The following distribution gives the state-wise teacher-student ratio in higher secondary schools of India. Find the mode and mean of this data. Interpret the two measures.
| Number of students per teacher | Number of states/U.T. |
|---|---|
| 15 - 20 | 3 |
| 20 - 25 | 8 |
| 25 - 30 | 9 |
| 30 - 35 | 10 |
| 35 - 40 | 3 |
| 40 - 45 | 0 |
| 45 - 50 | 0 |
| 50 - 55 | 2 |
Mode = 30.625 students per teacher
Mean = 29.21 students per teacher
Interpretation: The mode indicates that the most common teacher-student ratio across states/U.T. is approximately 31 students per teacher. The mean indicates that the average teacher-student ratio is approximately 29 students per teacher. Since mode is greater than mean, it shows that most states have a slightly higher ratio than the national average.
Explanation
This question tests understanding of calculating mode and mean for grouped data. The modal class is identified as the class with highest frequency (30-35 with frequency 10). The mean is calculated using the direct method with class marks. The interpretation requires comparing both measures to understand the distribution pattern.
Solution Steps
Step 1: Find the Modal Class - From the table, the highest frequency is 10 for class 30-35. So, modal class = 30-35.
Step 2: Apply Mode Formula - Mode = l + [(f₁ - f₀)/(2f₁ - f₀ - f₂)] × h, where l = 30, h = 5, f₁ = 10, f₀ = 9, f₂ = 3. Mode = 30 + [(10-9)/(20-9-3)] × 5 = 30 + [1/8] × 5 = 30.625
Step 3: Calculate Class Marks (xᵢ) - Class marks are: 17.5, 22.5, 27.5, 32.5, 37.5, 42.5, 47.5, 52.5
Step 4: Calculate fᵢxᵢ values - fᵢxᵢ values are: 52.5, 180, 247.5, 325, 112.5, 0, 0, 105
Step 5: Find Mean - Σfᵢ = 35, Σfᵢxᵢ = 1022.5. Mean = 1022.5/35 = 29.21
Step 6: Interpret - Mode (30.625) > Mean (29.21), indicating most states have higher ratio than average.