Question

Draw less than ogive' and 'more than ogive' on a single graph paper and hence find the median of the following data:

Class interval	5 – 10	10 – 15	15 – 20	20 – 25	25 – 30	30 – 35	35 – 40
Frequqncy	2	12	2	4	3	4	3

Answer 1

1. Cumulative frequency table

To construct the ogives, we first compute the cumulative frequencies for both types:

Class Interval	Frequqncy (f)	Upper Class Limits	Less than Cumulative Frequency (cf)	Lower Class Limits	More than Cumulative Frequency (cf)
5 – 10	2	Less than 10	2	More than or equal to 5	30
10 – 15	12	Less than 15	14	More than or equal to 10	28
15 – 20	2	Less than 20	16	More than or equal to 15	16
20 – 25	4	Less than 25	20	More than or equal to 20	14
25 – 30	3	Less than 30	23	More than or equal to 25	10
30 – 35	4	Less than 35	27	More than or equal to 30	7
35 – 40	3	Less than 40	30	More than or equal to 35	3

2. Plotting the Ogives and Locating median

Less than Ogive: Plot the points (10, 2), (15, 14), (20, 16), (25, 20), (30, 23), (35, 27), (40, 30) and join them with a smooth curve.

More than Ogive: Plot the points (5, 30), (10, 28), (15, 16), (20, 14), (25, 10), (30, 7), (35, 3) and join them with a smooth curve.

3. Mathematical verification

To verify the graphical result, we can compute the median algebraically using the standard formula:

Median = `L + ((N/2 - CF)/f) xx h`

Total frequency (N) = 30 ⇒ `N/2 = 15`

The cumulative frequency just greater than 15 is 16, corresponding to the median class 15 – 20.

Lower boundary of median class (L) = 15

Frequency of median class (f) = 2

Cumulative frequency of preceding class (CF) = 14

Class width (h) = 5

Median = `15 + ((15 - 14)/2) xx 5`

= `15 + (1/2) xx 5`

= 15 + 2.5

= 17.5