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 letters1−44−77−1010−1313−1616−19Number of surnames630401644Determine 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.

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Filo · Accepted Answer

No.ofletters No.ofsurnames fi​xi​ fi​xi​ cf 1−4 6 2.5 15 6 4−7 30 5.5 165 36 7−10 40 8.5 3407610−13 16 11.5 184 92 13−16 4 14.5 58 96 16−19 4 17.5 70 100  ∑fi​=100  ∑fi​xi​=832  ⇒  Mean(x)=∑fi​∑fi​xi​​=100832​=8.32Here, N=100 the, 2N​=2100​=50∴ Median class =7−10l=7,cf=36,f=40 and h=3⇒   Median=l+f2N​−cf​×h                       =7+4050−36​×3                       =7+1.05∴ Median=8.05Here, modal class =7−10l=7,f0​=30,f1​=40,f2​=16 and h=3⇒  Mode=l+2f1​−f0​−f2​f1​−f0​​×h                  =7+2×40−30−1640−30​×3                  =7+0.88                  =7.88

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

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

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

Question Text	$100$ 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: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.
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