Question

At a customer-service centre, the time required to serve customers is tabulated below.

Time required (in minutes )	31-40	41-50	51-60	61-70	71-80	81-90
No. of customers	12	18	27	20	17	6

Draw a less-than cumulative frequency curve for the data.

Answer 1

Here, the classes are to be made continuous. We find the less-than type cumulative frequencies by adding class frequencies from top to bottom and place them against the upper class boundaries .We take an additional point (lower boundary of the first class, 0)
Here, it is (30.5,0) (frequency 0).We plot the points (u_i ,F_i ) and join the successive points by a free-hand curve.

Classes	Extended classes	Frequencies	Upper class boundaries (ui )	Less-than cumulative frequencies (Fi )
-	-	-	30.5	0
31-40	30.5-40.5	12	40.5	12
41-50	40.5-50.5	18	50.5	30
51-60	50.5-60.5	27	60.5	57
61-70	60.5-70.5	20	70.5	77
71-80	70.5-80.5	17	80.5	94
81-90	80.5-90.5	6	90.5	100

We take the upper class boundaries (u_i ) on the X-axis and the less-than cumulative frequencies (F_i ) on the Y-axis.
Thus, we plot the points (30.5,0), (40.5,12) , (50.5,30), (60.5,57), (70.5,77), (80.5,94), (90.5,100)
We join these points by a smooth free-hand curve and get the less-than type frequency curve shown below.