1. 2 Chapter Descriptive Statistics 2012 Pearson Education, Inc.
All rights reserved.
1 of 149

2. Frequency Distributions and Their Graphs
Section 2.1
Frequency Distributions
and Their Graphs
© 2012 Pearson Education, Inc. All rights reserved.
2 of 149

3. Section 2.1 Objectives Construct frequency distributions
Construct frequency histograms, frequency polygons, relative frequency histograms, and ogives
© 2012 Pearson Education, Inc. All rights reserved.
3 of 149

4. Frequency Distribution
A table that shows classes or intervals of data with a count of the number of entries in each class.
The frequency, f, of a class is the number of data entries in the class.
Class
Frequency, f
1 – 5 5
6 – 10 8
11 – 15 6
16 – 20
21 – 25
26 – 30 4
Class width 6 – 1 = 5
Lower class
limits
Upper class
limits
© 2012 Pearson Education, Inc. All rights reserved.
4 of 149

5. Constructing a Frequency Distribution
Decide on the number of classes.
Usually between 5 and 20; otherwise, it may be difficult to detect any patterns.
Find the class width.
Determine the range of the data.
Divide the range by the number of classes.
Round up to the next convenient number.
© 2012 Pearson Education, Inc. All rights reserved.
5 of 149

6. Constructing a Frequency Distribution
Find the class limits.
You can use the minimum data entry as the lower limit of the first class.
Find the remaining lower limits (add the class width to the lower limit of the preceding class).
Find the upper limit of the first class. Remember that classes cannot overlap.
Find the remaining upper class limits.
© 2012 Pearson Education, Inc. All rights reserved.
6 of 149

7. Constructing a Frequency Distribution
Make a tally mark for each data entry in the row of the appropriate class.
Count the tally marks to find the total frequency f for each class.
© 2012 Pearson Education, Inc. All rights reserved.
7 of 149

8. Example: Constructing a Frequency Distribution
The following sample data set lists the prices )in dollars) of 30 portable global positioning system (GPS) navigators. Construct a frequency distribution that has seven classes
© 2012 Pearson Education, Inc. All rights reserved.
8 of 149

9. Solution: Constructing a Frequency Distribution
Number of classes = 7 (given)
Find the class width
Round up to 56
© 2012 Pearson Education, Inc. All rights reserved.
9 of 149

10. Solution: Constructing a Frequency Distribution
Use 59 (minimum value) as first lower limit. Add the class width of 56 to get the lower limit of the next class.
= 115
Find the remaining lower limits.
Lower limit
Upper limit 59
115
171
227
283
339
395
Class width = 56
© 2012 Pearson Education, Inc. All rights reserved.
10 of 149

11. Solution: Constructing a Frequency Distribution
The upper limit of the first class is 114 (one less than the lower limit of the second class). Add the class width of 56 to get the upper limit of the next class = 170 Find the remaining upper limits.
Lower limit
Upper limit 59
114
115
170
171
226
227
282
283
338
339
394
395
450
Class width = 56
© 2012 Pearson Education, Inc. All rights reserved.
11 of 149

12. Solution: Constructing a Frequency Distribution
Make a tally mark for each data entry in the row of the appropriate class.
Count the tally marks to find the total frequency f for each class.
Class
Tally
Frequency, f
59 – 114
IIII 5
115 – 170
IIII III 8
171 – 226
IIII I 6
227 – 282
283 – 338 II 2
339 – 394 I 1
395 – 450
III 3
© 2012 Pearson Education, Inc. All rights reserved.
12 of 149

13. Determining the Midpoint
Midpoint of a class
Class
Midpoint
Frequency, f
59 – 114 5
115 – 170 8
171 – 226 6
Class width = 56
© 2012 Pearson Education, Inc. All rights reserved.
13 of 149

14. Determining the Relative Frequency
Relative Frequency of a class
Portion or percentage of the data that falls in a particular class.
Class
Frequency, f
Relative Frequency
59 – 114 5
115 – 170 8
171 – 226 6
© 2012 Pearson Education, Inc. All rights reserved.
14 of 149

15. Determining the Cumulative Frequency
Cumulative frequency of a class
The sum of the frequency for that class and all previous classes.
Class
Frequency, f
Cumulative frequency
59 – 114 5
115 – 170 8
171 – 226 6 5 + 13 + 19
© 2012 Pearson Education, Inc. All rights reserved.
15 of 149

16. Expanded Frequency Distribution
Class
Frequency, f
Midpoint
Relative
frequency
Cumulative frequency
59 – 114 5
86.5
0.17
115 – 170 8
142.5
0.27 13
171 – 226 6
198.5
0.2 19
227 – 282
254.5 24
283 – 338 2
310.5
0.07 26
339 – 394 1
366.5
0.03 27
395 – 450 3
422.5
0.1 30
Σf = 30
© 2012 Pearson Education, Inc. All rights reserved.
16 of 149

17. Graphs of Frequency Distributions
Frequency Histogram
A bar graph that represents the frequency distribution.
The horizontal scale is quantitative and measures the data values.
The vertical scale measures the frequencies of the classes.
Consecutive bars must touch.
data values
frequency
© 2012 Pearson Education, Inc. All rights reserved.
17 of 149

18. Class Boundaries Class boundaries
The numbers that separate classes without forming gaps between them.
The distance from the upper limit of the first class to the lower limit of the second class is 115 – 114 = 1.
Half this distance is 0.5.
Class
Boundaries
Frequency, f
59 – 114 5
115 – 170 8
171 – 226 6
58.5 – 114.5
First class lower boundary = 59 – 0.5 = 58.5
First class upper boundary = = 114.5
© 2012 Pearson Education, Inc. All rights reserved.
18 of 149

19. Class Boundaries Class Class boundaries Frequency, f 59 – 114
58.5 – 114.5 5
115 – 170
114.5 – 170.5 8
171 – 226
170.5 – 226.5 6
227 – 282
226.5 – 282.5
283 – 338
282.5 – 338.5 2
339 – 394
338.5 – 394.5 1
395 – 450
394.5 – 450.5 3
© 2012 Pearson Education, Inc. All rights reserved.
19 of 149

20. Example: Frequency Histogram
Construct a frequency histogram for the Global Positioning system (GPS) navigators.
Class
Class boundaries
Midpoint
Frequency, f
59 – 114
58.5 – 114.5
86.5 5
115 – 170
114.5 – 170.5
142.5 8
171 – 226
170.5 – 226.5
198.5 6
227 – 282
226.5 – 282.5
254.5
283 – 338
282.5 – 338.5
310.5 2
339 – 394
338.5 – 394.5
366.5 1
395 – 450
394.5 – 450.5
422.5 3
© 2012 Pearson Education, Inc. All rights reserved.
20 of 149

21. Solution: Frequency Histogram (using Midpoints)
© 2012 Pearson Education, Inc. All rights reserved.
21 of 149

22. Solution: Frequency Histogram (using class boundaries)
You can see that more than half of the GPS navigators are priced below $
© 2012 Pearson Education, Inc. All rights reserved.
22 of 149

23. Graphs of Frequency Distributions
Frequency Polygon
A line graph that emphasizes the continuous change in frequencies.
data values
frequency
© 2012 Pearson Education, Inc. All rights reserved.
23 of 149

24. Example: Frequency Polygon
Construct a frequency polygon for the GPS navigators frequency distribution.
Class
Midpoint
Frequency, f
59 – 114
86.5 5
115 – 170
142.5 8
171 – 226
198.5 6
227 – 282
254.5
283 – 338
310.5 2
339 – 394
366.5 1
395 – 450
422.5 3
© 2012 Pearson Education, Inc. All rights reserved.
24 of 149

25. Solution: Frequency Polygon
The graph should begin and end on the horizontal axis, so extend the left side to one class width before the first class midpoint and extend the right side to one class width after the last class midpoint.
You can see that the frequency of GPS navigators increases up to $ and then decreases.
© 2012 Pearson Education, Inc. All rights reserved.
25 of 149

26. Graphs of Frequency Distributions
Relative Frequency Histogram
Has the same shape and the same horizontal scale as the corresponding frequency histogram.
The vertical scale measures the relative frequencies, not frequencies.
data values
relative frequency
© 2012 Pearson Education, Inc. All rights reserved.
26 of 149

27. Example: Relative Frequency Histogram
Construct a relative frequency histogram for the GPS navigators frequency distribution.
Class
Class boundaries
Frequency, f
Relative
frequency
59 – 114
58.5 – 114.5
86.5
0.17
115 – 170
114.5 – 170.5
142.5
0.27
171 – 226
170.5 – 226.5
198.5
0.2
227 – 282
226.5 – 282.5
254.5
283 – 338
282.5 – 338.5
310.5
0.07
339 – 394
338.5 – 394.5
366.5
0.03
395 – 450
394.5 – 450.5
422.5
0.1
© 2012 Pearson Education, Inc. All rights reserved.
27 of 149

28. Solution: Relative Frequency Histogram
From this graph you can see that 20% of GPS navigators are priced between $ and $
© 2012 Pearson Education, Inc. All rights reserved.
28 of 149

29. Graphs of Frequency Distributions
Cumulative Frequency Graph or Ogive
A line graph that displays the cumulative frequency of each class at its upper class boundary.
The upper boundaries are marked on the horizontal axis.
The cumulative frequencies are marked on the vertical axis.
data values
cumulative frequency
© 2012 Pearson Education, Inc. All rights reserved.
29 of 149

30. Constructing an Ogive
Construct a frequency distribution that includes cumulative frequencies as one of the columns.
Specify the horizontal and vertical scales.
The horizontal scale consists of the upper class boundaries.
The vertical scale measures cumulative frequencies.
Plot points that represent the upper class boundaries and their corresponding cumulative frequencies.
© 2012 Pearson Education, Inc. All rights reserved.
30 of 149

31. Constructing an Ogive Connect the points in order from left to right.
The graph should start at the lower boundary of the first class (cumulative frequency is zero) and should end at the upper boundary of the last class (cumulative frequency is equal to the sample size).
© 2012 Pearson Education, Inc. All rights reserved.
31 of 149

32. Example: Ogive
Construct an ogive for the GPS navigators frequency distribution.
Class
Class boundaries
Frequency, f
Cumulative frequency
59 – 114
58.5 – 114.5
86.5 5
115 – 170
114.5 – 170.5
142.5 13
171 – 226
170.5 – 226.5
198.5 19
227 – 282
226.5 – 282.5
254.5 24
283 – 338
282.5 – 338.5
310.5 26
339 – 394
338.5 – 394.5
366.5 27
395 – 450
394.5 – 450.5
422.5 30
© 2012 Pearson Education, Inc. All rights reserved.
32 of 149

33. Solution: Ogive
From the ogive, you can see that about 25 GPS navigators cost $300 or less. The greatest increase occurs between $ and $
© 2012 Pearson Education, Inc. All rights reserved.
33 of 149

34. Section 2.1 Summary Constructed frequency distributions
Constructed frequency histograms, frequency polygons, relative frequency histograms and ogives
© 2012 Pearson Education, Inc. All rights reserved.
34 of 149

35. More Graphs and Displays
Section 2.2
More Graphs and Displays
© 2012 Pearson Education, Inc. All rights reserved.
35 of 149

36. Section 2.2 Objectives
Graph quantitative data using stem-and-leaf plots and dot plots
Graph qualitative data using pie charts and Pareto charts
Graph paired data sets using scatter plots and time series charts
© 2012 Pearson Education, Inc. All rights reserved.
36 of 149

37. Graphing Quantitative Data Sets
Stem-and-leaf plot
Each number is separated into a stem and a leaf.
Similar to a histogram.
Still contains original data values. 26 2 3 4 5
Data: 21, 25, 25, 26, 27, 28, , 36, 36, 45
© 2012 Pearson Education, Inc. All rights reserved.
37 of 149

38. Example: Constructing a Stem-and-Leaf Plot
The following are the numbers of text messages sent last month by the cellular phone users on one floor of a college dormitory. Display the data in a stem-and-leaf plot.
© 2012 Pearson Education, Inc. All rights reserved.
38 of 149

39. Solution: Constructing a Stem-and-Leaf Plot
The data entries go from a low of 78 to a high of 159.
Use the rightmost digit as the leaf.
For instance,
78 = 7 | and = 15 | 9
List the stems, 7 to 15, to the left of a vertical line.
For each data entry, list a leaf to the right of its stem.
© 2012 Pearson Education, Inc. All rights reserved.
39 of 149

40. Solution: Constructing a Stem-and-Leaf Plot
Include a key to identify the values of the data.
From the display, you can conclude that more than 50% of the cellular phone users sent between 110 and 130 text messages.
© 2012 Pearson Education, Inc. All rights reserved.
40 of 149

41. Graphing Quantitative Data Sets
Dot plot
Each data entry is plotted, using a point, above a horizontal axis
Data: 21, 25, 25, 26, 27, 28, 30, 36, 36, 45 26
© 2012 Pearson Education, Inc. All rights reserved.
41 of 149

42. Example: Constructing a Dot Plot
Use a dot plot organize the text messaging data.
So that each data entry is included in the dot plot, the horizontal axis should include numbers between 70 and 160.
To represent a data entry, plot a point above the entry's position on the axis.
If an entry is repeated, plot another point above the previous point.
© 2012 Pearson Education, Inc. All rights reserved.
42 of 149

43. Solution: Constructing a Dot Plot
From the dot plot, you can see that most values cluster between 105 and 148 and the value that occurs the most is 126. You can also see that 78 is an unusual data value.
© 2012 Pearson Education, Inc. All rights reserved.
43 of 149

44. Graphing Qualitative Data Sets
Pie Chart
A circle is divided into sectors that represent categories.
The area of each sector is proportional to the frequency of each category.
© 2012 Pearson Education, Inc. All rights reserved.
44 of 149

45. Example: Constructing a Pie Chart
The numbers of earned degrees conferred (in thousands) in 2007 are shown in the table. Use a pie chart to organize the data. (Source: U.S. National Center for Educational Statistics)
Type of degree
Number
(thousands)
Associate’s
728
Bachelor’s
1525
Master’s
604
First professional 90
Doctoral 60
© 2012 Pearson Education, Inc. All rights reserved.
45 of 149

46. Solution: Constructing a Pie Chart
Find the relative frequency (percent) of each category.
Type of degree
Frequency, f
Relative frequency
Associate’s
728
Bachelor’s
1525
Master’s
604
First professional 90
Doctoral 60
3007
© 2012 Pearson Education, Inc. All rights reserved.
46 of 149

47. Solution: Constructing a Pie Chart
Construct the pie chart using the central angle that corresponds to each category.
To find the central angle, multiply 360º by the category's relative frequency.
For example, the central angle for cars is
360(0.24) ≈ 86º
© 2012 Pearson Education, Inc. All rights reserved.
47 of 149

48. Solution: Constructing a Pie Chart
Type of degree
Frequency, f
Relative frequency
Central angle
Associate’s
728
0.24
Bachelor’s
1525
0.51
Master’s
604
0.20
First professional 90
0.03
Doctoral 60
0.02
360º(0.02)≈7º
360º(0.24)≈86º
360º(0.51)≈184º
360º(0.20)≈72º
360º(0.03)≈11º
© 2012 Pearson Education, Inc. All rights reserved.
48 of 149

49. Solution: Constructing a Pie Chart
Type of degree
Relative frequency
Central angle
Associate’s
0.24
86º
Bachelor’s
0.51
184º
Master’s
0.20
72º
First professional
0.03
11º
Doctoral
0.02 7º
From the pie chart, you can see that most fatalities in motor vehicle crashes were those involving the occupants of cars.
© 2012 Pearson Education, Inc. All rights reserved.
49 of 149

50. Graphing Qualitative Data Sets
Pareto Chart
A vertical bar graph in which the height of each bar represents frequency or relative frequency.
The bars are positioned in order of decreasing height, with the tallest bar positioned at the left.
Frequency
Categories
© 2012 Pearson Education, Inc. All rights reserved.
50 of 149

51. Example: Constructing a Pareto Chart
In a recent year, the retail industry lost $36.5 billion in inventory shrinkage. Inventory shrinkage is the loss of inventory through breakage, pilferage, shoplifting, and so on. The causes of the inventory shrinkage are administrative error ($5.4 billion), employee theft ($15.9 billion), shoplifting ($12.7 billion), and vendor fraud ($1.4 billion). Use a Pareto chart to organize this data. (Source: National Retail Federation and Center for Retailing Education, University of Florida)
© 2012 Pearson Education, Inc. All rights reserved.
51 of 149

52. Solution: Constructing a Pareto Chart
Cause
$ (billion)
Admin. error
5.4
Employee theft
15.9
Shoplifting
12.7
Vendor fraud
1.4
From the graph, it is easy to see that the causes of inventory shrinkage that should be addressed first are employee theft and shoplifting.
© 2012 Pearson Education, Inc. All rights reserved.
52 of 149

53. Graphing Paired Data Sets
Each entry in one data set corresponds to one entry in a second data set.
Graph using a scatter plot.
The ordered pairs are graphed as points in a coordinate plane.
Used to show the relationship between two quantitative variables. y x
© 2012 Pearson Education, Inc. All rights reserved.
53 of 149

54. Example: Interpreting a Scatter Plot
The British statistician Ronald Fisher introduced a famous data set called Fisher's Iris data set. This data set describes various physical characteristics, such as petal length and petal width (in millimeters), for three species of iris. The petal lengths form the first data set and the petal widths form the second data set. (Source: Fisher, R. A., 1936)
© 2012 Pearson Education, Inc. All rights reserved.
54 of 149

55. Example: Interpreting a Scatter Plot
As the petal length increases, what tends to happen to the petal width?
Each point in the scatter plot represents the
petal length and petal width of one flower.
© 2012 Pearson Education, Inc. All rights reserved.
55 of 149

56. Solution: Interpreting a Scatter Plot
Interpretation
From the scatter plot, you can see that as the petal length increases, the petal width also tends to increase.
A complete discussion of types of correlation occurs in chapter 9. You may want, however, to discuss positive correlation, negative correlation, and no correlation at this point.
Be sure that students do not confuse correlation with causation.
© 2012 Pearson Education, Inc. All rights reserved.
56 of 149

57. Graphing Paired Data Sets
Time Series
Data set is composed of quantitative entries taken at regular intervals over a period of time.
e.g., The amount of precipitation measured each day for one month.
Use a time series chart to graph.
time
Quantitative data
© 2012 Pearson Education, Inc. All rights reserved.
57 of 149

58. Example: Constructing a Time Series Chart
The table lists the number of cellular telephone subscribers (in millions) for the years 1998 through Construct a time series chart for the number of cellular subscribers. (Source: Cellular Telecommunication & Internet Association)
© 2012 Pearson Education, Inc. All rights reserved.
58 of 149

59. Solution: Constructing a Time Series Chart
Let the horizontal axis represent the years.
Let the vertical axis represent the number of subscribers (in millions).
Plot the paired data and connect them with line segments.
© 2012 Pearson Education, Inc. All rights reserved.
59 of 149

60. Solution: Constructing a Time Series Chart
The graph shows that the number of subscribers has been increasing since 1998, with greater increases recently.
© 2012 Pearson Education, Inc. All rights reserved.
60 of 149

61. Section 2.2 Summary
Graphed quantitative data using stem-and-leaf plots and dot plots
Graphed qualitative data using pie charts and Pareto charts
Graphed paired data sets using scatter plots and time series charts
© 2012 Pearson Education, Inc. All rights reserved.
61 of 149