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 (30.625) indicates that most states/U.T. have approximately 30-31 students per teacher in higher secondary schools. The mean (29.21) indicates that on average, there are about 29 students per teacher across all states/U.T. The mode being slightly higher than the mean suggests that while the most common ratio is around 31, the overall average is pulled lower due to some states having fewer students per teacher.
Explanation
This question requires finding both mode and mean for grouped data and interpreting them. The modal class is identified as the class with maximum frequency (30-35 with frequency 10). The mean is calculated using the direct method with class marks (midpoints). The interpretation compares what each measure represents - mode shows the most frequent value while mean shows the average value.
Solution Steps
Step 1: Identify the modal class. The class 30-35 has the maximum frequency of 10, so it is the modal class.
Step 2: Apply the mode formula: Mode = l + [(f₁ - f₀)/(2f₁ - f₀ - f₂)] × h, where l = 30, f₁ = 10, f₀ = 9, f₂ = 3, h = 5
Step 3: Calculate Mode = 30 + [(10-9)/(-9-3)] × 5 = 30 + [1/8] × 5 = 30 + 0.625 = 30.625
Step 4: Calculate class marks (midpoints): 17.5, 22.5, 27.5, 32.5, 37.5, 42.5, 47.5, 52.5
Step 5: Calculate fᵢxᵢ for each class: 52.5, 180, 247.5, 325, 112.5, 0, 0, 105
Step 6: Find Σfᵢ = 35 and Σfᵢxᵢ = 1022.5
Step 7: Calculate Mean = Σfᵢxᵢ/Σfᵢ = 1022.5/35 = 29.21