Question

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 letters	1 − 4	4 − 7	7 − 10	10 − 13	13 − 16	16 − 19
Number of surnames	6	30	40	6	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.

Answer 1

The cumulative frequencies with their respective class intervals are as follows.

Number of letters	Frequency (fi)	Cumulative frequency
1 − 4	6	6
4 − 7	30	30 + 6 = 36
7 − 10	40	36 + 40 = 76
10 − 13	16	76 + 16 = 92
13 − 16	4	92 + 4 = 96
16 − 19	4	96 + 4 = 100
Total (n)	100

It can be observed that the cumulative frequency just greater than is 76, belonging to class interval 7 − 10.

Median class = 7 − 10

Lower limit (l) of median class = 7

Cumulative frequency (cf) of class preceding median class = 36

Frequency (f) of median class = 40

Class size (h) = 3

Median

= 8.05

To find the class marks of the given class intervals, the following relation is used.

Taking 11.5 as assumed mean (a), d_i, u_i, and f_iu_i are calculated according to step deviation method as follows.

Number of letters	Number of surnames fi	xi	di = xi− 11.5		fiui
1 − 4	6	2.5	− 9	− 3	− 18
4 − 7	30	5.5	− 6	− 2	− 60
7 − 10	40	8.5	− 3	− 1	− 40
10 − 13	16	11.5	0	0	0
13 − 16	4	14.5	3	1	4
16 − 19	4	17.5	6	2	8
Total	100				− 106

From the table, we obtain

∑f_iu_i = −106

∑f_i = 100

Mean,

= 11.5 − 3.18 = 8.32

The data in the given table can be written as

Number of letters	Frequency (fi)
1 − 4	6
4 − 7	30
7 − 10	40
10 − 13	16
13 − 16	4
16 − 19	4
Total (n)	100

From the table, it can be observed that the maximum class frequency is 40 belonging to class interval 7 − 10.

Modal class = 7 − 10

Lower limit (l) of modal class = 7

Class size (h) = 3

Frequency (f₁) of modal class = 40

Frequency (f₀) of class preceding the modal class = 30

Frequency (f₂) of class succeeding the modal class = 16

Therefore, median number and mean number of letters in surnames is 8.05 and 8.32 respectively while modal size of surnames is 7.88.