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.
(i) Median = 8.05 letters
(ii) Mean = 8.32 letters
(iii) Mode = 7.88 letters
Explanation
This question tests understanding of three measures of central tendency for grouped data: median, mean, and mode. Students need to construct cumulative frequency tables and apply the standard formulas from the textbook.
Solution Steps
Step 1: Construct Cumulative Frequency Table
| Number of letters | Number of surnames | 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 |
Total (n) = 100
Step 2: Find Median
n/2 = 100/2 = 50
The median class is 7-10 (cumulative frequency 76 50)
Using formula: Median = l + [(n/2 - cf)/f] × h
Where: l = 7, cf = 36, f = 40, h = 3
Median = 7 + [(50 - 36)/40] × 3 Median = 7 + (14/40) × 3 Median = 7 + 1.05 = 8.05
Step 3: Find Mean
Calculate class marks (x) and f·x:
| Class | f | x | f·x | | 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 | 58 | | 16-19 | 4 | 17.5 | 70 |
Σf·x = 832
Mean = Σf·x/Σf = 832/100 = 8.32
Step 4: Find Mode
The modal class is 7-10 (highest frequency = 40)
Using formula: Mode = l + [(f₁ - f₀)/(2f₁ - f₀ - f₂)] × h
Where: l = 7, f₁ = 40, f₀ = 30, f₂ = 16, h = 3
Mode = 7 + [(40 - 30)/(80 - 30 - 16)] × 3 Mode = 7 + [10/34] × 3 Mode = 7 + 0.88 = 7.88