1. Measures of Central Tendency

2. Definition Measures of Central Tendency (Mean, Median, Mode)

3. Central Tendency Refers to a characteristic where the frequency of a variable tends to cluster around the ‘center’

4. Measures of Central Tendency Arithmetic Mean Median Mode

5. Arithmetic Mean Data (units produced by workers) 10, 20, 30 Mean = Ungrouped data (1) =20

6. Arithmetic Mean Data (units produced by workers) 10, 20, 20, 25, 25, 25, 25, 30, 30, 50, 50, 50 Ungrouped data (2) Units (x)Worker(f) 101 202 254 302 503 Total 10 40 100 60 150 12360 fx

7. Arithmetic Mean Data (units produced by workers) 12, 24, 24, 25, 25, 25, 25, 32, 32, 45, 45, 45 Grouped data UnitsWorker(f) 10 – 201 20 – 306 30 – 402 40 – 503 Total Midpoint(m) 15 25 35 45 fm 15 150 70 135 12370

8. Arithmetic Mean Ungrouped data Grouped data

9. Features of Arithmetic Mean Commonly used Easily understood Greatly affected by extreme values

10. Median 1. Array 2. Median position 3. median

11. Median Data (units produced by workers) 20, 10, 30 (odd) Ungrouped data (1) ① Array 10, 20, 30 ② Median position ③ Median 20

12. Median Data (units produced by workers) 20, 10, 40, 30 (even) Ungrouped data (1) ① Array 10, 20, 30, 40 ② Median position ② Median

13. Median Data (units produced by workers) 10, 20, 20, 25, 25, 25, 25, 30, 30, 50, 50, 50 Median position= Ungrouped data (2) Units (x)Worker(f) 101 202 254 302 503 Total12 25 unitsMedian= c.f. 1 3 7 9 12

14. Median Data (units produced by workers) 12, 24, 24, 25, 25, 25, 25, 32, 32, 45, 45, 45 Median = Grouped data (2) UnitsWorker(f) 10 – 201 20 – 306 30 – 402 40 – 503 Total12 Median position = Median Class = 20-30 c.f. 1 7 9 12

15. Median Ungrouped data Grouped data

16. Features of Median Not affected by extreme values When data is skewed, the median is often a better indicator of “average” than the mean. Time consuming Unfamiliar to most people

17. Mode Data (units produced by workers) 10, 20, 20, 30 Mode = Ungrouped data (1) 20

18. Mode Data (units produced by workers) 10, 20, 20, 25, 25, 25, 25, 30, 30, 50, 50, 50 Ungrouped data (2) Units (x)Worker(f) 101 202 254 302 503 Total12 Mode = 25

19. Mode Data (units produced by workers) 12, 24, 24, 25, 25, 25, 25, 32, 32, 45, 45, 45 Mode = Grouped data (2) UnitsWorker(f) 10 – 201 20 – 306 30 – 402 40 – 503 Total12 The highest frequency: Modal group= 20-30 units

20. Mode Ungrouped data Grouped data Data with the highest frequency

21. Features of Mode Not affected by extreme values May be more than one mode, or no mode May not give a good indication of central values

22. Skewness of Data Distribution  Normal Mode = mean =median

23. Skewness of Data Distribution  Positively skewed Mode < median< mean

24. Skewness of Data Distribution  Negatively skewed Mean < median< mode

25. Arithmetic Mean ungrouped data grouped data

26. Median ungrouped data grouped data

27. Mode ungrouped data grouped data Data with the highest frequency

28. Measures of Dispersion

29. Definition Measures of Dispersion(Range, Quartile Deviation, Mean Deviation, Standard Deviation, Variance, Coefficient of Variation)

30. Dispersion It describes the level of variation and also indicates the level of consistency in the distribution.

31. Measures of Dispersion Range Quartile Deviation Mean Deviation Standard Deviation Variance Coefficient of Variation

32. Range It measures the difference between the highest and the lowest piece of data. Data1: Data2: 10, 20, 30 0, 20, 40 Range1 = x max – x min = 30 - 10 = 20 Range2 = x max – x min = 40 - 0 = 40

33. Feature It is easy to calculate and easy to understand. It is distorted by extreme values.

34. Quartile Deviation 1. Array 2. Quartile position 3. Quartile Value 4. IQR,QD

35. Quartile Deviation It excludes the first and last quarters of information and in doing so concentrates on the main core of data, ignoring extreme values. 45 46 50 55 60 65 67 69 69 70 71 72 73 74 76 78 78 79 80 82 83 85 90 95 Q1Q2Q3 Interquartile Range = Q 3 - Q1 Quartile Deviation =

36. Quartile Deviation (ungrouped) Q 1 position= Q 3 position= Q 1 value= Q 3 value=

37. Grouped data

38. Amount Spent ($)Number of Staff 0-10 2 10-20 3 20-30 4 30-40 3 40-50 1 Total13 c. f. 2 5 9 12 13

39. Amount Spent ($)Number of Staff 0-10 2 10-20 3 20-30 4 30-40 3 40-50 1 Total13 c. f. 2 5 9 12 13

40. Amount Spent ($)Number of Staff 0-10 2 10-20 3 20-30 4 30-40 3 40-50 1 Total13 c. f. 2 5 9 12 13

41. Amount Spent ($)Number of Staff 0-10 2 10-20 3 20-30 4 30-40 3 40-50 1 Total13 c. f. 2 5 9 12 13

42. 2 5 9 12 13 Amount Spent ($)Number of Staff 0-10 2 10-20 3 20-30 4 30-40 3 40-50 1 Total13 c. f.

43. Feature Not effected by extreme values. Not widely used or understood.

44. Quartile Deviation Q 1 = Q 3 = Ungrouped: I.Q.R= Q 3 value- Q 1 value Quartile Deviation =

45. Quartile Deviation Q 1 = Q 3 = Grouped: I.Q.R= Q 3 value- Q 1 value Quartile Deviation =

46. Mean Deviation The absolute distance of each score away from the mean.

47. Mean Deviation Ungrouped data

48. Mean Deviation Ungrouped data Team 1: 20 22 23 25 25 26 26 26 28 29 Team 2: 12 14 18 24 28 30 30 30 31 33

49. Mean Deviation Ungrouped data Team 1: 20 22 23 25 25 26 26 26 28 29 Team 2: 12 14 18 24 28 30 30 30 31 33

50. Mean Deviation Ungrouped data Team 1: 20 22 23 25 25 26 26 26 28 29 Team 2: 12 14 18 24 28 30 30 30 31 33

51. Mean Deviation Ungrouped data Team 1: 20 22 23 25 25 26 26 26 28 29 Team 2: 12 14 18 24 28 30 30 30 31 33 M.D. 1 = 2 M.D. 2 =6.4

52. Mean Deviation Grouped data

53. UnitsMidpoint(m)Worker(f)fmf|m – | 20-305 30-4010 40-5020 50-6015 Total 25 35 45 55 125 350 900 825 44 95 90 20 165 502,200 370

54. Mean Deviation Grouped data Ungrouped data

55. Standard Deviation/Variance Ungrouped data

56. Standard Deviation/Variance Ungrouped data Team 1: 20 22 23 25 25 26 26 26 28 29 Team 2: 12 14 18 24 28 30 30 30 31 33

57. Standard Deviation/Variance Ungrouped data Team 1: 20 22 23 25 25 26 26 26 28 29 Team 2: 12 14 18 24 28 30 30 30 31 33

58. Standard Deviation/Variance Ungrouped data Team 1: 20 22 23 25 25 26 26 26 28 29 Team 2: 12 14 18 24 28 30 30 30 31 33

59. Standard Deviation/Variance Ungrouped data Team 1: 20 22 23 25 25 26 26 26 28 29 Team 2: 12 14 18 24 28 30 30 30 31 33

60. Standard Deviation (Variance) Grouped data

61. Units (x) Worker (f) 20-305 30-4010 40-5020 50-6015 Total 25 35 45 55 Midpoint (m) fm 125 350 900 825 44 1,805 810 20 1,815 502,200 4,450

62. UnitsMidpoint (m) Worker (f) fmf(m – ) 2 20-30255125441,805 30-40351035044810 40-5045209004420 50-605515825441,815 Total-502,2004,450

63. Standard Deviation/Variance Ungrouped data

64. Standard Deviation (Variance) Grouped data

65. Coefficient of Variation

66. Coefficient of Variation (100 Students) Height: Weight: Height C.V.: Weight C.V.: Weight is more variant than Height.

67. Population & sample