1. **State the problem:** We are given class intervals and their corresponding frequencies. We need to find the mean, median, and mode of the grouped data. 2. **Given data:** Class intervals: 12-20, 21-29, 30-38, 39-47, 48-56, 57-65 Frequencies: 5, 6, 4, 12, 15, 8 3. **Calculate class midpoints ($x_i$):** $$\text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2}$$ - 12-20: $\frac{12+20}{2} = 16$ - 21-29: $\frac{21+29}{2} = 25$ - 30-38: $\frac{30+38}{2} = 34$ - 39-47: $\frac{39+47}{2} = 43$ - 48-56: $\frac{48+56}{2} = 52$ - 57-65: $\frac{57+65}{2} = 61$ 4. **Calculate mean:** Formula for mean of grouped data: $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$ Calculate $f_i x_i$ for each class: - $5 \times 16 = 80$ - $6 \times 25 = 150$ - $4 \times 34 = 136$ - $12 \times 43 = 516$ - $15 \times 52 = 780$ - $8 \times 61 = 488$ Sum of frequencies $\sum f_i = 5+6+4+12+15+8 = 50$ Sum of $f_i x_i = 80 + 150 + 136 + 516 + 780 + 488 = 2150$ Mean: $$\bar{x} = \frac{2150}{50} = 43$$ 5. **Calculate median:** Median class is the class where cumulative frequency reaches $\frac{N}{2} = 25$. Cumulative frequencies: - 5 - 5+6=11 - 11+4=15 - 15+12=27 (median class: 39-47) - 27+15=42 - 42+8=50 Median class: 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: $47-39+1=9$) Calculate: $$\text{Median} = 39 + \left(\frac{25 - 15}{12}\right) \times 9 = 39 + \frac{10}{12} \times 9 = 39 + 7.5 = 46.5$$ 6. **Calculate mode:** Mode class is the class with highest frequency: 15 (48-56) Mode formula: $$\text{Mode} = L + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$$ 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) - $h=9$ Calculate: $$\text{Mode} = 48 + \left(\frac{15 - 12}{2 \times 15 - 12 - 8}\right) \times 9 = 48 + \left(\frac{3}{30 - 20}\right) \times 9 = 48 + \frac{3}{10} \times 9 = 48 + 2.7 = 50.7$$ **Final answers:** - Mean = 43 - Median = 46.5 - Mode = 50.7