1. Understanding Basic Statistics Fourth Edition By Brase and Brase Prepared by: Lynn Smith Gloucester County College Chapter Two Organizing Data

2. Copyright © Houghton Mifflin Company. All rights reserved.2 | 2 A graphical display should Show the data. Induce the viewer to think about the substance of the graphic. Avoid distorting the message.

3. Copyright © Houghton Mifflin Company. All rights reserved.2 | 3 Frequency Table Partitions data into classes or intervals Shows how many data values are in each class Each data value falls into exactly one class

4. Copyright © Houghton Mifflin Company. All rights reserved.2 | 4 Frequency Table Shows the limits of each class Shows the frequency of each data value Shows the midpoint of each class

5. Copyright © Houghton Mifflin Company. All rights reserved.2 | 5 To make a frequency table: First determine the number of classes and determine the class width. Five to fifteen classes are most commonly used.

6. Copyright © Houghton Mifflin Company. All rights reserved.2 | 6 Finding class width 1.Compute: 2.Increase the value computed to the next highest whole number

7. Copyright © Houghton Mifflin Company. All rights reserved.2 | 7 Determining the Class Width Raw Data: 10.2 18.7 22.3 20.0 6.3 17.8 17.1 5.0 2.4 7.9 0.3 2.5 8.5 12.5 21.4 16.5 0.4 5.2 4.1 14.3 19.5 22.5 0.0 24.7 11.4 Use 5 classes. 24.7 – 0.0 5 = 4.94 Round class width up to 5.

8. Copyright © Houghton Mifflin Company. All rights reserved.2 | 8 Class limits The lower class limit is the lowest data value that can fit in a class. The upper class limit is the highest data value that can fit in a class.

9. Copyright © Houghton Mifflin Company. All rights reserved.2 | 9 Making a frequency table: Create the distinct classes. As a convenience, the lower class limit of the first class may be the smallest data value. Add the class width to the each lower class limit to get the lower class limits of successive classes. Fill in upper class limits to create distinct classes that accommodate all possible data values.

10. Copyright © Houghton Mifflin Company. All rights reserved.2 | 10 Creating the classes Raw Data: 10.2 18.7 22.3 20.0 6.3 17.8 17.1 5.0 2.4 7.9 0.3 2.5 8.5 12.5 21.4 16.5 0.4 5.2 4.1 14.3 19.5 22.5 0.0 24.7 11.4 Classes: 0.0 – 4.9 5.0 – 9.9 10.0 – 14.9 15.0 – 19.9 20.0 – 24.9

11. Copyright © Houghton Mifflin Company. All rights reserved.2 | 11 To make a frequency table: Tally the data into classes. Each data value falls into exactly one class. Total the tallies to obtain each class frequency.

12. Copyright © Houghton Mifflin Company. All rights reserved.2 | 12 Raw Data: 10.2 18.7 22.3 20.0 6.3 17.8 17.1 5.0 2.4 7.9 0.3 2.5 8.5 12.5 21.4 16.5 0.4 5.2 4.1 14.3 19.5 22.5 0.0 24.7 11.4 Classes: Tally 0.0 – 4.9 |||| | 5.0 – 9.9 |||| 10.0 – 14.9 |||| 15.0 – 19.9 |||| 20.0 – 24.9 |||| Tallying the data

13. Copyright © Houghton Mifflin Company. All rights reserved.2 | 13 Class frequencies Classes: Tallyf 0.0 – 4.9 |||| |6 5.0 – 9.9 ||||5 10.0 – 14.9 ||||4 15.0 – 19.9 ||||5 20.0 – 24.9 ||||5

14. Copyright © Houghton Mifflin Company. All rights reserved.2 | 14 To make a frequency table: Compute the midpoint for each class. The midpoint is also known as the class mark.

15. Copyright © Houghton Mifflin Company. All rights reserved.2 | 15 # of miles fclass midpoints 0.0 - 4.96 2.45 5.0 - 9.957.45 10.0 - 14.9412.45 15.0 - 19.9517.45 20.0 - 24.9522.45 Finding Class Midpoints

16. Copyright © Houghton Mifflin Company. All rights reserved.2 | 16 Class Boundaries

17. Copyright © Houghton Mifflin Company. All rights reserved.2 | 17 To make a frequency table: Determine the class boundaries. For integer data: Upper class boundary = upper class limit + 0.5 units. Lower class boundary = lower class limit  0.5 units.

18. Copyright © Houghton Mifflin Company. All rights reserved.2 | 18 # of miles fclass boundaries 0.0 - 4.96 -0.05 - 4.95 5.0 - 9.954.95 - 9.95 10.0 - 14.949.95 - 14.95 15.0 - 19.9514.95 - 19.95 20.0 - 24.9519.95 - 24.95 Finding Class Boundaries

19. Copyright © Houghton Mifflin Company. All rights reserved.2 | 19 Relative Frequency The relative frequency of a class is the proportion of all data that fall into that class. To find relative frequency of a class divide the class frequency (f) by the total of all frequencies (n).

20. Copyright © Houghton Mifflin Company. All rights reserved.2 | 20 Relative frequency

21. Copyright © Houghton Mifflin Company. All rights reserved.2 | 21 Finding relative frequencies # of miles fRelative frequencies 0.0 - 4.96 6/25 = 0.24 5.0 - 9.95 5/25 = 0.20 10.0 - 14.94 4/25 = 0.16 15.0 - 19.95 5/25 = 0.20 20.0 - 24.9 5 5/25 = 0.20 25

22. Copyright © Houghton Mifflin Company. All rights reserved.2 | 22 Histogram A visual display of data organized into a frequency table Bars represent each class Height of each bar represents class frequency (or relative frequency) Width of each bar represents class width

23. Copyright © Houghton Mifflin Company. All rights reserved.2 | 23 To construct a histogram Make a frequency table Place class boundaries on the horizontal axis Place frequencies or relative frequencies on the vertical axis For each class draw a bar whose width extends between corresponding class boundaries. The height of each bar is the appropriate frequency or relative frequency.

24. Copyright © Houghton Mifflin Company. All rights reserved.2 | 24 Histogram

25. Copyright © Houghton Mifflin Company. All rights reserved.2 | 25 Relative Frequency Histogram

26. Copyright © Houghton Mifflin Company. All rights reserved.2 | 26 Common Shapes of Histograms Symmetrical Uniform or rectangular Skewed left Skewed Right Bimodal

27. Copyright © Houghton Mifflin Company. All rights reserved.2 | 27 Typical Symmetrical Histogram

28. Copyright © Houghton Mifflin Company. All rights reserved.2 | 28 Typical Uniform or Rectangular Histogram

29. Copyright © Houghton Mifflin Company. All rights reserved.2 | 29 Typical Skewed Histograms

30. Copyright © Houghton Mifflin Company. All rights reserved.2 | 30 Typical Bimodal Histogram

31. Copyright © Houghton Mifflin Company. All rights reserved.2 | 31 Other Common Graphs Bar Graphs Pareto Charts Circle Graphs Time-Series Graphs

32. Copyright © Houghton Mifflin Company. All rights reserved.2 | 32 Bar Graph Bars are of uniform width and are uniformly spaced. Bars may be vertical or horizontal. Lengths of bars represent values being displayed, frequency or percentage of occurrence. Graph annotated with title, labels and scale or value for each bar

33. Copyright © Houghton Mifflin Company. All rights reserved.2 | 33 Bar Graph

34. Copyright © Houghton Mifflin Company. All rights reserved.2 | 34 Changing Scale Whenever you use a change of scale in a graphic, use a squiggle on the changed axis. A squiggle:

35. Copyright © Houghton Mifflin Company. All rights reserved.2 | 35 Bar Graph With a Change of Scale

36. Copyright © Houghton Mifflin Company. All rights reserved.2 | 36 Pareto Chart Tool of quality control Start with a bar chart Arrange bars in decreasing order of frequency Frequently used to investigate causes of problems

37. Copyright © Houghton Mifflin Company. All rights reserved.2 | 37 Pareto Chart

38. Copyright © Houghton Mifflin Company. All rights reserved.2 | 38 Circle Graph (Pie Chart) Shows division of whole into component parts Label parts with appropriate percentages of the whole

39. Copyright © Houghton Mifflin Company. All rights reserved.2 | 39 Circle Graph

40. Copyright © Houghton Mifflin Company. All rights reserved.2 | 40 Time Series Data sets composed of similar measurements taken at regular intervals over time

41. Copyright © Houghton Mifflin Company. All rights reserved.2 | 41 Time Series Graph Shows data values in chronological order Place time on horizontal scale Place other variable on vertical scale Connect data points with line segments

42. Copyright © Houghton Mifflin Company. All rights reserved.2 | 42 Time Series Graph

43. Copyright © Houghton Mifflin Company. All rights reserved.2 | 43 Rules For Any Graph Provide a title. Label axes. Identify units of measure. Present information clearly.

44. Copyright © Houghton Mifflin Company. All rights reserved.2 | 44 Exploratory Data Analysis Technique particularly useful for detecting patterns and extreme values. Also called EDA. Makes use of histograms and other graphics.

45. Copyright © Houghton Mifflin Company. All rights reserved.2 | 45 Stem and Leaf Display Organizes and groups data. Allows recovery of original data. Data values must have at least two digits.

46. Copyright © Houghton Mifflin Company. All rights reserved.2 | 46 To Construct a Stem and Leaf Display Divide digits of each data value into two parts: “stem” and “leaf.” Align stems in vertical column to left of a vertical line. Place leaves with same stem on same row as that stem, arranged in increasing order. Label to include magnitude or decimal point.

47. Copyright © Houghton Mifflin Company. All rights reserved.2 | 47 Stem and Leaf Display Raw Data: 35, 45, 42, 45, 41, 32, 25, 56, 67, 76, 65, 53, 53, 32, 34, 47, 43, 31

48. Copyright © Houghton Mifflin Company. All rights reserved.2 | 48 Stem and Leaf Display First data value = 35 Stem and Leaf Display First data value = 35 234567234567 stems 5leaf

49. Copyright © Houghton Mifflin Company. All rights reserved.2 | 49 Stem and Leaf Display Data value = 45 Stem and Leaf Display Data value = 45 234567234567 5 5

50. Copyright © Houghton Mifflin Company. All rights reserved.2 | 50 234567234567 5 5 2 Stem and Leaf Display Data value = 42 Stem and Leaf Display Data value = 42

51. Copyright © Houghton Mifflin Company. All rights reserved.2 | 51 234567234567 5 5 2 5 Stem and Leaf Display Data value = 45 Stem and Leaf Display Data value = 45

52. Copyright © Houghton Mifflin Company. All rights reserved.2 | 52 234567234567 5 5 2 5 1 Stem and Leaf Display Data value = 41 Stem and Leaf Display Data value = 41

53. Copyright © Houghton Mifflin Company. All rights reserved.2 | 53 234567234567 5 2 5 2 5 1 Stem and Leaf Display Data value = 32 Stem and Leaf Display Data value = 32

54. Copyright © Houghton Mifflin Company. All rights reserved.2 | 54 234567234567 5 2 5 2 5 1 5 Stem and Leaf Display Data value = 25 Stem and Leaf Display Data value = 25

55. Copyright © Houghton Mifflin Company. All rights reserved.2 | 55 234567234567 5 2 5 2 5 1 5 6 Stem and Leaf Display Data value = 56 Stem and Leaf Display Data value = 56

56. Copyright © Houghton Mifflin Company. All rights reserved.2 | 56 234567234567 5 2 5 2 5 1 5 6 7 Stem and Leaf Display Data value = 67 Stem and Leaf Display Data value = 67

57. Copyright © Houghton Mifflin Company. All rights reserved.2 | 57 234567234567 5 2 5 2 5 1 5 6 7 6 Stem and Leaf Display Data value = 76 Stem and Leaf Display Data value = 76

58. Copyright © Houghton Mifflin Company. All rights reserved.2 | 58 234567234567 5 2 5 2 5 1 5 6 7 5 6 Stem and Leaf Display Data value = 65 Stem and Leaf Display Data value = 65

59. Copyright © Houghton Mifflin Company. All rights reserved.2 | 59 234567234567 5 2 5 2 5 1 5 6 3 7 5 6 Stem and Leaf Display Data value = 53 Stem and Leaf Display Data value = 53

60. Copyright © Houghton Mifflin Company. All rights reserved.2 | 60 234567234567 5 2 5 2 5 1 5 6 3 3 7 5 6 Stem and Leaf Display Data value = 53 Stem and Leaf Display Data value = 53

61. Copyright © Houghton Mifflin Company. All rights reserved.2 | 61 234567234567 5 2 2 5 2 5 1 5 6 3 3 7 5 6 Stem and Leaf Display Data value = 32 Stem and Leaf Display Data value = 32

62. Copyright © Houghton Mifflin Company. All rights reserved.2 | 62 234567234567 5 2 2 4 5 2 5 1 5 6 3 3 7 5 6 Stem and Leaf Display Data value = 34 Stem and Leaf Display Data value = 34

63. Copyright © Houghton Mifflin Company. All rights reserved.2 | 63 234567234567 5 2 2 4 5 2 5 1 7 5 6 3 3 7 5 6 Stem and Leaf Display Data value = 47 Stem and Leaf Display Data value = 47

64. Copyright © Houghton Mifflin Company. All rights reserved.2 | 64 234567234567 5 2 2 4 5 2 5 1 7 3 5 6 3 3 7 5 6 Stem and Leaf Display Data value = 43 Stem and Leaf Display Data value = 43

65. Copyright © Houghton Mifflin Company. All rights reserved.2 | 65 234567234567 5 2 2 4 1 5 2 5 1 7 3 5 6 3 3 7 5 6 Stem and Leaf Display Data value = 31 Stem and Leaf Display Data value = 31

66. Copyright © Houghton Mifflin Company. All rights reserved.2 | 66 234567234567 5 2 2 4 1 5 2 5 1 7 3 5 6 3 3 7 5 6 2  5 indicates 25 years Finished Stem and Leaf Display