Draw frequency polygons for the following frequency distribution :

(a) using histogram

(b) without using histogram.

C.I.	5 - 15	15 - 25	25 - 35	35 - 45	45 - 55	55 - 65
f	8	16	18	14	8	2

(a) Using histogram

Steps:

1. Draw a histogram for the given data.

2. Mark the mid-point at the top of each rectangle of the histogram drawn.

3. Also, mark the mid-point of the immediately lower class-interval (in the given question, the immediately lower class-interval is 0 - 5) and mid-point of the immediately higher class-interval (in the given question the immediate upper class-interval is 65 - 75).

4. Join the consecutive mid-points marked by straight lines to obtain the required frequency polygon.

(b) Without using histogram

Steps:

1. Find the class-mark (mid-value) of each given class-interval.

   Class-mark = mid-value = $\dfrac{\text{Upper limit + Lower limit}}{2}$

2. On a graph paper, mark class-marks along x-axis and frequencies along y-axis

3. On this graph paper, mark points taking values of class-marks along x-axis and the values of their corresponding frequencies along y-axis.

4. Draw line segments joining the consecutive points marked in step (3) above.