The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate.
| Literacy rate (in %) | 45 - 55 | 55 - 65 | 65 - 75 | 75 - 85 | 85 - 95 |
|---|---|---|---|---|---|
| Number of cities | 3 | 10 | 11 | 8 | 3 |
The mean literacy rate is 69.43%.
To find the mean literacy rate, we use the direct method for grouped data. First, we calculate the class mark (xᵢ) for each class interval, which is the midpoint of the class. Then we multiply each class mark by its corresponding frequency (fᵢ) to get fᵢxᵢ.
Step 1: Calculate class marks (xᵢ) for each class:
- For 45-55: xᵢ = (45+55)/2 = 50
- For 55-65: xᵢ = (55+65)/2 = 60
- For 65-75: xᵢ = (65+75)/2 = 70
- For 75-85: xᵢ = (75+85)/2 = 80
- For 85-95: xᵢ = (85+95)/2 = 90
Step 2: Construct the calculation table:
| Literacy rate (%) | Number of cities (fᵢ) | Class mark (xᵢ) | fᵢxᵢ |
|---|---|---|---|
| 45 - 55 | 3 | 50 | 150 |
| 55 - 65 | 10 | 60 | 600 |
| 65 - 75 | 11 | 70 | 770 |
| 75 - 85 | 8 | 80 | 640 |
| 85 - 95 | 3 | 90 | 270 |
| Total | Σfᵢ = 35 | Σfᵢxᵢ = 2430 |
Step 3: Apply the formula for mean: Mean = Σfᵢxᵢ / Σfᵢ = 2430/35 = 69.43%
Therefore, the mean literacy rate of the 35 cities is approximately 69.43%.
Explanation
This question from Exercise 13.1 tests the student's understanding of finding the mean of grouped data using the direct method. The student needs to identify that class marks (midpoints) must be calculated first, then multiply each class mark by its frequency, sum all products, and divide by total frequency. The calculation is straightforward with evenly spaced class intervals of width 10.
Solution Steps
Step 1: Calculate class marks (xᵢ) as midpoints of each class interval
Step 2: Multiply each class mark by its frequency to get fᵢxᵢ values
Step 3: Sum all frequencies (Σfᵢ = 35) and all products (Σfᵢxᵢ = 2430)
Step 4: Apply mean formula: Mean = Σfᵢxᵢ/Σfᵢ = 2430/35 = 69.43%