1. **Problem Statement:** Find the mean, median, and mode of the grouped data representing scores of 100 students on a Math quiz. 2. **Given Data:** Class Intervals: 0-5, 6-10, 11-15, 16-20, 21-25, 26-30, 31-35 Frequencies: 3, 7, 15, 20, 18, 17, 13 3. **Formulas:** - Mean: $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$ where $f_i$ is frequency and $x_i$ is class midpoint. - Median: $$L + \left(\frac{\frac{N}{2} - F}{f_m}\right) \times h$$ where $L$ is lower boundary of median class, $N$ total frequency, $F$ cumulative frequency before median class, $f_m$ frequency of median class, $h$ class width. - Mode: $$L + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$$ where $L$ is lower boundary of modal class, $f_1$ frequency of modal class, $f_0$ frequency before modal class, $f_2$ frequency after modal class, $h$ class width. 4. **Calculate midpoints ($x_i$):** 0-5: 2.5, 6-10: 8, 11-15: 13, 16-20: 18, 21-25: 23, 26-30: 28, 31-35: 33 5. **Calculate mean:** $$\sum f_i x_i = 3\times2.5 + 7\times8 + 15\times13 + 20\times18 + 18\times23 + 17\times28 + 13\times33 = 7.5 + 56 + 195 + 360 + 414 + 476 + 429 = 1937.5$$ Total frequency $N = 3 + 7 + 15 + 20 + 18 + 17 + 13 = 93$ Mean $$= \frac{1937.5}{93} = 20.84$$ 6. **Calculate median:** $N/2 = 93/2 = 46.5$ Cumulative frequencies: 3, 10, 25, 45, 63, 80, 93 Median class is 21-25 (since cumulative frequency just before it is 45 and after is 63) $L=20.5$, $F=45$, $f_m=18$, $h=5$ $$\text{Median} = 20.5 + \left(\frac{46.5 - 45}{18}\right) \times 5 = 20.5 + \left(\frac{1.5}{18}\right) \times 5 = 20.5 + 0.42 = 20.92$$ 7. **Calculate mode:** Modal class is the class with highest frequency: 16-20 with frequency 20 $L=15.5$, $f_1=20$, $f_0=15$ (previous class frequency), $f_2=18$ (next class frequency), $h=5$ $$\text{Mode} = 15.5 + \left(\frac{20 - 15}{2\times20 - 15 - 18}\right) \times 5 = 15.5 + \left(\frac{5}{40 - 33}\right) \times 5 = 15.5 + \left(\frac{5}{7}\right) \times 5 = 15.5 + 3.57 = 19.07$$ **Final answers:** Mean = 20.84 Median = 20.92 Mode = 19.07