Descriptive Statistics: Graphical Representation of Data

Descriptive Statistics

Overview

Descriptive statistics involves methods for organizing, displaying, and summarizing data. This section focuses on graphical techniques for representing both quantitative and qualitative data, as well as paired data sets.

• Quantitative data: Numerical values that can be measured or counted.

• Qualitative data: Categorical data representing characteristics or attributes.

• Paired data: Two related sets of quantitative data, often analyzed together.

Graphing Quantitative Data Sets

Stem-and-Leaf Plots

A stem-and-leaf plot is a method for displaying quantitative data where each number is separated into a stem (all but the final digit) and a leaf (the final digit). This plot is similar to a histogram but retains the original data values, making it useful for sorting and identifying patterns.

• Stem: Represents the leading digit(s) of each data value.

• Leaf: Represents the last digit of each data value.

• Provides a quick visual of data distribution and retains actual data values.

Example: For the data set: 21, 25, 25, 26, 27, 28, 30, 36, 36, 45, the stem-and-leaf plot would be:

Stem	Leaf
2	1 5 5 6 7 8
3	0 6 6
4	5

Additional info: The plot shows clustering of values in the 20s and 30s.

Constructing a Stem-and-Leaf Plot: Step-by-Step

• Identify the stems (all but the last digit) and leaves (last digit) for each data value.

• List stems in a vertical column and write each leaf to the right of its stem.

• Example data set (number of text messages sent):

Number of Text Messages Sent
49, 104, 59, 88
75, 109, 68, 81
80, 78, 69, 55
114, 98, 73, 18
84, 46, 52, 25
26, 33, 25, 20
24, 43, 17, 49
32, 29, 29, 40
33, 30, 41, 35
36, 54, 30, 148

After organizing, the stem-and-leaf plot reveals that more than 50% of users sent between 20 and 50 messages.

Variations of Stem-and-Leaf Plots

Sometimes, each stem is split into two rows to provide more detail. For example, one row for leaves 0-4 and another for leaves 5-9. This helps to better visualize data distribution within each stem.

• First row: leaves 0-4

• Second row: leaves 5-9

Additional info: This method is useful for larger data sets with many values per stem.

Dot Plots

A dot plot is a simple way to display quantitative data. Each data value is represented by a dot above a number line. Multiple dots stacked above a value indicate frequency.

• Easy to construct and interpret for small data sets.

• Shows clusters, gaps, and outliers.

Example: For the data set above, most entries occur between 20 and 80, with only a few above 100, indicating 148 is an outlier.

Graphing Qualitative Data Sets

Pie Charts

A pie chart visually represents categorical data as sectors of a circle, with each sector's area proportional to the category's frequency or percentage of the whole.

• Useful for showing relative proportions of categories.

• Each sector's central angle is calculated as $360^{\circ} \times \text{relative frequency}$.

Example: Degrees conferred in 2019:

Type of Degree	Number (thousands)
Associate's	1037
Bachelor's	2013

Central angle for associate's degree: $360^{\circ} \times 0.255 = 91.8^{\circ}$

Additional info: Pie charts are best for data with few categories.

Pareto Charts

A Pareto chart is a vertical bar graph where bars represent frequencies or relative frequencies of categories, arranged in decreasing order from left to right. The tallest bar is on the left.

• Highlights the most significant categories.

• Useful for identifying major contributors in categorical data.

Example: Leading causes of death in the U.S. (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 chart shows heart disease as the leading cause, followed by cancer.

Graphing Paired Data Sets

Scatter Plots

A scatter plot displays paired quantitative data as points on a coordinate plane, with each point representing an ordered pair. Scatter plots are used to examine relationships between two variables.

• Each axis represents one variable.

• Patterns may indicate correlation (positive, negative, or none).

Example: Fisher's Iris data set plots petal length vs. petal width. As petal length increases, petal width tends to increase, indicating a positive relationship.

Time Series Charts

A time series chart graphs quantitative data collected at regular intervals over time. The horizontal axis represents time, and the vertical axis represents the measured variable.

• Useful for identifying trends, cycles, and patterns over time.

• Data points are connected by line segments.

Example: Number of motor vehicle thefts and burglaries in the U.S. from 2009 to 2019. The chart shows burglaries remained steady until 2011, then decreased.

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 distribution
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 and changes over time