Question

Draw a less-than type cumulative frequency curve for the following data and find the median graphically.

Body mass index	18-19	19-20	20-21	21-22	22-23	23-24
No. of persons	5	12	18	25	16	4

Answer 1

To draw the less-than type cumulative frequency curve, we prepare the following table .

Class interval	Frequencies ( fi)	Upper boundary (ui )	Less-than cumulative frequency (Fi )
-	-	18	0
18-19	5	19	5
19-20	12	20	17
20-21	18	21	35
21-22	25	22	60
22-23	16	23	76
23-24	4	24	80
Total	80 = N	-	-

We take the upper class boundaries (u_i ) on the X-axis and the less-than cumulative frequencies (F_i ) on the Y-axis.
We take an additional point preceding the first class interval with corresponding frequency zero.
Thus, we plot the points (18,0) ,(19,5), (20,17), (21,35), (22,60), (23,76), (24,80) .
We join these points by a smooth free-hand curve and get the less-than type frequency curve.



Median is the middle most item in the ordered arrangement of data.Therefore it is the value of $\frac{\mathrm{N}}{2}$th item.
Accordingly we mark the middle most observation i.e. 40 on Y-axis .
In the above graph it is A . Draw a line segment AB parallel to X-axis .Suppose the line segment cuts the less than cumulative frequency curve at B.
The point B has X-coordinate as median and Y-coordinate as $\frac{\mathrm{N}}{2}$.
We draw a line segment BC perpendicular to X-axis to get the foot of the perpendicular as the median.
Here the median is 21.1.