Question

100 surnames were randomly picked up from a local telephone directory and the distribution of number of letters of the English alphabet 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 and mean number of letters in the surname. Also, find the modal size of surnames.

Answer 1

We have the following:

Number of letters	Mid value$\left(x_i\right)$	Frequency$\left(f_i\right)$	Cumulative frequency	$f_i\times x_i$
1-4	2.5	6	6	15
4-7	5.5	30	36	165
7-10	8.5	40	76	340
10-13	11.5	16	92	184
13-16	14.5	4	96	58
16-19	17.5	4	100	70
		$f_i=100$		$\left(f_i\times x_i\right)=832$

Mean, $\overline{x}$=$\frac{\sum \left(f_i\times x_i\right)}{\sum f_i}$
=$\frac{832}{100}$
=8.32

$$\begin{gathered} \mathrm{Here},N=100 \\ \Rightarrow \frac{N}{2}=50 \end{gathered}$$
The cumulative frequency just greater than 50 is 76 and the corresponding class is 7-10.
Thus, the median class is 7-10.
$\therefore l=7,h=3,f=40,c=cf$of preceding class = 36 and $\frac{N}{2}=50$
Median, $M_e=l+\left\{h\times \frac{\left(\frac{N}{2}-c\right)}{f}\right\}$
$$\begin{gathered} =7+\left\{3\times \frac{\left(50-36\right)}{40}\right\} \\ =\left(7+3\times \frac{14}{40}\right) \\ =\left(7+\frac{42}{40}\right) \\ =7+1.05 \\ =8.05 \end{gathered}$$

∴ Mode = 3(Median) $-$ 2(Mean)
= (3$\times 8.05-2\times 8.32$)
= 7.51