100 surnames were randomly picked up from a local telephone directly 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 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.

Explanation

1) median class = 7 - 10
Class size, h = 3

Lower limit of median class, l = 7
Frequency of median class, f = 40
Cumulative frequency of class preceding median class,`cf = 36`

Median = `l + [(n/2 - cf)/f] xx h`
= `7 + [(50 - 36)/40] xx 3`
= `7 + (14/40) xx 3`

= `7 + 21/20`
= `7 + 1.05`
=`8.05`
2)Now, we will find the mean.
Class mark, xi= (Upper class limit + Lower class limit)/2
Taking assumed mean, a = 11.5

`Sigmafi` = 100 and` Sigmafiui` = -106
Class size, `h = 3`
Mean, (x) = `a + ((Sigmafiui)/(Sigmafi)) xx h`
`= 11.5 + (-106/100) xx 3`
`= 11.5 - 318/100`

`= 11.5 - 3.18`

`8.32`
3)it can be observed that the maximum class frequency is `40`, belonging to class interval `7 − 10`.

Class size, `h = 3`

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