The data given below represent the marks obtained by 35 students:

21 26 21 20 23 24 22 19 24

26 25 23 26 29 21 24 19 25

26 25 22 23 23 27 26 24 25

30 25 23 28 28 24 28 28

Taking class intervals 19 - 20, 21 - 22 etc., make a frequency distribution for the above data.

Construct a combined histogram and frequency polygon for the distribution.

From above data :

Least mark = 19

Greatest marks = 30

Range = 30 – 19 = 11.

Given,

We need to take class intervals as 19 - 20, 21 - 22 etc.

The following frequency distribution is discontinuous, to convert it into continuous frequency distribution,

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

= $\dfrac{21 - 20}{2} = \dfrac{1}{2}$

= 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.

Continuous frequency distribution for given data is :

Steps of construction of histogram :

1. Since, the scale on x-axis starts at 16.5, a break (kink) is shown near the origin on x-axis to indicate that the graph is drawn to scale beginning at 16.5.

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

3. Take 1 cm along y-axis = 2 units.

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

The required histogram is shown in the adjoining figure.

Steps of construction of frequency polygon :

1. Mark the mid-points of upper bases of rectangles of the histogram.

2. Join the consecutive mid-points by line segments.

3. Join the first end point with the mid-point of class 16.5 - 18.5 with zero frequency, and join the other end point with the mid-point of class 30.5 - 32.5 with zero frequency.

The required frequency polygon is shown by thick line segments in the diagram.