1. **State the problem:** We are given class intervals and their corresponding frequencies. We need to create a frequency table and calculate the mean, median, and mode. 2. **Create the frequency table:** | Class Interval | Frequency (f) | Midpoint (x) | f \times x | |---|---|---|---| | 12-20 | 5 | 16 | 80 | | 21-29 | 6 | 25 | 150 | | 30-38 | 4 | 34 | 136 | | 39-47 | 12 | 43 | 516 | | 48-56 | 15 | 52 | 780 | | 57-65 | 8 | 61 | 488 | 3. **Calculate the mean:** The formula for mean of grouped data is: $$\text{Mean} = \frac{\sum f x}{\sum f}$$ Calculate sums: $$\sum f = 5 + 6 + 4 + 12 + 15 + 8 = 50$$ $$\sum f x = 80 + 150 + 136 + 516 + 780 + 488 = 2150$$ So, $$\text{Mean} = \frac{2150}{50} = 43$$ 4. **Calculate the median:** Median class is the class where cumulative frequency reaches or exceeds $\frac{N}{2} = \frac{50}{2} = 25$. Cumulative frequencies: | Class Interval | Frequency (f) | Cumulative Frequency (CF) | |---|---|---| | 12-20 | 5 | 5 | | 21-29 | 6 | 11 | | 30-38 | 4 | 15 | | 39-47 | 12 | 27 | | 48-56 | 15 | 42 | | 57-65 | 8 | 50 | Median class is 39-47. Median formula: $$\text{Median} = L + \left(\frac{\frac{N}{2} - F}{f_m}\right) \times h$$ Where: - $L = 39$ (lower boundary of median class) - $N = 50$ - $F = 15$ (cumulative frequency before median class) - $f_m = 12$ (frequency of median class) - $h = 9$ (class width) Calculate: $$\text{Median} = 39 + \left(\frac{25 - 15}{12}\right) \times 9 = 39 + \frac{10}{12} \times 9 = 39 + 7.5 = 46.5$$ 5. **Calculate the mode:** Mode class is the class with highest frequency, which is 48-56 with frequency 15. Mode formula: $$\text{Mode} = L + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times h$$ Where: - $L = 48$ (lower boundary of modal class) - $f_1 = 15$ (frequency of modal class) - $f_0 = 12$ (frequency before modal class) - $f_2 = 8$ (frequency after modal class) - $h = 9$ (class width) Calculate: $$\text{Mode} = 48 + \frac{15 - 12}{2 \times 15 - 12 - 8} \times 9 = 48 + \frac{3}{30 - 20} \times 9 = 48 + \frac{3}{10} \times 9 = 48 + 2.7 = 50.7$$ **Final answers:** - Mean = 43 - Median = 46.5 - Mode = 50.7