100 sumames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabet 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.

For median:

We have, total frequency, N = 100

N/2 = 100/2 = 50

Observe, cf = 76 is just greater than 50.

Thus, median class = 7-10

Median is given by

Where,

L = Lower class limit of median class = 7

N/2 = 50

cf = cumulative frequency of the class preceding median class = 36

f = frequency of the median class = 40

h = class interval of the median class = 3

Substituting these values in the formula of median, we get

⇒

⇒ Median = 7 + 1.05

⇒ Median = 8.05

For mean:

Mean is given by

⇒ [Refer the values from the table given above]

⇒ Mean = 8.32

For mode:

Here, highest frequency is 40.

So, the modal class = 7-10

Mode is given by

Where,

L = Lower class limit of the modal class = 7

h = class interval of the modal class = 3

f₁ = frequency of the modal class = 40

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

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

Substituting values in the formula of mode,

⇒

⇒ Mode = 7 + 0.88

⇒ Mode = 7.88

Thus, median is 8.05, mean is 8.32 and mode is 7.88.