Age Band Students 0 - 10 4 10 - 20 5 20 - 30 15 30 - 40 4
To find the average age of the students, we need to calculate the mean of the grouped frequency distribution.
Calculating Average Age using Midpoints
First, find the midpoint for each age band. The midpoint is the average of the lower and upper bounds of the age band.
Next, multiply each midpoint by the number of students (frequency) in that band to get the sum of ages for that group.
Finally, divide the total sum of ages by the total number of students to find the average age.
Age Band Data Analysis
The provided data is organized as follows:
| Age Band | Students (Frequency) |
| 0 - 10 | 4 |
| 10 - 20 | 5 |
| 20 - 30 | 15 |
| 30 - 40 | 4 |
Step-by-Step Calculation
- Calculate Midpoints:
- 0 - 10: Midpoint = \(\frac{0 + 10}{2} = 5\)
- 10 - 20: Midpoint = \(\frac{10 + 20}{2} = 15\)
- 20 - 30: Midpoint = \(\frac{20 + 30}{2} = 25\)
- 30 - 40: Midpoint = \(\frac{30 + 40}{2} = 35\)
- Calculate Sum of (Midpoint \(\times\) Frequency):
- \((5 \times 4) = 20\)
- \((15 \times 5) = 75\)
- \((25 \times 15) = 375\)
- \((35 \times 4) = 140\)
- Total Sum = \(20 + 75 + 375 + 140 = 610\)
- Calculate Total Number of Students:
- Total Students = \(4 + 5 + 15 + 4 = 28\)
- Calculate Average Age:
- Average Age = \(\frac{\text{Total Sum of (Midpoint} \times \text{Frequency)}}{\text{Total Number of Students}}\)
- Average Age = \(\frac{610}{28}\)
- Average Age \(\approx 21.79\) years
Final Average Age Determination
The calculated average age is approximately 21.79 years. Comparing this value to the given options, the closest value is approximately 22 years.
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