6.   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.

 Class Number of surnames Cumulative Frequency Class mark 1-4 6 6 2.5 15 4-7 30 36 5.5 165 7-10 40 76 8.5 340 10-13 16 92 11.5 184 13-16 4 96 14.5 51 16-19 4 100 17.5 70 = 100 = 825

MEDIAN:

Median class = 7-10; Lower limit, l = 7;

Cumulative frequency of preceding class, c.f. = 36; f = 40; h = 3

Thus, the median of the data is 8.05

MODE:

The class having maximum frequency is the modal class.
The maximum frequency is 40 and hence the modal class = 7-10
Lower limit (l) of modal class = 7, class size (h) = 3
Frequency () of the modal class = 40; frequency () of class preceding the modal class = 30, frequency () of class succeeding the modal class = 16

Thus, Mode of the data is 7.88

MEAN:

Mean,

Thus, the Mean of the data is 8.25