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 :

Question :

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.

Solution :

Calculation of median :

First, we prepare the following table to compute the median :

We have : n = 100, so $$n\over 2$$ = 50

The cumulative frequency just greater than 50 is 76 and the corresponding class is (7 – 10). Thus, (7 – 10) is the median class such that

$$n\over 2$$ = 50, l = 7, cf = 36, f = 40 and h = 3.

Substituting these values in the formula,

Median = l + ($${n\over 2} – cf\over f$$)$$\times$$ h

= 7 + ($$50 – 26\over 40$$)(3) = 7 + 1.05 = 8.05

Calculation of mean :

Mean = $$\sum f_ix_i\over \sum f_i$$ = $$832\over 100$$ = 8.32

Calculation of mode :

The class (7 – 10) has the maximum frequency. Therefore, this is the modal class.

Here, l = 7, h = 3, $$f_1$$ = 40, $$f_0$$ = 30 and $$f_2$$ = 16

Now, let us substitute these values in the formula

Mode = l + ($$f_1 – f_0\over 2f_1 – f_0 – f_2$$)(h) = 7 + $$10\over 34$$ $$\times$$ 3 = 7 + 0.88 = 7.88