drawn to scale beginning at 40.

4. Plot the point (40, 0) as ogive starts from x-axis representing lower limit of first class.

5. Plot the points (45, 5), (50, 22), (55, 44), (60, 89), (65, 140), (70, 171), (75, 191) and (80, 200).

6. Join the points by a free hand curve.

7. Draw a line parallel to y-axis from point J(weight) = 55, touching the graph at point Q. From point Q draw a line parallel to x-axis touching y-axis at point K.

From graph, K = 44.

Hence, 44 students weight 55 kg or less.

Students weighing more than 55 kg = 200 - 44 = 156

Percentage of students weighing more than 55 kg = $\dfrac{156}{200} \times 100$ = 78%

Hence, percentage of students weighing more than 55 kg = 78%.

(ii) 30% of students = $\dfrac{30}{100} \times 200$ = 60.

Total students = 200

No. of students not in heaviest 30% = 200 - 60 = 140.

Draw a line parallel to x-axis from point I (no. of students) = 140, touching the graph at point R. From point R draw a line parallel to y-axis touching x-axis at point P.

From graph, P = 65

Hence, above 65 kg the heaviest 30% of the students fall.

(iii) Draw a line parallel to y-axis from point L (weight) = 55.70 kg, touching the graph at point M. From point M draw a line parallel to x-axis touching y-axis at point N.

(a) From graph,

N = 50.

∴ 50 students have weight less than 55.70 kg

Hence, 50 students are underweight.

(b) Since, 50 students have weight less than 55.70 kg

∴ 150 (200 - 50) students have weight more than 55.70 kg.

Hence, 150 students are overweight.