Question

In this $100$ surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in English alphabets in the surnames was obtained as follows: Determine the number of median letters in the surnames. Find the number of mean letters in the surnames and also, find the size of modal in the surnames.

Number of letters	$1-4$	$4-7$	$7-10$	$10-13$	$13-16$	$16-19$
Numbers of surnames	$6$	$30$	$40$	$16$	$4$	$4$

Answer 1

Step $1$: Compute the median.

Class interval	Frequency	Cumulative Frequency
$1-4$	$6$	$6$
$4-7$	$30$	$36$
$7-10$	$40$	$76$
$10-13$	$16$	$92$
$13-16$	$4$	$96$
$16-19$	$4$	$100$

Given: $n=100$ and $\frac{n}{2}=50$

Median class$=7-10$

Therefore, $l=7,C_f=36,f=40$ and $h=3$.

Median is given by :

Median$=l+\frac{{\displaystyle \frac{n}{2}}-C_f}{f}\times h$

$\Rightarrow$Median$=7+\frac{(50-36)}{40}\times 3$

$\Rightarrow$Median$=7+\frac{42}{40}$

$\Rightarrow$Median$=8.05$

Step $2$: Compute the mode.

Mode is given by:

mode$=l+\left(\frac{f_1-f_0}{2f_1-f_0-f_2}\right)\times h$

$\Rightarrow$mode$=7+\left(\frac{40-30}{(2\times 40)-30-16}\right)\times 3$

$\Rightarrow$mode$=7+\frac{30}{34}$

$\Rightarrow$mode$=7.88$

Step $3$: Compute the mean.

Class interval	${\mathrm{f}}_{\mathrm{i}}$	${\mathrm{x}}_{\mathrm{i}}$	${\mathrm{f}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}}$
$1-4$	$6$	$2.5$	$15$
$4-7$	$30$	$5.5$	$165$
$7-10$	$40$	$8.5$	$340$
$10-13$	$16$	$11.5$	$184$
$13-16$	$4$	$14.5$	$51$
$16-19$	$4$	$17.5$	$70$
Sum ${\mathrm{f}}_{\mathrm{i}}=100$	Sum ${\mathrm{f}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}}=825$

The mean is given by:

mean$=\frac{{\displaystyle \sum _{}}{f}_i{x}_i}{{\displaystyle \sum _{}^{}}{f}_i}$

$\Rightarrow$mean$=\frac{{\displaystyle 825}}{100}$

$\Rightarrow$mean$=8.25$

Hence, the number of median letters in the surnames is $8.05$, size of modal in the surnames is $7.88$ , and the number of mean letters in the surnames is $8.25$.