1. **Problem Statement:** Calculate the mean, mean deviation, and standard deviation for the marks of 30 students grouped in class intervals 61–70, 71–80, etc. 2. **Step 1: Organize data into frequency distribution with class intervals:** Class intervals: 61–70, 71–80, 81–90, 91–100 Count frequencies: 61–70: 7 (61,61,63,64,67,67,69,70,70,70) actually 10 values 71–80: 7 (72,73,73,74,75,76,76,78,78) actually 9 values 81–90: 4 (81,84,86,86,88,90) actually 6 values 91–100: 3 (91,91,98) actually 3 values Recount carefully: 61–70: 10 71–80: 9 81–90: 6 91–100: 3 3. **Step 2: Calculate midpoints ($x_i$) of each class:** $$x_1=\frac{61+70}{2}=65.5$$ $$x_2=\frac{71+80}{2}=75.5$$ $$x_3=\frac{81+90}{2}=85.5$$ $$x_4=\frac{91+100}{2}=95.5$$ 4. **Step 3: Calculate mean ($\bar{x}$):** Formula: $$\bar{x}=\frac{\sum f_i x_i}{\sum f_i}$$ Calculate numerator: $$\sum f_i x_i = 10\times65.5 + 9\times75.5 + 6\times85.5 + 3\times95.5 = 655 + 679.5 + 513 + 286.5 = 2134$$ Total frequency: $$\sum f_i = 10 + 9 + 6 + 3 = 28$$ Mean: $$\bar{x} = \frac{2134}{28} = 76.2143 \approx 76.2$$ 5. **Step 4: Calculate mean deviation (MD):** Formula: $$MD = \frac{\sum f_i |x_i - \bar{x}|}{\sum f_i}$$ Calculate $|x_i - \bar{x}|$: $$|65.5 - 76.2| = 10.7$$ $$|75.5 - 76.2| = 0.7$$ $$|85.5 - 76.2| = 9.3$$ $$|95.5 - 76.2| = 19.3$$ Calculate numerator: $$10 \times 10.7 + 9 \times 0.7 + 6 \times 9.3 + 3 \times 19.3 = 107 + 6.3 + 55.8 + 57.9 = 227$$ Mean deviation: $$MD = \frac{227}{28} = 8.107 \approx 8.1$$ 6. **Step 5: Calculate standard deviation (SD):** Formula: $$SD = \sqrt{\frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i}}$$ Calculate $(x_i - \bar{x})^2$: $$10.7^2 = 114.49$$ $$0.7^2 = 0.49$$ $$9.3^2 = 86.49$$ $$19.3^2 = 372.49$$ Calculate numerator: $$10 \times 114.49 + 9 \times 0.49 + 6 \times 86.49 + 3 \times 372.49 = 1144.9 + 4.41 + 518.94 + 1117.47 = 2785.72$$ Standard deviation: $$SD = \sqrt{\frac{2785.72}{28}} = \sqrt{99.49} = 9.974 \approx 10.0$$ **Final answers:** - Mean = 76.2 - Mean Deviation = 8.1 - Standard Deviation = 10.0