1. Measures of Variability OBJECTIVES To understand the different measures of variability To determine the range, variance, quartile deviation, mean deviation and standard deviation for ungrouped and grouped data

2. Measures of dispersion (variability or spread) consider the extent to which the observations vary

3. MEASURES OF VARIATION RANGE QUARTILE DEVIATION MEAN DEVIATION VARIANCE STANDARD DEVIATION

4. 1. Range, R The difference in value between the highest-valued data, H, and the lowest- valued data, L R = H – L Example: 3, 3, 5, 6, 8 R = H – L = 8 – 3 = 5

5. 2. Quartile Deviation, QD or semi-interquartile range obtained by getting one half the difference between the third and the first quartiles

6. SOLVE FOR Q 1 and Q 3

7. Problem The examination scores of 50 students in a statistics class resulted to the following values: Q 3 = 75.43 Q 1 = 54.24 Determine the value of the quartile deviation or semi-interquartile range.

8. Solution

9. Problem Compute the value of the semi- inter quartile range or quartile deviation The performance ratings of 100 faculty members of a certain college are presented in a frequency distribution as follows:

10. Class interval or Classes f <cf 71-74 3 3 75-78 10 13 79-82 13 26 1 st quartile class 83-86 18 44 87-90 25 69 91-94 19 88 3rd quartile class 95-98 12 100

11. Solution (Grouped data)

13. Solution cont’d…

14. 3. Mean Deviation, MD – based on all items in a distribution

15. For ungrouped data For grouped data

16. 4. Variance, s 2 - most commonly used measure of variability - the square of standard deviation

17. For ungrouped data

18. Note: The greater the variability of the observations in a data set, the greater the variance. If there is no variability of the observations, that is, if all are equal and hence, all are equal to the mean then s 2 = 0

19. For grouped data

20. 5. Standard Deviation, s - the positive square root of the variance

21. Problem: Find the (a) range, (b) quartile deviation, © mean deviation, (d) variance and (e) standard deviation Student Score 150 248 lowest value 372 467 571 6 65 773 highest value 862 964 1060

22. (a) Range, R R = H – L R = 73 – 48 = 25

23. (b) Quartile Deviation, QD Arrangement in ascending order 48 50 60 62 64 65 67 71 72 73 Using method 3 for finding Q n (ungrouped data) Q 1 is located at n/4 = 10/4 = 2.5 Q 1 = (50+60)/2 = 55 Q 3 is located at 3n/4 = 3(10)/4 = 7.5 Q 3 = (67+71)/2 =69

24. QD cont’d…

25. © Mean Deviation, MD

26. First, solve for the mean Ungrouped data

27. Data for mean deviation, MD Score, xx i = x- x i 2 73 9.896.04 72 8.877.44 71 7.860.84 67 3.814.44 65 1.8 3.24 64 0.8 0.64 62 -1.2 1.44 60 -3.210.24 50-13.274.24 48-15.2 231.04 TOTAL  x i  = 65.6 669.60

28. (d) Variance, s 2

29. (e) Standard Deviation, s

30. Problem: The following are marks obtained by a group of 40 students on an English examination Classesf<cf 95-99240 90-94238 85f-89436 80-84632 75-79526 70-74421 65-69517 60-64212 55-59210 50-5448 45-4914 40-4423 35-3911

31. Find the following: a. range b. quartile deviation c. mean deviation d. variance e. standard deviation

32. Solution a. Range, R = H – L = 99 – 35 = 64

33. b. Quartile Deviation, QD

34. Solve for Q 1

35. Substitute

36. c. Mean Deviation, MD

37. Data for mean deviation, MD Class intervalxffx|x i |f|x i | 95-999721942652 90-949221842142 85-898743481664 80-848264921166 75-79775385630 70-7472428814 65-69675335420 60-64622124918 55-595721141428 50-545242081976 45-49471 24 40-44422842958 35-39371 34 Total402840516

38. d. Variance, s 2

39. Data for the variance, s 2 Class intervalxffxxixi fx i 2 95-99972194261352 90-9492218421882 85-89874348161024 80-8482649211726 75-797753856180 70-7472428814 65-69675335-480 60-64622124-9162 55-59572114-14392 50-54524208-191444 45-49471 -24576 40-4442284-291682 35-39371 -341156 Total409660

40. e. Standard Deviation, s

41. New Topic…

42. Objectives To know the measures of skewness and kurtosis To find the Pearsonian coefficient of skewness

43. Measures of Skewness summarize the extent to which the observations are symmetrically distributed

44. Skewness the degree to which a distribution departs from symmetry about its mean value or refers to asymmetry (or "tapering") in the distribution of sample data

45. Positive skew the right tail is longer the mass of the distribution is concentrated on the left of the figure has a few relatively high values the distribution is said to be right-skewed mean > median > mode the skewness is greater than zero

47. Negative skew the left tail is longer the mass of the distribution is concentrated on the right of the figure has a few relatively low values the distribution is said to be left-skewed mean < median < mode the skewness is lower than zero

49. No skew the distribution is symmetric like the bell-shaped normal curve bell-shaped normal curve mean = median = mode

52. OR…

53. Exercise

54. Pearsonian coefficient of skewness

55. Skewness based on quartiles

56. Interpretation If skewness is positive, the data are positively skewed or skewed right, meaning that the right tail of the distribution is longer than the left. If skewness is negative, the data are negatively skewed or skewed left, meaning that the left tail is longer.

57. Interpretation cont’d… If skewness = 0, the data are perfectly symmetrical. But a skewness of exactly zero is quite unlikely for real-world data, so how can you interpret the skewness number? In the classic Principles of Statistics (1965), M.G. Bulmer suggests this rule of thumb:

58. Interpretation cont’d… If skewness is less than −1 or greater than +1, the distribution is highly skewed. If skewness is between −1 and −½ or between +½ and +1, the distribution is moderately skewed.

59. Interpretation cont’d… If skewness is between −½ and +½, the distribution is approximately symmetric. Example: With a skewness of −0.1082, the sample data are approximately symmetric.

60. Problem Find the Pearsonian coefficient of skewness of the set of data shown in the following table:

61. Scores of ten students in a mathematics ability test Student Score 150 248 372 467 571 6 65 773 862 964 1060

62. Computed values Refer to the previous computations

63. Interpretation Negative sign means the tail extends to the left the mean is less than the mode by 176% considered a substantial departure from symmetry

64. Problem Find the Pearsonian coefficient of skewness for the following set of data:

65. Class intervalxffx|x i |f|x i | 95-999721942652 90-949221842142 85-898743481664 80-848264921166 75-79775385630 70-7472428814 65-69675335420 60-64622124918 55-595721141428 50-545242081976 45-49471 24 40-44422842958 35-39371 34 Total402840516

67. Interpretation Negative (-) computed value means the mean is less than the mode by 76.1% considered quite negligible departure from symmetry given set of data is more or less evenly distributed

68. Problem Find the Pearsonian coefficient of skewness for the distribution whose

69. Solution

70. Interpretation Positive sign indicates the tail of the distribution extends to the right Computed value means the mean is greater than the mode by 38% considered negligible skewness

71. Measures of Kurtosis Kurtosis - the degree of peakedness (or flatness) of a distribution

72. Types of Kurtosis Mesokurtic distribution a normal distribution, neither too peaked nor too flat its kurtosis (Ku) is equal to 3

73. Leptokurtic distribution has a higher peak than the normal distribution with narrow humps and heavier tails its kurtosis (Ku) is higher than 3

74. Platykurtic distribution has a lower peak than a normal distribution flat distributions with values evenly distributed about the center with broad humps and short tails its kurtosis (Ku) is less than 3