Chapter Two

1. Chapter Two Organizing Data

2. A graphical display should • Show the data. • Induce the viewer to think about the substance of the graphic. • Avoid distorting the message.

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. Frequency Table • Shows the limits of each class • Shows the frequency of each data value • Shows the midpoint of each class

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. Finding class width • Compute: • Increase the value computed to the next highest whole number

7. 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. Determining the Class Width

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. 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. 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 Creating the classes Classes: 0.0 – 4.9 5.0 – 9.9 10.0 – 14.9 15.0 – 19.9 20.0 – 24.9

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. Tallying the data 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 ||||

13. Class frequencies Classes: Tally f 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. To make a frequency table: Compute the midpoint for each class. • The midpoint is also known as the class mark.

15. Finding Class Midpoints # of miles f class midpoints 0.0 - 4.9 6 2.45 5.0 - 9.9 5 7.45 10.0 - 14.9 4 12.45 15.0 - 19.9 5 17.45 20.0 - 24.9 5 22.45

16. Class Boundaries

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. Finding Class Boundaries # of miles f class boundaries 0.0 - 4.9 6 -0.05 - 4.95 5.0 - 9.9 5 4.95 - 9.95 10.0 - 14.9 4 9.95 - 14.95 15.0 - 19.9 5 14.95 - 19.95 20.0 - 24.9 5 19.95 - 24.95

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. Relative frequency

21. Finding relative frequencies # of miles f Relative frequencies 0.0 - 4.9 6 6/25 = 0.24 5.0 - 9.9 5 5/25 = 0.20 10.0 - 14.9 4 4/25 = 0.16 15.0 - 19.9 5 5/25 = 0.20 20.0 - 24.9 5 5/25 = 0.20 25

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. 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. Histogram

25. Relative Frequency Histogram

26. Common Shapes of Histograms • Symmetrical • Uniform or rectangular • Skewed left • Skewed Right • Bimodal

27. Typical Symmetrical Histogram

28. Typical Uniform or Rectangular Histogram

29. Typical Skewed Histograms

30. Typical Bimodal Histogram

31. Other Common Graphs • Bar Graphs • Pareto Charts • Circle Graphs • Time-Series Graphs

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. Bar Graph

34. Changing Scale • Whenever you use a change of scale in a graphic, use a squiggle on the changed axis. • A squiggle:

35. Bar Graph With a Change of Scale

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. Pareto Chart

38. Circle Graph (Pie Chart) • Shows division of whole into component parts • Label parts with appropriate percentages of the whole

39. Circle Graph

40. Time Series • Data sets composed of similar measurements taken at regular intervals over time

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. Time Series Graph

43. Rules For Any Graph • Provide a title. • Label axes. • Identify units of measure. • Present information clearly.

44. Exploratory Data Analysis • Technique particularly useful for detecting patterns and extreme values. • Also called EDA. • Makes use of histograms and other graphics.

45. Stem and Leaf Display • Organizes and groups data. • Allows recovery of original data. • Data values must have at least two digits.

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. 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. Stem and Leaf Display First data value = 35 2 3 4 5 6 7 5 leaf stems

49. Stem and Leaf Display Data value = 45 2 3 4 5 6 7 5 5

50. Stem and Leaf Display Data value = 42 2 3 4 5 6 7 5 5 2