Section 2.2 Notes: More Graphs and Displays

Chapter 2: Descriptive Statistics - Section 2.2: More Graphs and Displays

Objectives

• Learn how to graph and interpret quantitative data using stem-and-leaf plots and dot plots.
• Learn how to graph and interpret qualitative data using pie charts and Pareto charts.
• Learn how to graph and interpret paired data sets using scatter plots and time series charts.

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.
• Provides an easy way to sort data.

Example:
Data: 21, 25, 25, 26, 27, 28, 30, 36, 36, 45

Constructing a Stem-and-Leaf Plot: Example

A data set lists the numbers of text messages sent in one day by 50 cell phone users. Display the data in a stem-and-leaf plot and describe any patterns.

• Data entries range from 17 to 148.
• Use the rightmost digit as the leaf (e.g., 76 = 7 | 6 and 104 = 10 | 4).
• List the stems (7 to 14) to the left of a vertical line.
• For each data entry, list a leaf to the right of its stem.
• More than 50% of users sent between 20 and 50 text messages.

Constructing Variations of Stem-and-Leaf Plots: Example

Organize the data set using a stem-and-leaf plot with two rows for each stem and describe any patterns.

• List each stem twice.
• Use leaves 0-4 in the first stem row and leaves 5-9 in the second stem row.
• Using two rows per stem provides a more detailed picture of the data.
• Most users sent between 20 and 80 text messages.

Dot Plot

• Each data entry is plotted, using a point, above a horizontal axis.

Example:
Data: 21, 25, 25, 26, 27, 28, 30, 36, 36, 45

(A dot plot would visually represent each number with a dot above the corresponding value on the horizontal axis.)

Constructing a Dot Plot: Example

Use a dot plot to organize the data set from the previous example and describe any patterns.

• Most entries occur between 20 and 80.
• Only 4 people sent more than 100 text messages.
• 148 is an unusual data entry.

Technology (e.g., Minitab, StatCrunch) can be used to construct dot plots.

Graphing Qualitative Data Sets

Pie Chart

• Provides a convenient way to present qualitative data graphically as percents of a whole.
• A circle is divided into sectors that represent categories.
• The area of each sector is proportional to the frequency of each category.

Constructing a Pie Chart: Example

The numbers of earned degrees conferred (in thousands) in 2019 are shown. Use a pie chart to organize the data.

• Construct the pie chart using the central angle that corresponds to each category.
• To find the central angle, multiply 360^\circ by the category's relative frequency.
• For example, the central angle for associate’s degree is 360^\circ (0.255) \approx 91.8^\circ.
• Find the relative frequency (percent) of each category.
• Almost one-half of the degrees conferred in 2019 were bachelor’s degrees.

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.

Constructing a Pareto Chart: Example

In 2019, these were the leading causes of death in the United States:

• Accidents: 173,040
• Cancer: 599,601
• Chronic lower respiratory disease: 156,979
• Heart disease: 659,041
• Stroke (cerebrovascular diseases): 150,005

Use a Pareto chart to organize the data. What was the leading cause of death in the United States in 2019?

• Heart disease was the leading cause of death in the United States in 2019.
• Heart disease and cancer caused more deaths than the other three causes combined.

Graphing Paired Data Sets

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.

Interpreting a Scatter Plot: Example

The British statistician Ronald Fisher introduced 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.

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.
• As the petal length increases, the petal width also tends to increase.

Time Series

• Data set is composed of quantitative entries taken at regular intervals over a period of time (e.g., precipitation measured each day for one month).
• Use a time series chart to graph.

Constructing a Time Series Chart: Example

The table lists the number of motor vehicle thefts (in millions) and burglaries (in millions) in the United States for the years 2009 through 2019. Construct a time series chart for the number of motor vehicle thefts. Describe any trends.

• Let the horizontal axis represent the years and let the vertical axis represent the number of motor vehicle thefts (in millions).
• Then plot the paired data and connect them with line segments.
• The time series chart shows that the number of burglaries remained the same until 2011 and then decreased through 2019.