1. Graphical Displays of Data
Section 2.2
Graphical Displays of Data

2. Objectives
Create and interpret the basic types of graphs used to display data.
Distinguish between the basic shapes of a distribution.

3. Graphs Should be able to stand alone without the original data.
Must have a title and labels for both axes.
When appropriate, a legend, a source, and a date should be included.

4. Pie Charts
A pie chart shows how large each category is in relation to the whole. It is used to display qualitative or categorical data.
A pie chart uses the relative frequencies from the frequency distribution to divide the “pie” into different-sized wedges. The size, or central angle measure, of each wedge in the pie chart is calculated by multiplying 360° by the relative frequency of each class and rounding to the nearest whole degree.

5. Example 2.7: Creating a Pie Chart
Create a pie chart from the following data describing the distribution of housing types for students in a statistics class. Calculate the size of each wedge in the pie chart to the nearest whole degree.
Housing Types for Students in a Statistics Class
Type of Housing
Number of Students
Apartment
Dorm
House
Sorority/Fraternity House 20 15 9 5
Total 49

6. Example 2.7: Creating a Pie Chart (cont.)
Solution Relative Frequencies Central Angle Measures

7. Example 2.7: Creating a Pie Chart (cont.)

8. Types of Bar Graphs
A bar graph uses bars to represent the amount of data in each category. Displays qualitative data. Pareto charts are bar graphs in which the bars appear in descending order from largest to smallest. This type of bar graph is typically used with nominal data. A side-by-side bar graph is used when we want to create a bar graph that compares the same categories for different groups. A stacked bar graph can be a more efficient graph for displaying data from different samples.

9. Example 2.8: Creating a Bar Graph
Create a bar graph of the following data describing the distribution of housing types for students in a statistics class.
Housing Types for Students in a Statistics Class
Type of Housing
Number of Students
Apartment
Dorm
House
Sorority/Fraternity House 20 15 9 5

10. Example 2.8: Creating a Bar Graph (cont.)
Solution

11. Example 2.9: Creating a Pareto Chart
Create a Pareto chart, if appropriate, for the following data.
Fundraising Results
Class
Number of Fundraiser items sold
Ms. Boyd’s class
Ms. Willard’s class
Ms. Smith’s class
Ms. Carter’s class
Ms. Meredith’s class
200
157
176
181
143

12. Example 2.9: Creating a Pareto Chart (cont.)
Solution The data are nominal; therefore, it would be appropriate to create a Pareto chart from these data. Begin by rearranging the data from the largest frequency to the smallest frequency.
Ordered Fundraising Results
Class
Number of Fundraiser items sold
Ms. Boyd’s class
Ms. Carter’s class
Ms. Smith’s class
Ms. Willard’s class
Ms. Meredith’s class
200
181
176
157
143

13. Example 2.9: Creating a Pareto Chart (cont.)
Next, create a bar graph of the ordered data. The resulting Pareto chart is shown.

14. Example 2.10: Creating a Side-by-Side Bar Graph
Create a side-by-side bar graph of the following data describing the distribution of housing types for two different samples of students.
Housing Types for Students in a Statistics Class
Type of Housing
Number of Students from Sample A
Number of Students from Sample B
Apartment
Dorm
House
Sorority/Fraternity House 20 15 9 5 13 24 6 7

15. Example 2.10: Creating a Side-by-Side Bar Graph (cont.)
Solution

16. Example 2.11: Creating a Stacked Bar Graph
Create a stacked bar graph for the data in Example Solution With the stacked bar graph, it is easier to see that more students live in the dorms than in apartments.

17. Histograms
A frequency histogram, shortened to histogram, is a bar graph of a frequency distribution of quantitative data.
A relative frequency histogram is a histogram in which the heights of the bars represent the relative frequencies of each class rather than simply the frequencies.

18. Characteristics of Histograms
A bar graph of a frequency distribution.
The horizontal axis is a real number line.
The width of the bars represent the class width from the frequency table and should be uniform.
The bars in a histogram should touch.
The height of each bar represents the frequency of the class it represents.

19. Example 2.12: Constructing a Histogram
Construct a histogram of the 3-D TV prices from the previous section. The frequency distribution of the data is restated here.
3-D TV Prices
Class
Frequency
Midpoint
Class Boundaries
$ $1599
$ $1699
$ $1799
$ $1899
$ $1999 2 5 4
1549.5
1649.5
1749.5
1849.5
1949.5 – – – – –

20. Example 2.12: Constructing a Histogram (cont.)
Solution

21. Example 2.12: Constructing a Histogram (cont.)

22. Example 2.13: Constructing a Relative Frequency Histogram
Construct a relative frequency histogram of the 3-D TV prices from the previous section. The frequency distribution of the data is reprinted here.
3-D TV Prices
Class
Frequency
Relative Frequency
$ $1599
$ $1699
$ $1799 2 5 4

23. Example 2.13: Constructing a Relative Frequency Histogram (cont.)
3-D TV Prices (cont.)
Class
Frequency
Relative Frequency
$ $1899
$ $1999 5 4

24. Example 2.13: Constructing a Relative Frequency Histogram (cont.)
Solution

25. Frequency Polygons
A frequency polygon is a visual display created by plotting a point at the frequency of each class above each class midpoint and connecting the points using straight lines.

26. Constructing a Frequency Polygon
Frequency Polygons
Constructing a Frequency Polygon
1. Mark the class boundaries on the x‑axis and the frequencies on the y‑axis. Note, extra classes at the lower and upper ends will be added, each having a
frequency of 0. In our 3-D TV price example, these classes will be 1400–1499 at the lower end and 2000–2099 at the upper end.

27. Frequency Polygons
Constructing a Frequency Polygon (cont.) 2. Add the midpoints to the x-axis and plot a point at the frequency of each class directly above its midpoint. Notice that the class boundaries on the x‑axis have been lightened. This is due to the fact that frequency polygons represent the midpoints of the classes.

28. Frequency Polygons
Constructing a Frequency Polygon (cont.) 3. Join each point to the next with a line segment. Notice, this is not a smooth curve, but a polygon.

29. Ogive
An ogive (pronounced “oh-jive”) is a line graph which depicts the cumulative frequency of each class from a frequency table.
To create an ogive, begin by tabulating the cumulative frequency for each class. Unlike when creating a frequency polygon, we only include an extra class at the lower end for this graph, giving it a frequency of 0. Next, plot a point at the cumulative frequency for each class directly above its upper class boundary. The ogive is created by joining the points together with line segments.

30. Example 2.14: Creating an Ogive
Below is the frequency distribution of the 3-D TV prices with the cumulative frequency column included. Create an ogive of the data.

31. Example 2.14: Creating an Ogive (cont.)
3-D TV Prices
Class
Frequency
CumulativeFrequency
Class Boundaries
$ $1499
$ $1599
$ $1699
$ $1799
$ $1899
$ $1999 2 5 4 7 11 16 20 – – – – – –

32. Example 2.14: Creating an Ogive (cont.)
Solution Notice we have included the class 1400–1499 with a frequency of 0. The following is an ogive of the data.

33. Stem-and-Leaf Plots
A stem-and-leaf plot is a graph of quantitative data that is similar to a histogram in the way that it visually displays the distribution.
Characteristics:
A stem-and-leaf plot retains the original data.
The leaves are usually the last digit in each data value and the stems are the remaining digits.
A legend, sometimes called a key, should be included so that the reader can interpret the information.

34. Stem-and-Leaf Plots
Constructing a Stem‑and-Leaf Plot 1. Create two columns, one on the left for stems and one on the right for leaves. 2. List each stem that occurs in the data set in numerical order. Each stem is normally listed only once; however, the stems are sometimes listed two or more times if splitting the leaves would make the data set’s features clearer.

35. Stem-and-Leaf Plots
Constructing a Stem‑and-Leaf Plot (cont.) 3. List each leaf next to its stem. Each leaf will be listed as many times as it occurs in the original data set. There should be as many leaves as there are data values. Be sure to line up the leaves in straight columns so that the table is visually accurate. 4. Create a key to guide interpretation of the stem‑and-leaf plot. 5. If desired, put the leaves in numerical order to create an ordered stem-and-leaf plot.

36. Example 2.15: Creating a Stem-and-Leaf Plot
Create a stem-and-leaf plot of the following ACT scores from a group of college freshmen.
ACT Scores 18 27 35 23 26 17 24 22 29 21 31 32 25 19 20

37. Example 2.15: Creating a Stem-and-Leaf Plot (cont.)
Solution ACT Scores
Stem
Leaves 1 2 3
Key: 1|8 = 18

38. An Ordered Stem-and-Leaf Plot
The ordered stem-and-leaf plot from Example 2.15 is as follows. ACT Scores
Stem
Leaves 1 2 3
Key: 1|8 = 18

39. Example 2.16: Creating and Interpreting a Stem-and-Leaf Plot
Create a stem-and-leaf plot for the following starting salaries for entry-level accountants at public accounting firms. Use the stem-and-leaf plot that you create to answer the following questions.
Starting Salaries for Entry-Level Accountants
$51,500
$45,500
$45,800
$43,000
$42,900
$44,500
$48,300
$42,500
$46,300
$42,700
$46,500
$47,900
$40,900
$44,200
$50,000
$49,000
$47,700
$45,300
$40,700
$46,400
$50,700
$46,700
$48,000
$46,100
$48,200
$48,600
$44,300
$43,200
$45,000

40. Example 2.16: Creating and Interpreting a Stem-and-Leaf Plot (cont.)
a. What were the smallest and largest salaries recorded? b. Which salary appears the most often? c. How many salaries were in the range $41,000– $41,900? d. In which salary range did the most salaries lie: $40,000–$44,900, $45,000–$49,900, or $50,000 and above?

41. Example 2.16: Creating and Interpreting a Stem-and-Leaf Plot (cont.)
Solution This example is different than Example 2.15 in that the data have more than two digits and every data value ends in two zeros. It is important to choose the stems of the numbers so that the last significant digit in each data point will be the leaf in the chart

42. Example 2.16: Creating and Interpreting a Stem-and-Leaf Plot (cont.)
Listing the zeros as the leaves for each different stem would be of little use to anyone interpreting the stem-and-leaf plot. Instead, the key will denote that the salaries listed are in hundreds. Using the first two digits of each salary as a stem, we have the following plot. (Note: It is helpful to first list the salaries in numerical order.)

43. Example 2.16: Creating and Interpreting a Stem-and-Leaf Plot (cont.)

44. Example 2.16: Creating and Interpreting a Stem-and-Leaf Plot (cont.)
a. The smallest salary is $40,700; the largest salary is $51,500. b. $46,300 appears twice, which is more than any other salary. c. No salaries are in the range $41,000–$41,900 because there are no leaves listed for the stem 41. d. By counting the leaves in the given groups, we see that there are more salaries in the range $45,000– $49,900.

45. Dot Plots and Line Graphs
A dot plot is similar to a stem-and-leaf plot because it retains the original data, however instead of grouping certain points together, the data that are exactly the same appear together.
A line graph is used when data are measurements over time. The horizontal axis represents time. The vertical axis represents the variable being measured. Straight lines are used to connect points plotted at the value of each measurement above the time it was taken.

46. Example 2.17: Constructing a Line Graph
The Consumer Price Index (CPI) is a measure of the average change in value over time for a basket of goods and services. It is an index calculated by the US Bureau of Labor and Statistics. The table below shows the values of the CPI from several years. Construct a line graph of these data.

47. Example 2.17: Constructing a Line Graph (cont.)
Consumer Price Index
Year
CPI
1920
1930
1940
1950
1960
1970
1980
1990
2000
2010
20.0
16.7
14.0
24.1
29.6
38.8
82.4
130.7
172.2
218.1
Source: US Bureau of Labor Statistics. “Consumer Price Index History Table.” 19 Jan ftp://ftp.bls.gov/pub/special.requests/cpi/cpiai.txt (24 Jan. 2012).

48. Example 2.17: Constructing a Line Graph (cont.)
Solution

49. Summary Qualitative Data Type of Graph Definition Pie Chart Bar Graph
A pie chart shows how large each category
is in relation to the whole; that is, it uses
the relative frequencies from the
frequency distribution to divide the “pie”
into different-sized wedges. It can only be
used to display qualitative data.
In a bar graph, bars are used to represent
the amount of data in each category; one
axis displays the categories of qualitative
data and the other axis displays the
frequencies.

50. Side-by-Side Bar Graph
Summary (cont.)
Qualitative Data (cont.)
Type of Graph
Definition
Pareto Chart
Side-by-Side Bar Graph
A Pareto chart is a bar graph with the bars
in descending order of frequency. Pareto
charts are typically used with nominal
data.
A side-by-side bar graph is a bar graph
that compares the same categories for
different groups.

51. Summary (cont.) Qualitative Data (cont.) Type of Graph Definition
Stacked Bar Graph
A stacked bar graph is a bar graph that
compares the same categories for
different groups and shows category
totals.

52. Summary (cont.) Quantitative Data Type of Graph Definition Histogram
Frequency Polygon
A histogram is a bar graph of a frequency
distribution of quantitative data; the
horizontal axis is a number line.
A frequency polygon is a visual display of
the frequency of each class of
quantitative data that uses straight lines
to connect points plotted above the class
midpoints.

53. Summary (cont.) Quantitative Data (cont.) Type of Graph Definition
Ogive
Stem-and-Leaf Plot
An ogive displays the cumulative
frequency of each class of quantitative
data by using straight lines to connect
points plotted above the upper class
boundaries.
A stem-and-leaf plot retains the original
data; the leaves are the last significant
digit in each data value and the stems
are the remaining digits.
Stem
Leaves 32 33 34

54. Summary (cont.) Quantitative Data (cont.) Type of Graph Definition
Dot Plot
Line Graph
A dot plot retains the original data by
plotting a dot above each data value on
a number line.
A line graph uses straight lines to
connect points plotted at the value of
each measurement above the time it
was taken.