100 surnames were randomly picked from a local directory and the distribution of a number of letters of the English alphabet in the surname was obtained as follows:
Determine the median and mean a number of letters in the surname. Also, find the modal size of surnames

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Solution

$N o . o f l e t t e r s$ $N o . o f s u r n a m e s$
$f_{i}$ $x_{i}$ $f_{i} x_{i}$ $c f$
$1 - 4$ $6$ $2.5$ $15$ $6$
$4 - 7$ $30$ $5.5$ $165$ $36$
$7 - 10$ $40$ $8.5$ $340$ $76$
$10 - 13$ $16$ $11.5$ $184$ $92$
$13 - 16$ $4$ $14.5$ $58$ $96$
$16 - 19$ $4$ $17.5$ $70$ $100$
$\sum f_{i} = 100$ $\sum f_{i} x_{i} = 832$
$\Rightarrow$ $M e a n (¯ ¯ ¯ x) = \frac{\sum f_{i} x_{i}}{\sum f_{i}} = \frac{832}{100} = 8.32$
Here, $N = 100$ the, $\frac{N}{2} = \frac{100}{2} = 50$
$∴$ Median class $= 7 - 10$
$l = 7, c f = 36, f = 40$ and $h = 3$
$\Rightarrow$ $M e d i a n = l + \frac{\frac{N}{2} - c f}{f} \times h$

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

$= 7 + 1.05$
$∴$ $M e d i a n = 8.05$
Here, modal class $= 7 - 10$
$l = 7, f_{0} = 30, f_{1} = 40, f_{2} = 16$ and $h = 3$
$\Rightarrow$ $M o d e = l + \frac{f_{1} - f_{0}}{2 f_{1} - f_{0} - f_{2}} \times h$

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

$= 7 + 0.88$
$= 7.88$

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$N o . o f l e t t e r s$	$N o . o f s u r n a m e s$ $f_{i}$	$x_{i}$	$f_{i} x_{i}$	$c f$
$1 - 4$	$6$	$2.5$	$15$	$6$
$4 - 7$	$30$	$5.5$	$165$	$36$
$7 - 10$	$40$	$8.5$	$340$	$76$
$10 - 13$	$16$	$11.5$	$184$	$92$
$13 - 16$	$4$	$14.5$	$58$	$96$
$16 - 19$	$4$	$17.5$	$70$	$100$
	$\sum f_{i} = 100$		$\sum f_{i} x_{i} = 832$

100 surnames were randomly picked from a local directory and the distribution of a number of letters of the English alphabet in the surname was obtained as follows: Determine the median and mean a number of letters in the surname. Also, find the modal size of surnames

100 surnames were randomly picked from a local directory and the distribution of a number of letters of the English alphabet in the surname was obtained as follows:
Determine the median and mean a number of letters in the surname. Also, find the modal size of surnames