1. **Problem Statement:** Given the data set of student marks: 2, 4, 4, 8, 4, 7, 1, 8, 3, 9, find the Mean, Median, Q1, Q3, Mode, Range, Mean Deviation, Quartile Deviation, Variance, Standard Deviation, and the first four Moments about the mean. 2. **Step 1: Organize the data in ascending order:** $$1, 2, 3, 4, 4, 4, 7, 8, 8, 9$$ 3. **Step 2: Calculate the Mean (average):** Formula: $$\text{Mean} = \frac{\sum x_i}{n}$$ Sum of data: $$1+2+3+4+4+4+7+8+8+9=50$$ Number of data points: $$n=10$$ Mean: $$\frac{50}{10} = 5$$ 4. **Step 3: Calculate the Median (middle value):** Since $$n=10$$ (even), median is average of 5th and 6th values: 5th value = 4, 6th value = 4 Median: $$\frac{4+4}{2} = 4$$ 5. **Step 4: Calculate Q1 (first quartile) and Q3 (third quartile):** Q1 is median of first half (first 5 values): $$1, 2, 3, 4, 4$$ Median of these is 3 (3rd value) Q3 is median of second half (last 5 values): $$4, 7, 8, 8, 9$$ Median of these is 8 (3rd value) 6. **Step 5: Calculate Mode (most frequent value):** Value 4 appears 3 times, more than any other Mode = 4 7. **Step 6: Calculate Range:** Range = Max - Min = $$9 - 1 = 8$$ 8. **Step 7: Calculate Mean Deviation:** Formula: $$\text{Mean Deviation} = \frac{\sum |x_i - \bar{x}|}{n}$$ Calculate absolute deviations: $$|1-5|=4, |2-5|=3, |3-5|=2, |4-5|=1, |4-5|=1, |4-5|=1, |7-5|=2, |8-5|=3, |8-5|=3, |9-5|=4$$ Sum: $$4+3+2+1+1+1+2+3+3+4=24$$ Mean Deviation: $$\frac{24}{10} = 2.4$$ 9. **Step 8: Calculate Quartile Deviation:** Formula: $$\text{Quartile Deviation} = \frac{Q3 - Q1}{2}$$ $$\frac{8 - 3}{2} = \frac{5}{2} = 2.5$$ 10. **Step 9: Calculate Variance:** Formula: $$\text{Variance} = \frac{\sum (x_i - \bar{x})^2}{n}$$ Calculate squared deviations: $$(1-5)^2=16, (2-5)^2=9, (3-5)^2=4, (4-5)^2=1, (4-5)^2=1, (4-5)^2=1, (7-5)^2=4, (8-5)^2=9, (8-5)^2=9, (9-5)^2=16$$ Sum: $$16+9+4+1+1+1+4+9+9+16=70$$ Variance: $$\frac{70}{10} = 7$$ 11. **Step 10: Calculate Standard Deviation:** Formula: $$\text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{7} \approx 2.6458$$ 12. **Step 11: Calculate the first four Moments about the mean:** Moment of order $$k$$ about mean is $$\mu_k = \frac{1}{n} \sum (x_i - \bar{x})^k$$ - First moment about mean ($$k=1$$): $$\mu_1 = \frac{1}{10} \sum (x_i - 5) = 0$$ (by definition of mean) - Second moment about mean ($$k=2$$): $$\mu_2 = \text{Variance} = 7$$ - Third moment about mean ($$k=3$$): Calculate $$\sum (x_i - 5)^3$$: $$(1-5)^3 = (-4)^3 = -64$$ $$(2-5)^3 = (-3)^3 = -27$$ $$(3-5)^3 = (-2)^3 = -8$$ $$(4-5)^3 = (-1)^3 = -1$$ (three times) sum = -3 $$(7-5)^3 = 2^3 = 8$$ $$(8-5)^3 = 3^3 = 27$$ (two times) sum = 54 $$(9-5)^3 = 4^3 = 64$$ Sum: $$-64 -27 -8 -3 + 8 + 54 + 64 = 24$$ Third moment: $$\mu_3 = \frac{24}{10} = 2.4$$ - Fourth moment about mean ($$k=4$$): Calculate $$\sum (x_i - 5)^4$$: $$(1-5)^4 = 256$$ $$(2-5)^4 = 81$$ $$(3-5)^4 = 16$$ $$(4-5)^4 = 1$$ (three times) sum = 3 $$(7-5)^4 = 16$$ $$(8-5)^4 = 81$$ (two times) sum = 162 $$(9-5)^4 = 256$$ Sum: $$256 + 81 + 16 + 3 + 16 + 162 + 256 = 790$$ Fourth moment: $$\mu_4 = \frac{790}{10} = 79$$ **Final answers:** - Mean = 5 - Median = 4 - Q1 = 3 - Q3 = 8 - Mode = 4 - Range = 8 - Mean Deviation = 2.4 - Quartile Deviation = 2.5 - Variance = 7 - Standard Deviation = 2.6458 - First moment about mean = 0 - Second moment about mean = 7 - Third moment about mean = 2.4 - Fourth moment about mean = 79