1. **Problem Statement:** We have grouped data for weekly study hours of 50 students with frequency distribution. We will find the Mean, Median, Mode, Mean Deviation, Variance, and Standard Deviation. 2. **Step 1: Tabulate midpoints and frequency (f).** Calculate midpoints $x$ for each class interval: - For 21–25: $x=\frac{21+25}{2}=23$ - For 16–20: $x=\frac{16+20}{2}=18$ - For 11–15: $x=\frac{11+15}{2}=13$ - For 6–10: $x=\frac{6+10}{2}=8$ - For 1–5: $x=\frac{1+5}{2}=3$ 3. **Step 2: Compute $f x$ (frequency times midpoint) for each class.** - $5 \times 23 = 115$ - $10 \times 18 = 180$ - $18 \times 13 = 234$ - $11 \times 8 = 88$ - $6 \times 3 = 18$ Sum of $f x = 115 + 180 + 234 + 88 + 18 = 635$ 4. **Step 3: Calculate the Mean ($\bar{x}$).** $$\bar{x} = \frac{\sum f x}{\sum f} = \frac{635}{50} = 12.7$$ 5. **Step 4: Find the Median class.** Total students = 50, half = 25. Cumulative frequencies: - 1–5: 6 - 6–10: 6 + 11 = 17 - 11–15: 17 + 18 = 35 (median class since 35 > 25) Median class: 11–15 with $f_m=18$, lower boundary $L=10.5$, cumulative frequency before median class $F=17$, and class width $h=5$. 6. **Step 5: Compute Median:** $$\text{Median} = L + \left(\frac{\frac{N}{2} - F}{f_m}\right) \times h = 10.5 + \left(\frac{25 - 17}{18}\right) \times 5 = 10.5 + \frac{8}{18} \times 5 = 10.5 + 2.22 = 12.72$$ 7. **Step 6: Find the Mode class.** The class with highest frequency is 11–15 ($f=18$), so modal class is 11–15. Calculate: - $f_1 = 10$ (frequency before modal class) - $f_2 = 18$ (modal class frequency) - $f_3 = 11$ (frequency after modal class) - $L = 10.5$, $h=5$ Mode formula: $$\text{Mode} = L + \frac{f_2 - f_1}{2f_2 - f_1 - f_3} \times h = 10.5 + \frac{18 - 10}{2 \times 18 - 10 - 11} \times 5 = 10.5 + \frac{8}{15} \times 5 = 10.5 + 2.67 = 13.17$$ 8. **Step 7: Calculate $|x - \bar{x}|$ and $f |x - \bar{x}|$ for Mean Deviation:** - For 23: $|23 - 12.7|=10.3$, $5 \times 10.3=51.5$ - For 18: $|18 - 12.7|=5.3$, $10 \times 5.3=53$ - For 13: $|13 - 12.7|=0.3$, $18 \times 0.3=5.4$ - For 8: $|8 - 12.7|=4.7$, $11 \times 4.7=51.7$ - For 3: $|3 - 12.7|=9.7$, $6 \times 9.7=58.2$ Sum $f |x - \bar{x}| = 51.5 + 53 + 5.4 + 51.7 + 58.2 = 219.8$ Mean Deviation: $$MD = \frac{\sum f |x - \bar{x}|}{\sum f} = \frac{219.8}{50} = 4.40$$ 9. **Step 8: Calculate Variance and Standard Deviation:** Calculate $(x - \bar{x})^2$ and $f (x - \bar{x})^2$: - For 23: $(10.3)^2=106.09$, $5 \times 106.09 = 530.45$ - For 18: $(5.3)^2=28.09$, $10 \times 28.09 = 280.9$ - For 13: $(0.3)^2=0.09$, $18 \times 0.09 = 1.62$ - For 8: $(4.7)^2=22.09$, $11 \times 22.09 = 243.0$ - For 3: $(9.7)^2=94.09$, $6 \times 94.09 = 564.54$ Sum $f (x - \bar{x})^2 = 530.45 + 280.9 + 1.62 + 243.0 + 564.54 = 1620.51$ Variance: $$\sigma^2 = \frac{1620.51}{50} = 32.41$$ Standard Deviation: $$\sigma = \sqrt{32.41} = 5.69$$ 10. **Final Answers:** - Mean = 12.7 hours - Median = 12.72 hours - Mode = 13.17 hours - Mean Deviation = 4.40 hours - Variance = 32.41 hours² - Standard Deviation = 5.69 hours