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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.

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Solution

We have the following:
Number of letters Mid valuexi Frequencyfi Cumulative frequency fi×xi
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
fi=100 fi×xi=832
Mean, x¯=fi×xifi
=832100

=8.32

Here, N=100N2=50
The cumulative frequency just greater than 50 is 76 and the corresponding class is 7-10.
Thus, the median class is 7-10.
l=7, h=3, f=40, c=cf of preceding class = 36 and N2=50
Median, Me=l+h×N2-cf
=7+3×50-3640=7+3×1440=7+4240=7+1.05=8.05


∴ Mode = 3(Median) - 2(Mean)
= (3×8.05-2×8.32)
= 7.51

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