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.

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.