Question

Following is the age distribution of a group of students. Draw the cumulative frequency polygon, cumulative frequency curve (less than type) and hence obtain the median value.

Age	Frequency	Age	Frequency
5-6	40	11-12	92
6-7	56	12-13	80
7-8	60	13-14	64
8-9	66	14-15	44
9-10	84	15-16	20
10-11	96	16-17	8

Answer 1

We first prepare the cumulative frequency table by less than method as given below.

Age	Frequency	Age less than	Cumulative frequency
5-6	40	6	40
6-7	56	7	96
7-8	60	8	156
8-9	66	9	222
9-10	84	10	306
10-11	96	11	402
11-12	92	12	498
12-13	80	13	574
13-14	64	14	638
14-15	44	15	682
15-16	20	16	702
16-17	8	17	710

Other than the given class intervals, we assume a class $4-5$ before first class-interval $5-6$ with zero frequency.
Now we, mark the upper class limits (including the imagined class) along X-axis on a suitable scale and the cumulative frequencies along Y-axis on suitable scale.
Thus, we plot the points $(5,0),(6,40),(7,96),(8,156),(9,222),(10,306),(11,402),(12,494),(13,574),(14,638),(15,682),(16,702),and(17,710)$.
These points are marked and pointed by line segments to obtain the cumulative frequency polygon shown in Fig. 7.7.
In order to obtain the cumulative frequency curve, we draw a smooth curve passing through the points discussed above.
The graph (Fig. 7.8) shows the total number of students as $710$. The median is the age corresponding to $\frac{N}{2}=\frac{710}{2}=335$ students. In order to find the median, we first locate the point corresponding to ${355}^{th}$ student on Y-axis. Let the point be P. From this point draw a line parallel to the X- axis cutting the curve at Q. From this point Q draw a line parallel to Y-axis and meeting X-axis at the point M. The x-coordinate of M. The x-coordinate of M is $10.5$ (See Fig.7.8). Hence, median is $10.5$.