Question

Find the mode and median of the following frequency distribution:

$x$	$10$	$11$	$12$	$13$	$14$	$15$
$f$	$1$	$4$	$7$	$5$	$9$	$8$

Answer 1

Step 1: Calculate the mode of the date

Mode is the observation that occurs most frequently or has highest frequency.

From the given distribution table, $14$ has the highest frequency of nine.

So, mode is $14$.

Step 2: Calculate the median of the data

Median is the middle observation of a frequency distribution.

Construct the cumulative frequency distribution table:

Cumulative frequency is calculated by adding the present and past frequency(s) for each given observation.

$x$	$10$	$11$	$12$	$13$	$14$	$15$
$f$	$1$	$4$	$7$	$5$	$9$	$8$
$c.f.$	$1$	$5$	$12$	$17$	$26$	$34$

The sum of all frequencies will be $\sum f\left(n\right)=1+4+7+5+9=34$ , which is an even number.

We know that, median for even number of frequencies is $\mathrm{median}=\frac{{\left({\displaystyle \frac{\mathrm{n}}{2}}\right)}^{\mathrm{th}}\mathrm{term}+{\left({\displaystyle \frac{\mathrm{n}+2}{2}}\right)}^{\mathrm{th}}\mathrm{term}}{2}$

So, median term will be,

$$\begin{gathered} =\frac{{\left({\displaystyle \frac{34}{2}}\right)}^{\mathrm{th}}\mathrm{term}+{\left({\displaystyle \frac{34+2}{2}}\right)}^{\mathrm{th}}\mathrm{term}}{2} \\ =\frac{{17}^{\mathrm{th}}\mathrm{term}+{18}^{\mathrm{th}}\mathrm{term}}{2} \end{gathered}$$

Now, according to the c.f. distribution table (above), ${17}^{th}term$ is $13$ and ${18}^{th}term$ is $14$.

So, the median will be $\left(\frac{13+14}{2}\right)=13.5$.

Therefore, for the given data, mode is $14$ and median is $13.5$.