Descriptive Statistics: Graphical Methods for Quantitative and Qualitative Data

Descriptive Statistics: Graphical Methods

Section 2.2: More Graphs and Displays

This section introduces graphical techniques for summarizing and interpreting both quantitative and qualitative data. Understanding these methods is essential for effective data analysis and communication in statistics.

Objectives

• How to graph and interpret quantitative data using stem-and-leaf plots and dot plots.

• How to graph and interpret qualitative data using pie charts and Pareto charts.

Graphing Quantitative Data Sets

Stem-and-Leaf Plot

A stem-and-leaf plot is a method for organizing quantitative data by splitting each value into a "stem" (all but the final digit) and a "leaf" (the final digit). This plot retains the original data values and provides a quick way to sort and visualize small data sets.

• Stem: All digits except the last one.

• Leaf: The last digit of the number.

• Useful for small sets of quantitative data.

• Impractical for large data sets.

Example: For the number 124, the stem is 12 and the leaf is 4.

Example: Constructing a Stem-and-Leaf Plot

Given the following data set (number of text messages sent in one day by 50 users):

75	49	104	59	88
123	75	109	68	81
66	80	80	69	51
42	84	18	52	25
24	36	41	25	25
32	23	20	17	49
33	29	21	39	35
38	36	54	30	146

The stem-and-leaf plot organizes these values by tens (stems) and units (leaves). For example, the row for stem '2' (20s) might look like: 0, 1, 3, 4, 5, 5, 5, 9.

Interpretation: More than 50% of users sent between 20 and 50 messages.

Dot Plot

A dot plot displays each data value as a dot above a number line. It is especially useful for small to moderate-sized data sets and for visualizing the distribution and frequency of individual values.

• Each dot represents one observation.

• Clusters, gaps, and outliers are easily identified.

Example: For the data set 21, 25, 25, 26, 27, 28, 30, 36, 36, 45, the dot plot shows two dots above 25, one above 26, etc.

Graphing Qualitative Data Sets

Pie Chart

A pie chart is a circular graph divided into sectors, each representing a category's proportion of the whole. It is ideal for displaying the relative frequencies of qualitative data.

• Each sector's angle is proportional to the category's frequency.

• Useful for showing parts of a whole.

Example: Earned degrees conferred in 2019:

Type of degree	Number (in thousands)
Associate's	1037
Bachelor's	2013
Master's	834
Doctoral	188

To construct the pie chart, calculate the relative frequency and corresponding angle for each category:

Type of degree	f	Relative frequency	Angle
Associate's	1037	0.255	91.8°
Bachelor's	2013	0.494	177.8°
Master's	834	0.205	73.8°
Doctoral	188	0.046	16.6°

Formula: To find the angle for each sector: $\text{Angle} = 360^\circ \times \text{Relative Frequency}$

Bar Graph

A bar graph uses bars to represent the frequency or relative frequency of categories. Bars can be arranged in any order, and there is always a space between bars to emphasize the categorical nature of the data.

• x-axis: Qualitative categories

• y-axis: Frequency or relative frequency

• Bars are separated by spaces

Pareto Chart

A Pareto chart is a special type of bar graph where categories are ordered from highest to lowest frequency. It is particularly useful for identifying the most significant factors in a data set.

• Bars are arranged in descending order of frequency.

• Helps prioritize categories by importance.

Example: Leading causes of death in the United States (2019):

Cause	Number of deaths
Heart disease	659,041
Cancer	599,601
Accidents	173,040
Chronic lower respiratory disease	156,979
Stroke	150,005

The Pareto chart visually shows that heart disease and cancer are the leading causes, accounting for more deaths than the other three causes combined.

Summary Table: Graph Types and Their Uses

Graph Type	Data Type	Main Purpose
Stem-and-leaf plot	Quantitative	Sort and display small data sets, retain original values
Dot plot	Quantitative	Show frequency and distribution of individual values
Pie chart	Qualitative	Show parts of a whole as percentages
Bar graph	Qualitative	Compare frequencies of categories
Pareto chart	Qualitative	Highlight most significant categories in descending order

Additional info: StatCrunch is a statistical software tool that can be used to construct these graphs efficiently, as shown in the examples.