Moments And Measures Of Skewness And Kurtosis.docx r5v2m

MOMENTS AND MEASURES OF SKEWNESS AND KURTOSIS Q1. The first moment of a distribution about the value 2 is 2. Find the mean of the distribution. Q2. (i) The first two moments of a distribution about the value 5 are 4 and 52. Find the mean and the variance. (ii) The first two moments of a distribution about the value 5 of the variable are 2 and 20. Find the mean and the variance. Q3. (i) The first three moments about the value 2 are 4, 65 and 325. Find its mean, S.D. and the moment measure of skewness. (ii) If the first, second and third moments of a distribution about 2 are 1, 16 and 40 respectively, then find first, second and third central moments and moment measure of skewness of the distribution. Q4. The mean of a certain distribution is 3 and the second moments about the mean is 10. Find the first and second moment about 5. Q5. The A.M of a certain distribution is 5. The second and third moments about the mean are 20 and 140 respectively. Find the third moment of the distribution about 10. Q6. (i) If the second and third central moments of a distribution be 4 and 12 respectively, find the skewness of the distribution. (ii) For a moderately skewed distribution, mean = 172, Median =167 and S.D. = 60. Find the co-efficient of skewness and mode. Q7. The karl-Pearson’s coefficient of skweness of a distribution is 0.32. Its standard deviation is 6.5 and mean 29.6. Find the mode. Q8. (i) Given coefficient of skewness = -0.475, Mean = 64 and Median = 66. Find the value of the standard deviation. (ii) Given Pearson’s co efficient of skewness = 0.3, Median = 42, Mode= 36; Find C.V. Q9. (I) If Q1= 26, Q3= 76, and coefficient of skewness = 0.2, find the median. (ii) In a frequency distribution, the coefficient of skewness based upon the quartiles is 0.25. If the sum of upper and lower quartile is 166 and the median is 77, find the value of the lower and upper quartiles. (iii) For a symmetrical distribution (where skewness = 0), Q1=36 and Q3= 63. Using formula of Bowley’s measure of skewness , find the median of the distribution. Q10. For a given distribution, CV = 30%, Variance = 36 and Pearson’s coefficient of skewness = -0.25. Find its median and mode. Q11. The first two moments of a distribution about the value 1 are 2 and 25. Find the mean and standard deviation of the distribution. Q12. (i) If the second and third central moments of a distribution be 4 and 12. Find the skewness of the distribution. (ii) For some symmetrical distribution, Q1= 24 and Q3 = 42, find median. (iii) If the 2nd and 4th central moments of a distribution be 5 and 75 respectively, find the kurtosis of the distribution. Q13. For a moderately skewed data A.M = 100, coefficient of variation = 35, Karl Pearson’s coefficient of skewness = 0.2, Find mode and median. Q14. (i) In a group of 10 observations ∑x = 452, ∑x2 = 24270 and mode = 43.7. Find the Pearson’s Coefficient of skewness. (ii) In a distribution Mean=65: Median = 70 and coefficient of skewness is -0.6. Find (a) Mode (b) Coefficient of variation.

Q15. The first four raw moments obtained from a frequency distribution are -4, 22, -105 and 444. Find the coefficients of skewness and Kurtosis. What is nature of the distribution? Q16. Find the first, second and third moments about 9 from the following data . values 2 4 6 8 10 12 Frequency 6 9 7 15 7 6 Hence, Find the first, second and third central moments and moment measure of skewness of the distribution. Q17. Calculate first three central moments from the following distribution. X : 5 7 10 12 17 f : 1 2 4 2 1 Q18. Find the first four central moments of the frequency distribution given below. Hence, find a measure of central tendency, dispersion, skewness and kurtosis. CI 110-119 120-129 130-139 140-149 150-159 160-169 170-179 f

16

25

38

52

37

19

13

Q19. (i) Find the first two moment about zero for the set of number 2,4,6,8 and hence find mean and S.D. (ii) The first two moments about 5 are 2 are 20. Find the mean and variance. (ii) If the first moments of distribution about the value 2 is 4, find the mean of the distribution. Q20. (i) If the second and third central moments of distribution and 4 and 10. Find the skewness of the distribution. (ii) If the 2nd, 3rd and 4th central moments of a distribution be 9, 12 and 125 respectively, Find coefficient of skewness and kurtosis. (iii) The first three moments of a distribution about the value 1 are 2, 5, and 80. Find its mean, S.D and the moments measure of Skewness.