More Graphs and Displays

Section 2.2

StatisticsMrs. SpitzFall 2008

Larson/Farber Ch 2

Objectives:

§ How to graph quantitative data sets using stem-and-leaf plots and dot plots

§ How to graph qualitative data using pie charts and Pareto charts

§ How to graph paired data sets using scatter plots and time series charts.

Assignment: pp. 51-54 #1-30

Larson/Farber Ch 2

Graphing Quantitative Data Sets – don’t write this down.

§ In Section 2.1, you learned several traditional ways to display quantitative data graphically. In this section, you will learn a newer way to display quantitative data called a stem-and-leaf plot. Stem-and-leaf plots are examples of exploratory data analysis (EDA), which was developed by John Tukey in 1977.

Larson/Farber Ch 2

Graphing Quantitative Data Sets – don’t write this down.

§ In a stem and leaf plot, each number is separated into a stem (the entry’s leftmost digits) and a leaf (the rightmost digit). A stem-and-leaf plot is similar to a histogram but has the advantage that the graph still contains the original data values. Another advantage is that it provides an easy way to sort data.

Larson/Farber Ch 2

Stem-and-Leaf Plot

6 | 7 | 8 | 9 |10 |11 | 12 |

Lowest value is 67 and highest value is 125, so list stems from 6 to 12.

102 124 108 86 103 82

2

4

8

6

3

2

Stem LeafTo see complete display, go to next slide.

Divide each data value into a stem and a leaf. The leaf is the rightmost significant digit. The stem consists of the digits to the left. The data shown represent the first line of the ‘minutes on phone’ data used earlier. The complete stem and leaf will be shown on the next slide.

Larson/Farber Ch 2

6 | 7

7 | 1 8

8 | 2 5 6 7 7

9 | 2 5 7 9 9

10 | 0 1 2 3 3 4 5 5 7 8 9

11 | 2 6 8

12 | 2 4 5

Stem-and-Leaf Plot

Key: 6 | 7 means 67

It is very important to use a key to explain the plot. 6|7 could mean 6700 or .067 for a different problem. A stem and leaf should not be used with data when values are very different such as 3, 34,900, 24 etc. The stem-and leaf has the advantage over a histogram of retaining the original values.

Larson/Farber Ch 2

Stem-and-Leaf with two lines per stem

6 | 7 7 | 1

7 | 8 8 | 2 8 | 5 6 7 7 9 | 2 9 | 5 7 9 9 10 | 0 1 2 3 3

4 10 | 5 5 7 8 9 11 | 2 11 | 6 8 12 | 2 4 12 | 5

Key: 6 | 7 means 67

1st line digits 0 1 2 3 4

2nd line digits 5 6 7 8 9

1st line digits 0 1 2 3 4

2nd line digits 5 6 7 8 9 With two lines per stem the data is more finely “chopped”. Class width is 5 times the leaf unit. All stems except possibly the first and last must have two lines even if one is blank. For this data set, the first line for the stem 6 can be blank because there are no data values from 60 to 64.

Larson/Farber Ch 2

Dot Plots

§ You can also use a dot plot to graph quantitative data. In a dot plot, each data entry is plotted, using a point, above a horizontal axis. Like a stem-and-leaf plot, a dot plot allows you to see how data are distributed and determine specific data entries.

Larson/Farber Ch 2

Dot Plot

66 76 86 96 106 116 126

Phone

minutes

Dot plots also allow you to retain original values.

Larson/Farber Ch 2

Graphing Qualitative Data Sets

§ Pie charts provide a convenient way to present qualitative data graphically. A pie chart is a circle graph that shows relationships of parts to a whole.

Larson/Farber Ch 2

NASA budget (billions of $) divided among 3 categories.

Pie Chart• Used to describe parts of a whole• Central Angle for each segment

Construct a pie chart for the data.

Human Space Flight 5.7Technology 5.9Mission Support 2.7

Billions of $

Pie charts help visualize the relative proportion of each category. Find the relative frequency for each category and multiply it by 360 degrees to find the central angle.

Larson/Farber Ch 2

NASA Budget(Billions of $)

HumanSpace Flight

40%

Total

Pie ChartBillions of $

Human Space Flight 5.7Technology 5.9Mission Support 2.7

14.3

Degrees

143149 68

360MissionSupport

19%

Technology41%

Larson/Farber Ch 2

You will need a protractor and a compass.

1. Take the part and divide it by the whole, and you will get a decimal. Multiply by 360 and you will get the number of degrees in that part of the circle. Do this until you have 360 degrees.

2. Draw a circle. Starting from the middle of the circle, draw a line from the center of the circle to the side of the circle. This is your starting point for 0 degrees.

3. Use your protractor to measure the number of degrees required, mark and draw another line from the center to the edge.

4. Start at the edge of the next to begin measuring your next cut in the pie. Continue until you are done.

5. Yes, you have to do a few by hand so you get the idea. Later, we will use Excel to create pie charts.

Larson/Farber Ch 2

Pareto Chart

§ Another way to graph qualitative data is to use a Pareto chart. A Pareto chart is 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. Such positioning helps highlight important data and is used frequently in business.

Larson/Farber Ch 2

Example Pareto Chart

§ Last year, the retail industry lost $40.9 million in inventory shrinkage. The causes of the inventory shrinkage are administrative error ($7.8 million), employee theft ($15.6 million), shoplifting ($14.7 million), and vendor fraud ($2.9 million). If you were a retailer, which causes of inventory shrinkage would you address first? Construct a Pareto chart to show which causes would be addressed first.

Larson/Farber Ch 2

You can use Excel to make a Pareto Chart

Causes of Inventory Shrinkage

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

Employee theft Shoplifting Admin Error Vendor Fraud

Cause

Mill

ion

s o

f D

olla

rs

Larson/Farber Ch 2

Graphing Paired Data Sets

§ If two data sets have the same number of entries, and each entry in the first data set corresponds to one entry in the second data set, the sets are called paired data sets. For instance, suppose, a data set contains the costs of an item and a second data set contains the sales amounts for the item at each cost. Because each cost corresponds to a sales amount, the data sets are paired. One way to graph paired data sets is to use a scatter plot, where the ordered pairs are graphed as points in a coordinate plane.

Larson/Farber Ch 2

Scatter PlotAbsences Grade

0 2 4 6 8 10 12 14 16

404550556065707580859095

Absences (x)

x825

121596

y78929058437481

Finalgrade

(y)

Larson/Farber Ch 2

Time Series

§ A data set that is composed of entries taken at regular intervals over a period of time is a time series. For instance, the amount of precipitation measured each day for one month is an example of a time series. You can use a time series chart to graph a time series.

Larson/Farber Ch 2

Constructing a Time Series Chart

§ The table lists a number of cellular telephone subscribers in millions, and a subscriber’s average local monthly bill for service, in dollars, for the years 1987 through 1996. Construct a time series chart for the number of cellular subscribers. What can you conclude?

Year Subscribers Average bill

1987 1.2 96.83

1988 2.1 98.02

1989 3.5 89.30

1990 5.3 80.90

1991 7.6 72.74

1992 11.0 68.68

1993 16.0 61.48

1994 24.1 56.21

1995 33.8 51.00

1996 44.0 47.70

Larson/Farber Ch 2

Constructing a Time Series Chart

§ Let the horizontal axis represent the years and the vertical axis represent the number of subscribers in millions. Then plot the paired data. From the graph, you can see the number of subscribers has been increasing since 1987. Recent years show greater increases.

Cellular Phone Subscribers (1987-1996)

0

5

10

15

20

25

30

35

40

45

50

1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997

Year

Su

bsc

irb

ers

(in

Mill

ion

s)