Chapter 2: Summarizing Data in Tables and Graphs

2.1 Organizing Qualitative Data

Qualitative data, also known as categorical data, must be organized to be useful for analysis. This section covers methods for organizing such data using tables and graphical displays.

• Raw Data: Data collected from surveys or experiments before any organization.

• Ways to Organize Data: Tables, graphs, and numerical summaries.

2.1.1 Organize Qualitative Data in Tables

A frequency distribution lists each category of data and the number of occurrences for each category.

• Frequency Table: A table that displays the frequency (count) of each category.

• Relative Frequency: The proportion or percentage of observations within a category, calculated as:

• Relative Frequency Distribution: Lists each category with its relative frequency.

Example: A physical therapist records the body part requiring rehabilitation for 30 patients. The data is organized into a frequency and relative frequency table to summarize the types of injuries.

2.1.2 Construct Bar Graphs

A bar graph visually represents categorical data. Each category is represented by a rectangle, with the height corresponding to the frequency or relative frequency.

• Categories are labeled on one axis; frequencies or relative frequencies on the other.

• Bars are of equal width and do not touch.

• Pareto Chart: A bar graph with bars ordered from highest to lowest frequency.

• Side-by-Side Bar Graphs: Used to compare two or more groups, typically using relative frequencies for fair comparison.

• Horizontal Bar Graphs: Useful when category names are lengthy.

Example: Comparing educational attainment in 1990 and 2017 using side-by-side relative frequency bar graphs.

2.1.3 Construct Pie Charts

A pie chart is a circular graph divided into sectors, where each sector represents a category. The area of each sector is proportional to the frequency or relative frequency of the category.

• Useful for showing the proportion of categories within a whole.

Example: Constructing a pie chart for educational attainment data from the U.S. Census Bureau.

2.2 Organizing Quantitative Data: The Popular Displays

Quantitative data can be discrete or continuous. The method of organization depends on the type and range of data values.

• Discrete Data: Data with a countable number of values.

• Continuous Data: Data that can take any value within a range.

2.2.1 Organize Discrete Data in Tables

Discrete data with few values can be organized similarly to qualitative data, using frequency and relative frequency tables.

Example: Number of customers arriving at a restaurant in 40 intervals is summarized in a frequency table.

2.2.2 Construct Histograms of Discrete Data

A histogram is a graphical representation of the distribution of numerical data. For discrete data, each bar represents a value or class, and bars touch each other to indicate continuity.

• Height of each bar shows frequency or relative frequency.

2.2.3 Organize Continuous Data in Tables

Continuous data or discrete data with many values are grouped into classes (intervals).

• Lower Class Limit: Smallest value in a class.

• Upper Class Limit: Largest value in a class.

• Class Width: Difference between consecutive lower class limits.

Guidelines for Creating Classes:

• Choose the lower class limit as the smallest observation or a convenient number below it.

• Decide on the number of classes (typically 5–20).

• Calculate class width:

• Round up to a convenient number.

Example: Organizing parking fine data into classes and constructing frequency and relative frequency tables.

2.2.4 Construct Histograms of Continuous Data

Histograms for continuous data use class intervals on the horizontal axis and frequency or relative frequency on the vertical axis. Bars touch to indicate continuous data.

Example: Constructing a histogram for parking fine data using a class width of 25.

2.2.5 Draw Dot Plots

A dot plot displays each data value as a dot above its position on a number line. Multiple dots are stacked for repeated values.

Example: Drawing a dot plot for the number of arrivals at a restaurant.

2.2.6 Identify the Shape of a Distribution

The shape of a data distribution provides insight into the nature of the data.

• Uniform Distribution: Frequencies are evenly spread.

• Bell-Shaped Distribution: Highest frequency in the middle, tails off symmetrically.

• Skewed Right: Tail on the right is longer.

• Skewed Left: Tail on the left is longer.

Example: Identifying the shape of a histogram for parking fines.

2.2.7 Draw Time Series Graphs

Time series data are values measured at different points in time. A time-series plot has time on the horizontal axis and the variable's value on the vertical axis, with points connected by line segments.

Example: Plotting the Partisan Conflict Index over time.

2.3 Graphical Misrepresentations of Data

Graphs can be misleading or deceptive, either unintentionally or intentionally, by creating incorrect impressions of the data.

2.3.1 Describe What Can Make a Graph Misleading or Deceptive

• Misleading Graph: Unintentionally creates a false impression.

• Deceptive Graph: Purposely creates a false impression.

• Common Issues: Inconsistent scales, misplaced origins, truncated axes, and excessive or misleading design elements.

Guidelines for Good Graphics:

• Title and label axes clearly, including units and data sources.

• Avoid distortion and minimize white space.

• Avoid clutter and unnecessary 3D effects.

• Use consistent design and avoid relative graphs without data or scales.

Example: A bar graph starting at a value other than zero can exaggerate differences between categories.