(Q8) 100 surnames 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.

Number of letters 1-4 4-7 7-10 10-13 13-16 16-19
Number of surnames 6 30 40 16 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 :

Number of letters in the surnames.

Also find the modal size of the surnames.

a =
highest value
2
 =
2
 =
Number of Letters (C.I.) Number of surnames(fi) c.f. Class marks(xi) di = xi - a fidi
1-4 6 6 2.5 - -
4-7 30 cf 36 5.5 - -
7-10 (l) 40 f 76 8.5 (a)
10-13 16 92 11.5
13-16 4 96 14.5
16-19 4 100 17.5
- ∑fi = 100 - - - ∑fidi = -

Total observations are n = .

n
2
 =
2
 =

50 lies in the class 7-10

∴ Median Class = 7-10

l - lower boundary =

f - frequency of the median class = 40

cf = 36 , Class size h = 3

Median = l +
(
n
2
 - cf )
f
 × h
Median = +
-
40
 ×
Median = +
40
 ×

Median = +

∴ Median =

Assumed mean, a = 8.5

Mean = x̄ = a +
∑fidi
∑fi
Mean x̄ = 8.5 +
-
100
 = 8.5 - =

∴ Mean =

Mode :

Maximum number of surnames = 40

Modal class = 7-10

l - lower boundary of the modal class =

l = ; f1 = 40, f0 = 30, f2 = 16, h = 3

f1 - f0
2f1 - f0 - f2
 × h
∴Mode = +
-
2 × - ( + )
 × = +
× = .882

Median = ; Mean = ; Modal size = .88.