STATISTICSStatistics is the branch of mathematics that involves the collection, analysis, interpretation, presentation, and organization of data. It is used to understand and draw conclusions from data by using mathematical theories and methodologies.

STATSTICS LECTURE Statistics is the branch of mathematics that involves the collection, analysis, interpretation, presentation, and organization of data. It is used to understand and draw conclusions from data by using mathematical theories and methodologies.

1. Lecture 3 Measures of Central Tendency and Measures of Dispersion Benjamin Alvarez Dillena Jr., Ed.D.

2. Measures of Central Tendency  

3. Measures of Central Tendency   Advantage Used for both continuous and discrete numeric data Disadvantage Cannot be used for qualitative data Influence by outliers

4. Measures of Central Tendency    

5. Measures of Central Tendency   Advantage Less affected by outliers Disadvantage Cannot be identified for categorical nominal data  

6. Measures of Central Tendency Mode (Mo) Similar to the Statistical mean and median, Mode is a way of representing important information about random variables or populations in a single number. In a normal Distribution, the value of Mode or modal value is the same as the mean and median whereas the value of Mode in a highly skewed Distribution may be very different. Mode is the most useful measure of central tendency while observing the categorical data such as the most preferred flavors of soda or Models of bikes for which average median values based on Order cannot be calculated.

7. Measures of Central Tendency Mode (Mo) Types of Mode The different types of Mode are Unimodal, Bimodal, Trimodal, and Multimodal. Unimodal Mode - A set of data with one Mode is known as a Unimodal Mode. For example , the Mode of data set A = { 14, 15, 16, 17, 15, 18, 15, 19} is 15 as there is only one value repeating itself. Hence, it is a Unimodal data set.

8. Measures of Central Tendency Mode (Mo) Types of Mode The different types of Mode are Unimodal, Bimodal, Trimodal, and Multimodal. Bimodal Mode - A set of data with two Modes is known as a Bimodal Mode. This means that there are two data values that are having the highest frequencies. For example , the Mode of data set A = { 8,13,13,14,15,17,17,19} is 13 and 17 because both 13 and 17 are repeating twice in the given set. Hence, it is a Bimodal data set.

9. Measures of Central Tendency Mode (Mo) Types of Mode The different types of Mode are Unimodal, Bimodal, Trimodal, and Multimodal. Trimodal Mode - A set of data with three Modes is known as a Trimodal Mode. This means that there are three data values that are having the highest frequencies. For example , the Mode of data set A = {2, 2, 2, 3, 4, 4, 5, 6, 5,4, 7, 5, 8} is 2 , 4 , and 5 because all the three values are repeating thrice in the given set. Hence, it is a Trimodal data set. Multimodal Mode - A set of data with four or more than four Modes is known as a Multimodal Mode. For example, The Mode of data set A = {100, 80, 80, 95, 95, 100, 90, 90,100 ,95 } is 80, 90, 95, and 100 because both all the four values are repeated twice in the given set. Hence, it is a Multimodal data set.

10. Measures of Central Tendency Mode (Mo) Types of Mode The different types of Mode are Unimodal, Bimodal, Trimodal, and Multimodal. Multimodal Mode - A set of data with four or more than four Modes is known as a Multimodal Mode. For example , The Mode of data set A = {100, 80, 80, 95, 95, 100, 90, 90} is 80 , 90 , 95 , and 100 because both all the four values are repeated twice in the given set. Hence, it is a Multimodal data set.

11. Measures of Central Tendency     Advantage Available for both numerical and categorical data Disadvantage May not effect the center of distribution very well

12. Summary of when to Use the Mean, Median and Mode Type of Variable Best Measure of Central Tendency Nominal Mode Ordinal Median Interval/Ratio (not skewed) Mean Interval/Ratio (skewed) Median Mean < Median Negative (Left-skewness) Mean = Median Symmetry (Zero-skewness) Mean > Median Positive (Right-skewness)

13. Fractiles/ Other Measures of Central Tendency  

14. Fractiles/ Other Measures of Central Tendency  

15. Fractiles/ Other Measures of Central Tendency  

16. Measures of Variation / Measures of Dispersion A measure of variation shows the extent to which numerical values tend to spread out over the average. A suitable measure should be large when the value vary over a wide range and should be small when the range of variation is not too great. A measure of dispersion is a method of measuring the degree by which numerical data or values tend to spread from or cluster about a central point of average.

17. Measures of Variation / Measures of Dispersion The Range The Mean Absolute Deviation The Percentile Deviation The Decile Deviation The Quartile Deviation The Inter-quartile Range The Variance The Standard Deviation

18. The Range      

19. The Mean Absolute Deviation (MAD)    

20. The Quantile Deviation Another way of measuring the variability of an observation is through quantile deviation, these are: Percentile Deviation (PD) Decile Deviation (DD) Quartile Deviation (QD) Inter-quartile Range (IR)

21. The Percentile Deviation (PD)    

22. The Decile Deviation (DD)    

23. The Quartile Deviation (QD)    

24.     The Inter-quartile Range (IR)

25.       Variance

26. Standard Deviation  

27. Measures of Central Tendency  

28. Measures of Central Tendency Example No. 1 Find the measures of central tendency (Mean, Median, and Mode) and the other measures of central tendency (P 80 , Q 3 , and D 3 ) of the following data: 67, 90, 77, 33, 30, 33, 65, 66, 80, & 70. N 1 2 3 4 5 6 7 8 9 10 Σx 611 Solution for Mean:       30 33 33 65 66 67 70 77 80 90

29. N 1 2 3 4 5 6 7 8 9 10 Σx 611 30 33 33 65 66 67 70 77 80 90 Solution for Median:     Since N = 10, the median is between 5 th and 6 th scores.   Measures of Central Tendency Example No. 1 Find the measures of central tendency (Mean, Median, and Mode) and the other measures of central tendency (P 80 , Q 3 , and D 3 ) of the following data: 67, 90, 77, 33, 30, 33, 65, 66, 80, & 70.

30. N 1 2 3 4 5 6 7 8 9 10 Σx 611 30 33 33 65 66 67 70 77 80 90 Solution for Mode:   Observed the given data: If mode is 1 = unimodal; 2 = bimodal; 3 = Trimodal; & 4 & ↑ = multimodal Measures of Central Tendency Example No. 1 Find the measures of central tendency (Mean, Median, and Mode) and the other measures of central tendency (P 80 , Q 3 , and D 3 ) of the following data: 67, 90, 77, 33, 30, 33, 65, 66, 80, & 70.

31. N 1 2 3 4 5 6 7 8 9 10 Σx 611 30 33 33 65 66 67 70 77 80 90 Solution for Percentile 80:         Measures of Central Tendency Example No. 1 Find the measures of central tendency (Mean, Median, and Mode) and the other measures of central tendency (P 80 , Q 3 , and D 3 ) of the following data: 67, 90, 77, 33, 30, 33, 65, 66, 80, & 70.

32. N 1 2 3 4 5 6 7 8 9 10 Σx 611 30 33 33 65 66 67 70 77 80 90 Solution for Quartile 3:         Measures of Central Tendency Example No. 1 Find the measures of central tendency (Mean, Median, and Mode) and the other measures of central tendency (P 80 , Q 3 , and D 3 ) of the following data: 67, 90, 77, 33, 30, 33, 65, 66, 80, & 70.

33. N 1 2 3 4 5 6 7 8 9 10 Σx 611 30 33 33 65 66 67 70 77 80 90 Solution for Decile 3:         Measures of Central Tendency Example No. 1 Find the measures of central tendency (Mean, Median, and Mode) and the other measures of central tendency (P 80 , Q 3 , and D 3 ) of the following data: 67, 90, 77, 33, 30, 33, 65, 66, 80, & 70.

34. Measures of Central Tendency Example No. 2 Find the measures of central tendency (Mean, Median, and Mode) and the other measures of central tendency (P 60 , Q 1 , and D 9 ) of the following data: 16, 24, 11, 30, 30, 27, 18, 32, 25. N 1 2 3 4 5 6 7 8 9 Σx 213 11 16 18 24 25 27 30 30 32 Solution for Mean:      

35. Measures of Central Tendency Example No. 2 Find the measures of central tendency (Mean, Median, and Mode) and the other measures of central tendency (P 60 , Q 1 , and D 9 ) of the following data: 16, 24, 11, 30, 30, 27, 18, 32, 25. N 1 2 3 4 5 6 7 8 9 Σx 213 11 16 18 24 25 27 30 30 32 Solution for Median:    

36. Measures of Central Tendency Example No. 2 Find the measures of central tendency (Mean, Median, and Mode) and the other measures of central tendency (P 60 , Q 1 , and D 9 ) of the following data: 16, 24, 11, 30, 30, 27, 18, 32, 25. N 1 2 3 4 5 6 7 8 9 Σx 213 11 16 18 24 25 27 30 30 32 Solution for Mode:  

37. Measures of Central Tendency Example No. 2 Find the measures of central tendency (Mean, Median, and Mode) and the other measures of central tendency (P 60 , Q 1 , and D 9 ) of the following data: 16, 24, 11, 30, 30, 27, 18, 32, 25. N 1 2 3 4 5 6 7 8 9 Σx 213 11 16 18 24 25 27 30 30 32 Solution for Percentile 60:          

38. Measures of Central Tendency Example No. 2 Find the measures of central tendency (Mean, Median, and Mode) and the other measures of central tendency (P 60 , Q 1 , and D 9 ) of the following data: 16, 24, 11, 30, 30, 27, 18, 32, 25. N 1 2 3 4 5 6 7 8 9 Σx 213 11 16 18 24 25 27 30 30 32 Solution for Quartile 1:          

39. Measures of Central Tendency Example No. 2 Find the measures of central tendency (Mean, Median, and Mode) and the other measures of central tendency (P 60 , Q 1 , and D 9 ) of the following data: 16, 24, 11, 30, 30, 27, 18, 32, 25. N 1 2 3 4 5 6 7 8 9 Σx 213 11 16 18 24 25 27 30 30 32 Solution for Decile 9:          

40. Measures of Central Tendency Example No. 3 Find the measures of central tendency (Mean, Median, and Mode) and the other measures of central tendency (Q 3 , P 65 , and D 8 ) of the following data: Class Interval 32 - 39 11 40 - 47 19 48 - 55 13 56 - 63 22 64 - 71 15

41. Measures of Central Tendency Example No. 3 Solution for Mean:       Class Interval Class Boundaries <CF 32 - 39 11 40 - 47 19 48 - 55 13 56 - 63 22 64 - 71 15 i = Σf = Σfx = 80 8 35.5 43.5 51.5 59.5 67.5 390.5 826.5 669.5 1309 1012.5 4208 31.5 – 39.5 39.5 – 47.5 47.5 – 55.5 55.5 – 63.5 63.5 – 71.5 Mean, Median, Mode, Q 3 , P 65 , and D 8 11 30 43 65 80

42. Class Interval Class Boundaries <CF 32 - 39 11 40 - 47 19 48 - 55 13 56 - 63 22 64 - 71 15 i = Σf = Σfx = 80 8 35.5 43.5 51.5 59.5 67.5 4208 31.5 – 39.5 39.5 – 47.5 47.5 – 55.5 55.5 – 63.5 63.5 – 71.5 11 Solution for Median:         30 43 65 80 390.5 826.5 669.5 1309 1012.5 80 69 50 37 15     >CF   Measures of Central Tendency Example No. 3 Mean, Median, Mode, Q 3 , P 65 , and D 8

43. Class Interval Class Boundaries <CF 32 - 39 11 40 - 47 19 48 - 55 13 56 - 63 22 64 - 71 15 i = Σf = Σfx = 80 8 35.5 43.5 51.5 59.5 67.5 4208 31.5 – 39.5 39.5 – 47.5 47.5 – 55.5 55.5 – 63.5 63.5 – 71.5 11 Solution for Median:   30 43 65 80 390.5 826.5 669.5 1309 1012.5 80 69 50 37 15     >CF   Measures of Central Tendency Example No. 3 Mean, Median, Mode, Q 3 , P 65 , and D 8

44. Class Interval Class Boundaries <CF 32 - 39 11 40 - 47 19 48 - 55 13 56 - 63 22 64 - 71 15 i = Σf = Σfx = 80 8 35.5 43.5 51.5 59.5 67.5 4208 31.5 – 39.5 39.5 – 47.5 47.5 – 55.5 55.5 – 63.5 63.5 – 71.5 11 30 43 65 80 Solution for Mode:         390.5 826.5 669.5 1309 1012.5 Measures of Central Tendency Example No. 3 Mean, Median, Mode, Q 3 , P 65 , and D 8

45. Class Interval Class Boundaries <CF 32 - 39 11 40 - 47 19 48 - 55 13 56 - 63 22 64 - 71 15 i = Σf = Σfx = 80 8 35.5 43.5 51.5 59.5 67.5 4208 31.5 – 39.5 39.5 – 47.5 47.5 – 55.5 55.5 – 63.5 63.5 – 71.5 11 30 43 65 80 Solution for Quartile 3:         390.5 826.5 669.5 1309 1012.5 Measures of Central Tendency Example No. 3 Mean, Median, Mode, Q 3 , P 65 , and D 8

46. Class Interval Class Boundaries <CF 32 - 39 11 40 - 47 19 48 - 55 13 56 - 63 22 64 - 71 15 i = Σf = Σfx = 80 8 35.5 43.5 51.5 59.5 67.5 4208 31.5 – 39.5 39.5 – 47.5 47.5 – 55.5 55.5 – 63.5 63.5 – 71.5 11 30 43 65 80 Solution for Percentile 65:         390.5 826.5 669.5 1309 1012.5 Measures of Central Tendency Example No. 3 Mean, Median, Mode, Q 3 , P 65 , and D 8

47. Class Interval Class Boundaries <CF 32 - 39 11 40 - 47 19 48 - 55 13 56 - 63 22 64 - 71 15 i = Σf = Σfx = 80 8 35.5 43.5 51.5 59.5 67.5 4208 31.5 – 39.5 39.5 – 47.5 47.5 – 55.5 55.5 – 63.5 63.5 – 71.5 11 30 43 65 80 Solution for Decile 8:         390.5 826.5 669.5 1309 1012.5 Measures of Central Tendency Example No. 3 Mean, Median, Mode, Q 3 , P 65 , and D 8

48. Measures of Central Tendency Example No. 4 Below are the responses of 130 students in a survey conducted: Question Response T 5 4 3 2 1 Gives concise but clear directions for students to follow. 25 44 33 15 13 130 125 176 99 30 13 443     Using the descriptive ratings below, determine their weighted mean response. 5 – Very Effective 4.20 – 5.00 4 – Significantly Effective 3.40 – 4.19 3 – Moderately Effective 2.60 – 3.39 2 – Slightly Effective 1.80 – 2.59 1 – Not Effective 1.00 – 1.79 Solution: Verbal Description: Significantly Effective

49. Measures of Dispersion Example No. 5 Given: Find the following: The Range The Mean Absolute Deviation The Percentile Deviation/The Decile Deviation The Quartile Deviation/The Inter-quartile Range The Variance (s 2 and σ 2 ) The Standard Deviation (s and σ) 25 78 42 23 18 18 68 77 83 80 48 36

50.  N  1  2  3  4  5  6  7  8  9  10  11  12  Σ 18 18 23 25 36 42 48 68 77 78 80 83 596 31.67 31.67 26.67 24.67 13.67 7.67 1.67 18.33 27.33 28.33 30.33 33.33 275.34 1002.9889 1002.9889 711.2889 608.6089 186.8689 58.8289 2.7889 335.9889 746.9289 802.5889 919.9089 1110.8889 7490.6668 Solution: The Mean          

51.  N  1  2  3  4  5  6  7  8  9  10  11  12  Σ 18 18 23 25 36 42 48 68 77 78 80 83 596 31.67 31.67 26.67 24.67 13.67 7.67 1.67 18.33 27.33 28.33 30.33 33.33 275.34 1002.9889 1002.9889 711.2889 608.6089 186.8689 58.8289 2.7889 335.9889 746.9289 802.5889 919.9089 1110.8889 7490.6668 Solution: The MAD      

52.  N  1  2  3  4  5  6  7  8  9  10  11  12  Σ 18 18 23 25 36 42 48 68 77 78 80 83 596 31.67 31.67 26.67 24.67 13.67 7.67 1.67 18.33 27.33 28.33 30.33 33.33 275.34 1002.9889 1002.9889 711.2889 608.6089 186.8689 58.8289 2.7889 335.9889 746.9289 802.5889 919.9089 1110.8889 7490.6668 Solution: Percentile Deviation                    

53.  N  1  2  3  4  5  6  7  8  9  10  11  12  Σ 18 18 23 25 36 42 48 68 77 78 80 83 596 31.67 31.67 26.67 24.67 13.67 7.67 1.67 18.33 27.33 28.33 30.33 33.33 275.34 1002.9889 1002.9889 711.2889 608.6089 186.8689 58.8289 2.7889 335.9889 746.9289 802.5889 919.9089 1110.8889 7490.6668 Solution: Decile Deviation                    

54.  N  1  2  3  4  5  6  7  8  9  10  11  12  Σ 18 18 23 25 36 42 48 68 77 78 80 83 596 31.67 31.67 26.67 24.67 13.67 7.67 1.67 18.33 27.33 28.33 30.33 33.33 275.34 1002.9889 1002.9889 711.2889 608.6089 186.8689 58.8289 2.7889 335.9889 746.9289 802.5889 919.9089 1110.8889 7490.6668 Solution: Quartile Deviation                    

55.  N  1  2  3  4  5  6  7  8  9  10  11  12  Σ 18 18 23 25 36 42 48 68 77 78 80 83 596 31.67 31.67 26.67 24.67 13.67 7.67 1.67 18.33 27.33 28.33 30.33 33.33 275.34 1002.9889 1002.9889 711.2889 608.6089 186.8689 58.8289 2.7889 335.9889 746.9289 802.5889 919.9089 1110.8889 7490.6668 Solution:    

56.  N  1  2  3  4  5  6  7  8  9  10  11  12  Σ 18 18 23 25 36 42 48 68 77 78 80 83 596 31.67 31.67 26.67 24.67 13.67 7.67 1.67 18.33 27.33 28.33 30.33 33.33 275.34 1002.9889 1002.9889 711.2889 608.6089 186.8689 58.8289 2.7889 335.9889 746.9289 802.5889 919.9089 1110.8889 7490.6668 Solution: Population Variance (σ 2 )  

57. Example No. 2 # of CI Class Interval Class Boundaries <CF 1 37 – 43 8 2 44 – 50 9 3 51 – 57 7 4 58 – 64 11 5 65 – 71 10 6 72 – 78 19 7 79 – 85 14 8 86 – 92 9 9 93 - 99 13 40 47 54 61 68 75 82 89 96 i = 7 100 320 423 378 671 680 1425 1148 801 1248 7094 36.5 – 43.5 43.5 – 50.5 50.5 – 57.5 57.5 – 64.5 64.5 -71.5 71.5 – 78.5 78.5 – 85.5 85.5 – 92.5 92.5 – 99.5 8 17 24 35 45 64 78 87 100 30.94 23.94 16.94 9.94 2.94 4.06 11.06 18.06 25.06 247.52 215.46 118.58 109.34 29.40 77.14 154.84 162.54 325.78 7658.2688 5158.1124 2008.7452 1086.8396 86.4360 313.1884 1712.5304 2935.4724 8164.0468 29123.64 Solution :       1440.60

58. Example No. 2 # of CI Class Interval Class Boundaries <CF 1 37 – 43 8 2 44 – 50 9 3 51 – 57 7 4 58 – 64 11 5 65 – 71 10 6 72 – 78 19 7 79 – 85 14 8 86 – 92 9 9 93 - 99 13 40 47 54 61 68 75 82 89 96 i = 7 100 320 423 378 671 680 1425 1148 801 1248 7094 36.5 – 43.5 43.5 – 50.5 50.5 – 57.5 57.5 – 64.5 64.5 -71.5 71.5 – 78.5 78.5 – 85.5 85.5 – 92.5 92.5 – 99.5 8 17 24 35 45 64 78 87 100 30.94 23.94 16.94 9.94 2.94 4.06 11.06 18.06 25.06 247.52 215.46 118.58 109.34 29.40 77.14 154.84 162.54 325.78 7658.2688 5158.1124 2008.7452 1086.8396 86.4360 313.1884 1712.5304 2935.4724 8164.0468 29123.64 1440.60 The Range      

59. # of CI Class Interval Class Boundaries <CF 1 37 – 43 8 2 44 – 50 9 3 51 – 57 7 4 58 – 64 11 5 65 – 71 10 6 72 – 78 19 7 79 – 85 14 8 86 – 92 9 9 93 - 99 13 40 47 54 61 68 75 82 89 96 i = 7 100 320 423 378 671 680 1425 1148 801 1248 7094 36.5 – 43.5 43.5 – 50.5 50.5 – 57.5 57.5 – 64.5 64.5 -71.5 71.5 – 78.5 78.5 – 85.5 85.5 – 92.5 92.5 – 99.5 8 17 24 35 45 64 78 87 100 30.94 23.94 16.94 9.94 2.94 4.06 11.06 18.06 25.06 247.52 215.46 118.58 109.34 29.40 77.14 154.84 162.54 325.78 7658.2688 5158.1124 2008.7452 1086.8396 86.4360 313.1884 1712.5304 2935.4724 8164.0468 29123.64 1440.60 The MAD      

60. Solution: Percentile Deviation                                    

61. Solution: Decile Deviation                                  

62. Solution: Quartile Deviation                                      

63. # of CI Class Interval Class Boundaries <CF 1 37 – 43 8 2 44 – 50 9 3 51 – 57 7 4 58 – 64 11 5 65 – 71 10 6 72 – 78 19 7 79 – 85 14 8 86 – 92 9 9 93 - 99 13 40 47 54 61 68 75 82 89 96 i = 7 100 320 423 378 671 680 1425 1148 801 1248 7094 36.5 – 43.5 43.5 – 50.5 50.5 – 57.5 57.5 – 64.5 64.5 -71.5 71.5 – 78.5 78.5 – 85.5 85.5 – 92.5 92.5 – 99.5 8 17 24 35 45 64 78 87 100 30.94 23.94 16.94 9.94 2.94 4.06 11.06 18.06 25.06 247.52 215.46 118.58 109.34 29.40 77.14 154.84 162.54 325.78 7658.2688 5158.1124 2008.7452 1086.8396 86.4360 313.1884 1712.5304 2935.4724 8164.0468 29123.64 1440.60 Sample Variance        

64. # of CI Class Interval Class Boundaries <CF 1 37 – 43 8 2 44 – 50 9 3 51 – 57 7 4 58 – 64 11 5 65 – 71 10 6 72 – 78 19 7 79 – 85 14 8 86 – 92 9 9 93 - 99 13 40 47 54 61 68 75 82 89 96 i = 7 100 320 423 378 671 680 1425 1148 801 1248 7094 36.5 – 43.5 43.5 – 50.5 50.5 – 57.5 57.5 – 64.5 64.5 -71.5 71.5 – 78.5 78.5 – 85.5 85.5 – 92.5 92.5 – 99.5 8 17 24 35 45 64 78 87 100 30.94 23.94 16.94 9.94 2.94 4.06 11.06 18.06 25.06 247.52 215.46 118.58 109.34 29.40 77.14 154.84 162.54 325.78 7658.2688 5158.1124 2008.7452 1086.8396 86.4360 313.1884 1712.5304 2935.4724 8164.0468 29123.64 1440.60 Sample Standard Deviation  

65. # of CI Class Interval Class Boundaries <CF 1 37 – 43 8 2 44 – 50 9 3 51 – 57 7 4 58 – 64 11 5 65 – 71 10 6 72 – 78 19 7 79 – 85 14 8 86 – 92 9 9 93 - 99 13 40 47 54 61 68 75 82 89 96 i = 7 100 320 423 378 671 680 1425 1148 801 1248 7094 36.5 – 43.5 43.5 – 50.5 50.5 – 57.5 57.5 – 64.5 64.5 -71.5 71.5 – 78.5 78.5 – 85.5 85.5 – 92.5 92.5 – 99.5 8 17 24 35 45 64 78 87 100 30.94 23.94 16.94 9.94 2.94 4.06 11.06 18.06 25.06 247.52 215.46 118.58 109.34 29.40 77.14 154.84 162.54 325.78 7658.2688 5158.1124 2008.7452 1086.8396 86.4360 313.1884 1712.5304 2935.4724 8164.0468 29123.64 1440.60 Population Variance        

66. # of CI Class Interval Class Boundaries <CF 1 37 – 43 8 2 44 – 50 9 3 51 – 57 7 4 58 – 64 11 5 65 – 71 10 6 72 – 78 19 7 79 – 85 14 8 86 – 92 9 9 93 - 99 13 40 47 54 61 68 75 82 89 96 i = 7 100 320 423 378 671 680 1425 1148 801 1248 7094 36.5 – 43.5 43.5 – 50.5 50.5 – 57.5 57.5 – 64.5 64.5 -71.5 71.5 – 78.5 78.5 – 85.5 85.5 – 92.5 92.5 – 99.5 8 17 24 35 45 64 78 87 100 30.94 23.94 16.94 9.94 2.94 4.06 11.06 18.06 25.06 247.52 215.46 118.58 109.34 29.40 77.14 154.84 162.54 325.78 7658.2688 5158.1124 2008.7452 1086.8396 86.4360 313.1884 1712.5304 2935.4724 8164.0468 29123.64 1440.60 Population Standard Deviation  

67. Activity Find the Mean, Median, Mode , Variance (Sample & Population) and Standard Deviation (Sample & Population) of the following data: 16, 24, 11, 30, 30, 27, 18, 32, 25. N 1 2 3 4 5 6 7 8 9 Σx 213 11 16 18 24 25 27 30 30 32