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.
(Mean) 8.32 letters
(Median) 8.05 letters
(Mode) 7.88 letters (approximately)
Explanation
This question requires calculating all three measures of central tendency for grouped data. The median is found using the formula L + [(n/2 - cf)/f] × h, where we first identify the median class using cumulative frequency. The mean is calculated using the direct method by finding class marks and computing Σfᵢxᵢ/Σfᵢ. The mode is found using the formula L + [(f₁ - f₀)/(2f₁ - f₀ - f₂)] × h, where the modal class is the one with maximum frequency.
Solution Steps
Step 1: Calculate cumulative frequencies. | Class | Frequency (f) | Cumulative Frequency (cf) | | 1 - 4 | 6 | 6 | | 4 - 7 | 30 | 36 | | 7 - 10 | 40 | 76 | | 10 - 13 | 16 | 92 | | 13 - 16 | 4 | 96 | | 16 - 19 | 4 | 100 |
Step 2: Find Median. n = 100, n/2 = 50. Median class is 7 - 10 (since cf = 76 50). L = 7, cf = 36, f = 40, h = 3. Median = L + [(n/2 - cf)/f] × h Median = 7 + [(50 - 36)/40] × 3 = 7 + (14/40) × 3 = 7 + 1.05 = 8.05
Step 3: Find Mean using direct method. Class marks (xᵢ): 2.5, 5.5, 8.5, 11.5, 14.5, 17.5. fᵢxᵢ: =15, =165, =340, =184, =58, =70. Σfᵢxᵢ = 15 + 165 + 340 + 184 + 58 + 70 = 832. Mean = Σfᵢxᵢ/Σfᵢ = 832/100 = 8.32
Step 4: Find Mode. Modal class = 7 - 10 (maximum frequency = 40). L = 7, f₁ = 40, f₀ = 30, f₂ = 16, h = 3. Mode = L + [(f₁ - f₀)/(2f₁ - f₀ - f₂)] × h Mode = 7 + [(40 - 30)/(80 - 30 - 16)] × 3 = 7 + (10/34) × 3 = 7 + 0.88 = 7.88 (approx.)