1. **Problem Statement:** We have the ages of 20 teachers: 23, 20, 24, 26, 29, 39, 35, 37, 39, 40, 25, 24, 20, 28, 30, 34, 39, 45, 54, 58. We need to create a frequency distribution table with class intervals of 5 years, then find the mean and median age. 2. **Create Frequency Distribution Table:** - Class intervals of width 5 starting from 20 (minimum age) to 60 (covering maximum age 58): 20-24, 25-29, 30-34, 35-39, 40-44, 45-49, 50-54, 55-59 - Count the number of ages in each interval: - 20-24: 20, 20, 23, 24, 24 → 5 - 25-29: 25, 26, 28, 29 → 4 - 30-34: 30, 34 → 2 - 35-39: 35, 37, 39, 39, 39 → 5 - 40-44: 40 → 1 - 45-49: 45 → 1 - 50-54: 54 → 1 - 55-59: 58 → 1 3. **Frequency Table:** | Class Interval | Frequency (f) | |----------------|--------------| | 20 - 24 | 5 | | 25 - 29 | 4 | | 30 - 34 | 2 | | 35 - 39 | 5 | | 40 - 44 | 1 | | 45 - 49 | 1 | | 50 - 54 | 1 | | 55 - 59 | 1 | 4. **Calculate Midpoints (x) for each class:** - Midpoint = (Lower limit + Upper limit) / 2 - 20-24: 22, 25-29: 27, 30-34: 32, 35-39: 37, 40-44: 42, 45-49: 47, 50-54: 52, 55-59: 57 5. **Calculate Mean:** - Formula: $$\bar{x} = \frac{\sum f x}{\sum f}$$ - Calculate $f x$ for each class: - 5 \times 22 = 110 - 4 \times 27 = 108 - 2 \times 32 = 64 - 5 \times 37 = 185 - 1 \times 42 = 42 - 1 \times 47 = 47 - 1 \times 52 = 52 - 1 \times 57 = 57 - Sum of $f x = 110 + 108 + 64 + 185 + 42 + 47 + 52 + 57 = 665$ - Sum of frequencies $\sum f = 20$ - Mean: $$\bar{x} = \frac{665}{20} = 33.25$$ 6. **Calculate Median:** - Total frequency $N = 20$ - Median class is the class where cumulative frequency reaches $\frac{N}{2} = 10$ - Cumulative frequencies: - 20-24: 5 - 25-29: 5 + 4 = 9 - 30-34: 9 + 2 = 11 (median class) - Median class: 30-34 - Use median formula: $$\text{Median} = L + \left(\frac{\frac{N}{2} - F}{f_m}\right) \times h$$ where - $L = 29.5$ (lower boundary of median class 30-34) - $F = 9$ (cumulative frequency before median class) - $f_m = 2$ (frequency of median class) - $h = 5$ (class width) - Calculate median: $$\text{Median} = 29.5 + \left(\frac{10 - 9}{2}\right) \times 5 = 29.5 + \frac{1}{2} \times 5 = 29.5 + 2.5 = 32$$ **Final answers:** - Mean age = 33.25 - Median age = 32