Frequency Moments Skewness 8C22Da

1. Problem 1: Draw cumulative frequency curve, frequency polygon, and ogive for the given marks distribution. 2. Given data: Marks-group: 0–10, 10–20, 20–30, 30–40, 40–50, 50–60, 60–70 Frequencies: 4, 8, 11, 15, 12, 6, 3 3. Calculate cumulative frequencies (CF): CF: 4, 4+8=12, 12+11=23, 23+15=38, 38+12=50, 50+6=56, 56+3=59 4. Frequency polygon: plot midpoints vs frequencies. Midpoints: 5, 15, 25, 35, 45, 55, 65 Frequencies: 4, 8, 11, 15, 12, 6, 3 5. Ogive curves: plot upper class boundaries vs cumulative frequencies. Upper boundaries: 10, 20, 30, 40, 50, 60, 70 CF: 4, 12, 23, 38, 50, 56, 59 6. Plot cumulative frequency curve using CF and upper boundaries. --- 7. Problem 2: Calculate first four moments about the mean and find \beta_1 and \beta_2. 8. Given data: x: 0,1,2,3,4,5,6,7,8 f: 1,8,28,56,70,56,28,8,1 9. Calculate mean \bar{x}: $$\bar{x} = \frac{\sum f x}{\sum f} = \frac{(1\cdot0)+(8\cdot1)+\cdots+(1\cdot8)}{1+8+28+56+70+56+28+8+1} = \frac{4\cdot224}{256} = 4$$ 10. Calculate central moments \mu_r = \frac{\sum f (x - \bar{x})^r}{\sum f} for r=2,3,4. 11. Compute \mu_2 (variance): $$\mu_2 = \frac{\sum f (x-4)^2}{256} = \frac{448}{256} = 1.75$$ 12. Compute \mu_3: $$\mu_3 = \frac{\sum f (x-4)^3}{256} = 0$$ (symmetry) 13. Compute \mu_4: $$\mu_4 = \frac{\sum f (x-4)^4}{256} = 6.125$$ 14. Calculate \beta_1 (skewness squared): $$\beta_1 = \frac{\mu_3^2}{\mu_2^3} = \frac{0^2}{(1.75)^3} = 0$$ 15. Calculate \beta_2 (kurtosis): $$\beta_2 = \frac{\mu_4}{\mu_2^2} = \frac{6.125}{(1.75)^2} \approx 2.0$$ --- 16. Problem 3: Find Mean, Median, Mode for given series. 17. Given data: Size (below): 5,10,15,20,25,30,35 Frequency: 1,3,13,17,27,36,38 18. Calculate class intervals and frequencies: Classes: 0–5, 5–10, 10–15, 15–20, 20–25, 25–30, 30–35 Frequencies: 1, 3, 13, 17, 27, 36, 38 19. Calculate total frequency $N=1+3+13+17+27+36+38=135$ 20. Calculate midpoints: 2.5, 7.5, 12.5, 17.5, 22.5, 27.5, 32.5 21. Calculate Mean: $$\bar{x} = \frac{\sum f x}{N} = \frac{1\cdot2.5 + 3\cdot7.5 + \cdots + 38\cdot32.5}{135} = \frac{2700}{135} = 20$$ 22. Calculate Median: Median class is where cumulative frequency exceeds $\frac{N}{2} = 67.5$. Cumulative frequencies: 1,4,17,34,61,97,135 Median class: 25–30 23. Median formula: $$\text{Median} = l + \left(\frac{\frac{N}{2} - F}{f_m}\right) \times h$$ where $l=25$, $F=61$, $f_m=36$, $h=5$ 24. Substitute: $$= 25 + \left(\frac{67.5 - 61}{36}\right) \times 5 = 25 + 0.9 = 25.9$$ 25. Calculate Mode: Modal class is class with highest frequency: 30–35 (frequency 38) 26. Mode formula: $$\text{Mode} = l + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times h$$ where $l=30$, $f_1=38$, $f_0=36$, $f_2=0$, $h=5$ 27. Substitute: $$= 30 + \frac{(38 - 36)}{(2\times38 - 36 - 0)} \times 5 = 30 + \frac{2}{40} \times 5 = 30 + 0.25 = 30.25$$ --- 28. Problem 4: Calculate Karl Pearson’s coefficient of skewness. 29. Given data: Size: 1,2,3,4,5,6,7 Frequency: 10,18,30,25,12,3,2 30. Calculate mean: $$\bar{x} = \frac{\sum f x}{\sum f} = \frac{10\cdot1 + 18\cdot2 + \cdots + 2\cdot7}{100} = 3.22$$ 31. Calculate median: Total frequency $N=100$, median class is where cumulative frequency exceeds 50. Cumulative frequencies: 10,28,58,83,95,98,100 Median class: 3–4 32. Median formula: $$\text{Median} = l + \left(\frac{\frac{N}{2} - F}{f_m}\right) \times h$$ where $l=3$, $F=28$, $f_m=30$, $h=1$ 33. Substitute: $$= 3 + \left(\frac{50 - 28}{30}\right) \times 1 = 3 + 0.73 = 3.73$$ 34. Calculate mode: Modal class is 3–4 (frequency 30) 35. Mode formula: $$\text{Mode} = l + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times h$$ where $l=3$, $f_1=30$, $f_0=18$, $f_2=25$, $h=1$ 36. Substitute: $$= 3 + \frac{(30 - 18)}{(2\times30 - 18 - 25)} \times 1 = 3 + \frac{12}{17} = 3.71$$ 37. Calculate standard deviation $\sigma$: $$\sigma = \sqrt{\frac{\sum f (x - \bar{x})^2}{N}} = 1.34$$ 38. Calculate Karl Pearson’s coefficient of skewness: $$\text{Skewness} = \frac{\bar{x} - \text{Mode}}{\sigma} = \frac{3.22 - 3.71}{1.34} = -0.37$$ --- 39. Problem 5: Short note on Moments and Kurtosis. 40. Moments are quantitative measures related to shape of distribution. - First moment: mean - Second moment: variance - Third moment: skewness (asymmetry) - Fourth moment: kurtosis (peakedness) 41. Kurtosis measures tail heaviness and peak sharpness. - High kurtosis: heavy tails, sharp peak - Low kurtosis: light tails, flat peak 42. Moments help describe distribution shape beyond mean and variance. Final answers are provided with detailed steps for each problem.