1. Chapter 3 Data Description Reference: Allan G. Bluman (2007) Elementary Statistics: A Step-by Step Approach . New York : McGraw Hill

5. The Sample Mean

6. The Sample Mean - Example

7. The Population Mean

8. The Population Mean - Example

9. The Sample Mean for an Ungrouped Frequency Distribution The mean for a ungrouped frequency distribution is given by X f X n = ( )  

10. The Sample Mean for an Ungrouped Frequency Distribution - Example Score, X Frequency, f 0 2 1 4 2 12 3 4 4 3 5 Score, X 0 2 1 4 2 12 3 4 4 3 5 Frequency, f

11. The Sample Mean for an Ungrouped Frequency Distribution - Example 5 Score, X  X 0 2 0 1 4 4 2 12 24 3 4 12 4 3 12 5 Frequency, f f Score, X Frequency, f f  X 0 2 0 1 4 4 2 12 24 3 4 12 4 3 12

12. The Sample Mean for a Grouped Frequency Distribution The mean for a grouped frequency distribution is given by X f X n Here X is the correspond ing class midpoint. m m = ( )   .

13. The Sample Mean for a Grouped Frequency Distribution - Example Class Frequency, f 15.5 - 20.5 3 20.5 - 25.5 5 25.5 - 30.5 4 30.5 - 35.5 3 35.5 - 40.5 2 5 Class 15.5 - 20.5 3 20.5 - 25.5 5 25.5 - 30.5 4 30.5 - 35.5 3 35.5 - 40.5 2 5 Frequency, f

14. The Sample Mean for a Grouped Frequency Distribution - Example 35.5 - 40.5 2 38 76 5 Class X  X 15.5 - 20.5 3 18 54 20.5 - 25.5 5 Frequency, f m f m Class Frequency, f X m f  X m 15.5 - 20.5 3 18 54 20.5 - 25.5 5 23 115 25.5 - 30.5 4 28 112 30.5 - 35.5 3 33 99 5 23 115 25.5 - 30.5 4 28 112 30.5 - 35.5 3 33 99 35.5 - 40.5 2 38 76

15. The Sample Mean for a Grouped Frequency Distribution - Example

28. The Median-Ungrouped Frequency Distribution - Example This class contains the 5th through the 13th values.

29. The Median for a Grouped Frequency Distribution class median the of boundary lower L class median the of width w class median the of frequency f class median the preceding immediately class the of frequency cumulative cf frequencies the of sum n Where L w f cf n MD can be computed from: median The m m     ) ( ) 2 (    

30. The Median for a Grouped Frequency Distribution - Example Class Frequency, f 15.5 - 20.5 3 20.5 - 25.5 5 25.5 - 30.5 4 30.5 - 35.5 3 35.5 - 40.5 2 5 Class 15.5 - 20.5 3 20.5 - 25.5 5 25.5 - 30.5 4 30.5 - 35.5 3 35.5 - 40.5 2 5 Frequency, f

31. The Median for a Grouped Frequency Distribution - Example 5 Class Cumulative 15.5 - 20.5 3 3 20.5 - 25.5 5 8 25.5 - 30.5 4 12 30.5 - 35.5 3 15 35.5 - 40.5 2 17 5 Frequency, f Frequency Class Frequency, f Cumulative Frequency 15.5 - 20.5 3 3 20.5 - 25.5 5 8 25.5 - 30.5 4 12 30.5 - 35.5 3 15 35.5 - 40.5 2 17

33. The Median for a Grouped Frequency Distribution = 17 = = = – 20.5 = 5 ( ) ( ) = (17 / 2) – 8 4 = 26.125. n cf f w L MD n cf f w L m m 8 4 25.5 25 5 2 5 25 5      . ( ) .

38. The Mode for an Ungrouped Frequency Distribution - Example Values Frequency, f 15 3 20 5 25 8 30 3 35 2 5 Values 15 3 20 5 25 8 30 3 35 2 5 Mode Frequency, f

40. The Mode for a Grouped Frequency Distribution - Example Modal Class

44. The Weighted Mean

46. Positively Skewed X Y M o d e < M e d i a n < M e a n P o s i t i v e l y S k e w e d

47. Symmetrical n Y X S y m m e t r i c a l M e a n = M e d i a = M o d e

48. Negatively Skewed < M e d i a n < M o d e Y X N e g a t i v e l y S k e w e d M e a n

50. Measures of Variation - Population Variance

51. Measures of Variation - Population Standard Deviation

53. Measures of Variation - Example

54. Measures of Variation - Sample Variance The unbias ed estimat or of the population variance o r the samp le varianc e is a statistic whose valu e approxim ates the expected v alue of a population variance. It is deno ted by s 2 , ( ) , where s X X n and X sample mean n sample size = = 2 2 1    

55. Measures of Variation - Sample Standard Deviation The sample standard deviation is the squ are root of t he sample variance. = 2 s s X X n     ( ) . 2 1

56. Shortcut Formula for the Sample Variance and the Standard Deviation = = X X n n s X X n n 2 2 2 2 1 1         ( ) / ( ) / s 2

58. Sample Variance - Example = 1263  (79) = 3.7 = 3.7 2 s X X n n s 2 2 2 1 5 4 1 9       ( ) / / . .

60. Sample Variance for Grouped and Ungrouped Data The sample variance for groupe d data: = s f X f X n n m m 2 2 2 1       [( ) / ] . For ungrouped data, replace X m with the observe X value.

61. Sample Variance for Ungrouped Data - Example X f f  X f  X 2 5 2 10 50 6 3 18 108 7 8 56 392 8 1 8 64 9 6 54 486 10 4 40 400 n = 24  f  X = 186  f  X 2 = 1500 X f f  X f  X 2 5 2 10 50 6 3 18 108 7 8 56 392 8 1 8 64 9 6 54 486 10 4 40 400 n = 24  f  X = 186  f  X 2 = 1500

62. Sample Variance for Ungrouped Data - Example The sample variance and standa rd deviati on: = = 1500  [(186) 2 s f X f X n n s 2 2 2 1 24 23 2 54 2 54 1 6          [( ) / ] / ] . . . . .

68. The Empirical (Normal) Rule        -- 95%                                

87. Exploratory Data Analysis – Stem and Leaf Plot - Example Leading Digit (Stem) Trailing Digit (Leaf) 0 2 1 3 4 2 0 3 5 3 1 2 2 2 2 3 6 4 3 4 4 5 5 1 2 7

91. Exploratory Data Analysis - Box Plot - Example (Cardiograms data)