Construct a frequency distribution table for the numbers given below, using the class intervals 21 - 30, 31 - 40, ….., etc.

75, 67, 57, 50, 26, 33, 44, 58, 67, 75, 78, 43, 41, 31, 21, 32, 40, 62, 54, 69, 48, 47, 51, 38, 39, 43, 61, 63, 68, 53, 56, 49, 59, 37, 40, 68, 23, 28, 36 and 47.

Use the table obtained to draw :

(i) a histogram

(ii) an ogive

Frequency distribution table :

(i) The above distribution is discontinuous, converting into continuous distribution, we get :

Adjustment factor = (Lower limit of one class - Upper limit of previous class) / 2

= $\dfrac{31 - 30}{2} = \dfrac{1}{2}$

= 0.5

Subtract the adjustment factor (0.5) from all the lower limits and add the adjustment factor (0.5) to all the upper limits.

Steps of construction of histogram :

1. Since, the scale on x-axis starts at 20.5, a break (kink) is shown near the origin on x-axis to indicate that the graph is drawn to scale beginning at 20.5

2. Take 2 cm along x-axis = 10 units.

3. Take 1 cm along y-axis = 1 unit.

4. Construct rectangles corresponding to the above continuous frequency distribution table.

The required histogram is shown in the below figure:

(ii) Steps of construction of ogive :

1. Since, the scale on x-axis starts at 20.5, a break (kink) is shown near the origin on x-axis to indicate that the graph is drawn to scale beginning at 20.5

2. Take 2 cm along x-axis = 10 units.

3. Take 1 cm along y-axis = 5 units.

4. Ogive always starts from a point on x-axis representing the lower limit of first class. Mark point (20.5, 0).

5. Take upper class limits along x-axis, corresponding cumulative frequencies along y-axis, mark the points (30.5, 4), (40.5, 13), (50.5, 22), (60.5, 29), (70.5, 37) and (80.5, 40).

6. Join the points marked by a free hand curve.

The required ogive is shown in the below figure: