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 | 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
The cumulative frequency of the given data can be calculated as:
Class interval | frequency | Cumulative frequency |
1-4 | 6 | 6 |
4-7 | 30 | 36 |
7-10 | 40 | 76 |
10-13 | 16 | 92 |
13-16 | 4 | 96 |
16-19 | 4 | 100 |
As per the question,
N= 100
Hence,
Median class = 7-10
Lower limit, l = 7
cf = 36
f = 40
h = 3
Now,
Median can be calculated as:
= 8.05
Now, mode can be calculated as:
Modal class = 7-10
l = 7
h = 3
f1 = 40
f0 = 30
f2 = 16
= 
= 7.88
Now, mean of the following data can be calculated as:
Class interval | fi | xi | fixi |
1-4 | 6 | 2.5 | 15 |
4-7 | 30 | 5.5 | 165 |
7-10 | 40 | 8.5 | 340 |
10-13 | 16 | 11.5 | 184 |
13-16 | 4 | 14.5 | 51 |
16-19 | 4 | 17.5 | 70 |
∑fi = 100 | ∑fixi = 825 |
= 8.25