A Mathematical View of Our World

1. A Mathematical View of Our World 1st ed. Parks, Musser, Trimpe, Maurer, and Maurer

2. Chapter 8 Descriptive Statistics – Data and Patterns

3. Section 8.1Organizing and Picturing Data • Goals • Study visual displays of data • Dot plots • Stem-and-leaf plots • Histograms • Bar graphs • Line graphs • Pie charts

4. 8.1 Initial Problem • You need to give a sales report showing that: • District A had $135,000 in sales. • District B had $85,000 in sales. • District C had $115,000 in sales. • How can you present this data clearly to compare the 3 districts? • The solution will be given at the end of the section.

5. Obtaining Data • Data sets are sets of numbers collected from the real world. • Data can be obtained from: • Previously published research • A designed experiment • An observational study • A survey

6. Obtaining Data, cont’d • Once data has been collected, exploratory data analysis takes an initial look at data to see what patterns might emerge or what further questions need to be asked. • One way to carry out exploratory data analysis is to represent data pictorially.

7. Dot Plots • A dot plot is a graph in which: • The horizontal axis represents the data values. • The vertical axis represents the frequency of the data values. • One dot is placed for each occurrence of each data value.

8. Example 1 • Create a dot plot for the test scores: 26, 32, 54, 62, 67, 70, 71, 71, 74, 76, 80, 81, 84, 87, 87, 87, 89, 93, 95, 96. • Solution: Notice the scores have been arranged in order. a. 50 b. 51 c. 52 d. 53

9. Stem-and-Leaf Plot • A stem-and-leaf plot is a graph in which: • The digit furthest to the right is called the leaf. • The other digits are called the stem. • The stems and leaves are placed in vertical columns, with the leaves arranged in numerical order.

10. Example 2 • Create a stem-and-leaf plot for the test scores: 26, 32, 54, 62, 67, 70, 71, 71, 74, 76, 80, 81, 84, 87, 87, 87, 89, 93, 95, 96. • Solution: The tens digits will be the stems and the ones digits will be the leaves.

11. Example 2, cont’d • Solution, cont’d: The plot at right shows: • A cluster of values between 54 and 96. • A gap between 54 and 32. • The values 32 and 26 are outliers, separated from the other scores by a large gap.

12. Example 3 • Create and interpret a stem-and-leaf plot for the pizza prices:$9.20, $10.50, $10.70, $10.80, $12.00, $12.10, $12.20, $12.20, $12.30. • Solution: The dollar amounts will be the stems and the tens of cents will be the leaves.

13. Example 3, cont’d • Solution, cont’d: The plot at right shows: • Two clusters of prices separated by a gap. • The price $9.20 may be considered an outlier.

14. Histograms • A histogram is a graph in which: • The data is separated into intervals called measurement classes or bins. • Various interval sizes can be chosen, depending on the situation. • A frequency table, showing the number of data values in each bin, can be created to aid in drawing a histogram.

15. Example 4 • Create a histogram for the test scores: 26, 32, 54, 62, 67, 70, 71, 71, 74, 76, 80, 81, 84, 87, 87, 87, 89, 93, 95, 96. • Solution: Make a frequency table first, using bins of width 10.

16. Example 4, cont’d • Solution, cont’d: Create the histogram. • The height of each bar is equal to the frequency of the bin.

17. Example 4, cont’d • Note: The choice of bin size affects the appearance of the graph. • A histogram of the same data set with a bin size of 5 is shown next.

18. Question: Why is the histogram with bin size 5 not the best choice to represent the data set from the previous example? Choose the best answer.

19. Question cont’d: a. There are outliers. b. There are a wide range of values. c. It is hard to see the overall pattern of the scores. d. The bars are too narrow.

20. Example 4, cont’d • A histogram of the same data set with a bin size of 20 is shown next.

21. Question: Why is the histogram with bin size 20 not the best choice to represent the data set from the previous example? Choose the best answer. a. The bars are too tall. b. A lot of information about the data is lost. c. There are a wide range of values. d. The frequencies of the bins are not the same.

22. Relative Frequency Histograms • A relative frequency histogram is a graph in which: • The data is separated into bins. • The relative frequency (percent of the whole data set) of each bin is calculated. • The height of each bar is equal to the relative frequency of the bin. • A relative frequency table can be created to aid in drawing a relative frequency histogram.

23. Example 5 • Create a relative frequency histogram for the test scores: 26, 32, 54, 62, 67, 70, 71, 71, 74, 76, 80, 81, 84, 87, 87, 87, 89, 93, 95, 96, using a bin size of 10. • Solution: Find the relative frequency of each bin.

24. Example 5, cont’d • Solution, cont’d: The graph is shown below.

25. Bar Graphs • A bar graph is any graph in which the height or length of bars is used to represent quantities. • A histogram is a special type of bar graph.

26. Example 6 • Create a bar graph to display the data in the table.

27. Example 6, cont’d

28. Line Graphs • A line graph is used to graph data values that occur over time. • The horizontal axis represents the time. • The vertical axis represents the data value. • Each data value is plotted and the dots are connected by a line.

29. Example 7 • Create a line graph for the data shown in the bar graph below.

30. Example 7, cont’d • Solution:

31. Example 8 • Interpret the line graph shown here.

32. Example 8, cont’d • Solution: The general trend in the graph is an increase in the number completing college, although there were a few years with decreases. • In 1980, about 17.5% completed. • In 2002, about 26.7% completed. • In order to emphasize the most recent statistic, the percentage for 2002 was highlighted in the graph.

33. Pie Charts • A pie chart is used to graph relative proportions of quantities. • Pie charts are also called circle graphs. • Each quantity is graphed as a wedge-shaped portion of the circle.

34. Example 9 • The pie chart shows the average number of hours of sleep for a certain group of adults. • Interpret the chart.

35. Example 9, cont’d • Solution: Most of the people sleep 7 or 8 hours per night. • Also, 6% of the people get 5 hours of sleep or less per night.

36. Choosing a Graph • The different types of graphs and their uses are summarized below.

37. Example 10 • The table shows the average number of hours worked in different countries. • What type of graph would be most effective?

38. Example 10, cont’d • Solution: • We do not need to show a trend over time or percentages, so rule out line graphs and pie charts. • A bar graph would make comparison between countries easy. • The categories are the countries. • The height of each bar will represent the number of hours worked per year.

39. Example 10, cont’d • Solution, cont’d: A bar graph for the data is shown below.

40. 8.1 Initial Problem Solution • You need to give a sales report showing that: • District A had $135,000 in sales. • District B had $85,000 in sales. • District C had $115,000 in sales. • How can you present this data clearly to compare the 3 districts? • Either a bar graph or a pie chart allows for easy comparison between categories.

41. Initial Problem Solution, cont’d • A pie chart will clearly show the difference in proportions of sales from the different districts. • Calculate the total sales. • Find what portion of a circle represents each district’s sales. • The results are shown at right.

42. Section 8.2Comparisons • Goals • Study comparison graphs • Double-stem-and-leaf plots • Comparison histograms • Multiple bar graphs • Multiple line graphs • Multiple pie charts • Proportional bar graphs

43. 8.2 Initial Problem • How can the monthly sales of the 3 items be presented to show and compare the sales trends? • The solution will be given at the end of the section.

44. Double-Stem-and-Leaf Plots • A double-stem-and-leaf plot compares two data sets. • The stems are placed in the middle column. • The leaves of one data set are placed on the left, and the leaves of the other set on the right.

45. Example 1 • Create a double-stem-and-leaf plot to compare scores from the two classes. • Class 1: 26, 32, 54, 62, 67, 70, 71, 71, 74, 76, 80, 81, 84, 87, 87, 87, 93, 95, 96 • Class 2: 34, 45, 52, 57, 63, 65, 68, 70, 71, 72, 74, 76, 76, 78, 83, 85, 85, 87, 92, 99

46. Example 1, cont’d • Solution: Since more leaves are at the top on the left than on the right, it appears that Class 1 did somewhat better on the test than Class 2.

47. Question: Choose the statement that is not true. a. Class 1 had more low scores than Class 2. b. Class 2 has a larger gap than Class 1. c. Class 2 has fewer scores in the 80s and 80s than Class 1. d. Class 2 has a higher score than Class 1.

48. Comparison Histogram • A comparison histogramcompares two data sets. • The same bin size is chosen for both sets. • Bars for both sets are placed side-by-side in each interval, where necessary.

49. Example 2 • Create a comparison histogram to compare the scores from the two classes. • Class 1: 26, 32, 54, 62, 67, 70, 71, 71, 74, 76, 80, 81, 84, 87, 87, 87, 93, 95, 96 • Class 2: 34, 45, 52, 57, 63, 65, 68, 70, 71, 72, 74, 76, 76, 78, 83, 85, 85, 87, 92, 99

50. Example 2, cont’d • Solution: A bin size of 10 was used.