Descriptive Statistics: Graphical Methods for Data Representation

Descriptive Statistics: Graphical Methods for Data Representation

Section 2.2: More Graphs and Displays

This section introduces various graphical techniques for representing both quantitative and qualitative data. These methods help in visualizing data distributions, identifying patterns, and interpreting relationships between variables.

Graphing Quantitative Data Sets

Stem-and-Leaf Plot

• Definition: A stem-and-leaf plot separates each data value into a stem (all but the final digit) and a leaf (the final digit).

• Purpose: Similar to a histogram, but retains original data values and provides an easy way to sort data.

• Construction Steps:

  1. Identify the stems (e.g., tens place) and list them vertically.

  2. For each data entry, write the leaf (units place) to the right of its stem.

• Example: For the data set of text messages sent by 50 users, stems range from 1 to 14, and leaves are the rightmost digits.

Table: Example Stem-and-Leaf Plot

Stem	Leaves
1	6 9
2	0 2 3 3 3 3 4 6 6 6 8 9 9 9 9
3	0 0 1 1 3 3 4 5 6 6 8 9
4	1 3 5 8
5	2 3 8
6	6 7 8 9
7	6 6 8 9
8	0 0 6 7 8 9
9	2 9
10	2
11	5 9
12	2 6
13	3
14	9

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

Variations of Stem-and-Leaf Plots

• Each stem can be split into two rows: one for leaves 0-4, another for leaves 5-9, providing more detail.

• Example: Most users sent between 20 and 80 messages.

Dot Plot

• Definition: Each data entry is represented by a dot above a horizontal axis at its value.

• Purpose: Useful for small data sets and for identifying clusters, gaps, and outliers.

• Example: In the text message data, most entries are between 20 and 80; 149 is an outlier.

Graphing Qualitative Data Sets

Pie Chart

• Definition: A circular chart divided into sectors, each representing a category's proportion of the whole.

• Construction:

  1. Calculate the relative frequency for each category.

  2. Multiply each relative frequency by 360° to find the central angle for each sector.

• Example: Degrees conferred in 2014 (in thousands):

Type of Degree	Number (thousands)	Relative Frequency	Angle
Associate's	1003	0.264	95°
Bachelor's	1870	0.491	177°
Master's	754	0.198	71°
Doctoral	178	0.047	17°

• Interpretation: Nearly half of degrees conferred were bachelor's degrees.

Pareto Chart

• Definition: A vertical bar graph with bars ordered by decreasing height, representing frequency or relative frequency of categories.

• Purpose: Highlights the most significant categories.

• Example: Leading causes of death in the U.S. (2014):

Cause	Deaths
Heart disease	614,348
Cancer	591,699
Chronic lower respiratory disease	147,101
Accidents	136,053
Stroke	133,103

• Interpretation: Heart disease was the leading cause of death; heart disease and cancer together caused more deaths than the other three causes combined.

Graphing Paired Data Sets

Scatter Plot

• Definition: Plots ordered pairs as points in a coordinate plane to show the relationship between two quantitative variables.

• Example: Fisher's Iris data set: petal length vs. petal width for three iris species.

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

Time Series Chart

• Definition: Plots quantitative data collected at regular intervals over time, connecting points with line segments.

• Example: Motor vehicle thefts in the U.S. (2005-2015):

Year	Motor Vehicle Thefts (millions)	Burglaries (millions)
2005	1.24	2.16
2006	1.20	2.19
2007	1.10	2.19
2008	0.96	2.22
2009	0.88	2.20
2010	0.74	2.15
2011	0.72	2.11
2012	0.72	2.09
2013	0.70	1.93
2014	0.66	1.71
2015	0.71	1.58

• Interpretation: Motor vehicle thefts decreased until 2011, then remained stable through 2015.

Key Formulas

• Central Angle for Pie Chart: $\text{Central Angle} = 360^\circ \times \text{Relative Frequency}$

Summary Table: Graph Types and Their Uses

Graph Type	Data Type	Main Purpose
Stem-and-Leaf Plot	Quantitative	Sort and display original data values
Dot Plot	Quantitative	Show frequency and identify outliers
Pie Chart	Qualitative	Show proportions of categories
Pareto Chart	Qualitative	Highlight most significant categories
Scatter Plot	Paired Quantitative	Show relationship between two variables
Time Series Chart	Quantitative over time	Show trends over time

Additional info: These graphical methods are foundational for exploring and interpreting data in statistics, aiding in both descriptive analysis and preparation for further inferential techniques.