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 surname was obtained as follows:

Determine the median number of letters in the surnames. Find the mean number of letter in the surnames? Also, find the modal size of the surnames.

We have n = 100

So,

The cumulative Frequency just greater than is 36 then the median class is 7 – 10 such that

the lower limit (l) = 7

cumulative frequency of the class preceding 7 – 10 (cf) = 36

frequency of the median class 7 – 10 = 40,

class size (h) = 3

Using the formula,,we have

= 7 + 1.05

= 8.05

Now, we calculate the Mean

Now,

⇒ = 11.5 – 3.18

⇒ = 8.32

Now, we have to find the mode

Here, the maximum class frequency is 40, and the class corresponding to this frequency is 7 – 10.

So, the modal class is 7 – 10.

Now, modal class = 7 – 10, lower limit (l) of modal class = 7, class size(h) = 3

frequency (f₁) of the modal class = 40

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

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

Now, let us substitute these values in the formula

= 7 + 0.88

= 7.88