Question

Draw a less than ogive for the following frequency distribution. Hence, estimate the median from the ogive.

Class interval	$6500-7000$	$7000-7500$	$7500-8000$	$8000-8500$	$8500-9000$	$9000-9500$	$9500-10000$
Frequency	$10$	$18$	$22$	$25$	$17$	$10$	$8$

Answer 1

Step 1: Making a less than cumulative frequency distribution

A less than cumulative frequency distribution can be calculated by adding all the frequencies less than the considered class interval.

We prepare the less than cumulative frequency distribution as follows:

Class interval	Less than $7000$	Less than $7500$	Less than $8000$	Less than $8500$	Less than $9000$	Less than $9500$	Less than $10000$
Frequency	$10$	$28$	$50$	$75$	$92$	$102$	$110$

Now, we plot the points $\left(7000,10\right),\left(7500,28\right),\left(8000,50\right),\left(8500,75\right),\left(9000,92\right),\left(9500,102\right),\left(10000,110\right)$ on the graph to get the required ogive.

Here,

$$\begin{gathered} n=110 \\ \Rightarrow \frac{n}{2}=55 \end{gathered}$$

Therefore from the graph, the median is the ${55}^{th}$ observation.

Step 2: Drawing a less than ogive

Let, N be the point on the $y$-axis representing $55$. From N, draw a line parallel to the $x$-axis meeting the ogive at H. From H, draw HM perpendicular to $x$-axis, meeting $x$-axis at M, M represents $8200$.

From the ogive, the ${55}^{th}$ observation is at $8200$.

Hence, the estimated median is $8200$.