Question

$100$ surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabets in the surnames was obtained as follows:

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

Determine the median number of letters in the surnames. Find the mean number of letters in the surnames? Also, find the modal size of the surnames.

Answer 1

$No.ofletters$	$No.ofsurnames$ $f_i$	$x_i$	$f_i{x}_i$	$cf$
$1-4$	$6$	$2.5$	$15$	$6$
$4-7$	$30$	$5.5$	$165$	$36$
$7-10$	$40$	$8.5$	$340$	$76$
$10-13$	$16$	$11.5$	$184$	$92$
$13-16$	$4$	$14.5$	$58$	$96$
$16-19$	$4$	$17.5$	$70$	$100$
	$\sum f_i=100$		$\sum f_i{x}_i=832$

$\Rightarrow$ $Mean\left(\overline{x}\right)=\frac{\sum f_i{x}_i}{\sum f_i}=\frac{832}{100}=8.32$

Here, $N=100$ the, $\frac{N}{2}=\frac{100}{2}=50$

$\therefore$ Median class $=7-10$

$l=7,cf=36,f=40$ and $h=3$

$\Rightarrow$ $Median=l+\frac{\frac{N}{2}-cf}{f}\times h$

$=7+\frac{50-36}{40}\times 3$

$=7+1.05$

$\therefore$ $Median=8.05$

Here, modal class $=7-10$

$l=7,f_0=30,f_1=40,f_2=16$ and $h=3$

$\Rightarrow$ $Mode=l+\frac{f_1-f_0}{2f_1-f_0-f_2}\times h$

$=7+\frac{40-30}{2\times 40-30-16}\times 3$

$=7+0.88$

$=7.88$