Exploring Data with Tables and Graphs: Elementary Statistics Study Notes

Chapter 2: Exploring Data with Tables and Graphs

Introduction

This chapter introduces essential graphical and tabular methods for organizing, summarizing, and interpreting statistical data. Understanding these visual tools is crucial for effective data analysis and communication in statistics.

Frequency Distributions

Definition and Purpose

• Frequency distribution is a table that displays the number of data values (frequency) within specific intervals or categories.

• Helps organize large data sets and reveals patterns such as central tendency, spread, and shape.

Graphs that Enlighten: Dotplots

Dotplots

• A dotplot is a graph of quantitative data in which each data value is plotted as a point (dot) above a horizontal scale of values.

• Dots representing equal values are stacked vertically.

Features of a Dotplot

• Displays the shape of the distribution of data.

• It is usually possible to recreate the original list of data values from the plot.

Example

A dotplot of pulse rates for males shows the distribution and frequency of different pulse rate values.

Graphs that Enlighten: Stemplots

Stemplots (Stem-and-Leaf Plots)

• A stemplot (or stem-and-leaf plot) represents quantitative data by separating each value into two parts: the stem (such as the leftmost digit) and the leaf (such as the rightmost digit).

• Allows for quick visualization of data distribution and retention of original data values.

Features of a Stemplot

• Shows the shape of the distribution of the data.

• Retains the original data values.

• The sample data are sorted (arranged in order).

Example

A stemplot of pulse rates might have stems for tens digits and leaves for units digits, e.g., stem '6' with leaves '1 0 0 0 0 0 2 2 2 2 2 2 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 9 0 9 8 0'.

Time-Series Graphs

Definition and Application

• A time-series graph is a graph of time-series data, which are quantitative values collected at different points in time (e.g., monthly or yearly).

• Used to analyze trends, cycles, and patterns over time.

Feature of a Time-Series Graph

• Reveals information about trends over time, such as increases, decreases, or periodic fluctuations.

Example

A time-series graph of law enforcement fatalities from 1985 to 2015 shows how the number changes year by year.

Bar Graphs

Definition and Features

• A bar graph uses bars of equal width to show frequencies of categories of categorical (qualitative) data.

• Bars may or may not be separated by small gaps.

Feature of a Bar Graph

• Shows the relative distribution of categorical data, making it easier to compare different categories.

Pareto Charts

Definition and Features

• A Pareto chart is a bar graph for categorical data, with bars arranged in descending order according to frequencies.

• Bars decrease in height from left to right, highlighting the most significant categories.

Features of a Pareto Chart

• Shows the relative distribution of categorical data for easy comparison.

• Draws attention to the more important categories.

Example Table: Causes of Fatal Plane Crashes (Pareto Chart)

Cause	Frequency
Pilot Error	Highest
Mechanical Sabotage	Medium
Weather	Lower
Other	Lowest

Pie Charts

Definition and Features

• A pie chart depicts categorical data as slices of a circle, with the size of each slice proportional to the frequency count for the category.

Feature of a Pie Chart

• Shows the distribution of categorical data in a commonly used, visually intuitive format.

Example Table: Causes of Fatal Plane Crashes (Pie Chart)

Cause	Proportion
Pilot Error	Largest slice
Mechanical Sabotage	Medium slice
Weather	Smaller slice
Other	Smallest slice

Frequency Polygons

Definition and Features

• A frequency polygon is a graph using line segments connected to points located directly above class midpoint values.

• Similar to a histogram, but uses line segments instead of bars.

Relative Frequency Polygon

• A variation that uses relative frequencies (proportions or percentages) for the vertical scale.

Example Table: Frequency Polygon of Commute Times

Commute Time (min)	Frequency
0-10	2
10-20	5
20-30	7
30-40	3

Additional info: Table values inferred for illustration.

Graphs That Deceive

Nonzero Vertical Axis

• Using a vertical scale that starts at a value greater than zero can exaggerate differences between groups.

• Always examine a graph carefully to see whether the vertical axis begins at zero; otherwise, differences may be misleading.

Pictographs

• Pictographs use drawings of objects to represent data, which can be misleading if the data are one-dimensional but depicted with two- or three-dimensional objects.

• Doubling the sides of a square increases its area by a factor of four, not two; doubling the sides of a cube increases its volume by a factor of eight, not two.

• Such representations can grossly distort differences in the data.

Example Table: Pictograph of NSA Collected Phone Records

Year	Records Collected (millions)
Year 1	151
Year 2	534

Best Practices for Graphical Display

Principles for Effective Graphs

• For small data sets (20 values or fewer), use a table instead of a graph.

• A graph should focus on the true nature of the data, not on distracting design features.

• Do not distort data; construct graphs to reveal the true nature of the data.

• Most of the ink in a graph should be used for the data, not for other design elements.

Additional info: Principles adapted from Edward Tufte's guidelines for data visualization.

Summary Table: Types of Graphs and Their Uses

Graph Type	Data Type	Main Use
Dotplot	Quantitative	Distribution shape, individual values
Stemplot	Quantitative	Distribution shape, retains data
Time-Series Graph	Quantitative (over time)	Trends and patterns
Bar Graph	Categorical	Compare categories
Pareto Chart	Categorical	Highlight most important categories
Pie Chart	Categorical	Show proportions
Frequency Polygon	Quantitative	Distribution shape

Key Formulas

Relative Frequency

• Relative frequency of a class: $\text{Relative Frequency} = \frac{\text{Frequency of the class}}{\text{Total frequency}}$

Class Midpoint

• Class midpoint: $\text{Class Midpoint} = \frac{\text{Lower class limit} + \text{Upper class limit}}{2}$

Conclusion

Effective use of tables and graphs is fundamental in statistics for summarizing, analyzing, and communicating data. Understanding both enlightening and deceptive graphical techniques is essential for accurate interpretation and presentation of statistical information.