Chapter 2: Summarizing Data in Tables and Graphs

2.1 Organizing Qualitative Data

Organizing qualitative (categorical) data is essential for meaningful analysis. This section covers methods for summarizing and displaying qualitative data using tables and graphical representations.

2.1.1 Organize Qualitative Data in Tables

• Frequency Distribution: A table that lists each category of data and the number of occurrences for each category.

• Relative Frequency: The proportion (or percent) of observations within a category, calculated as:

• Relative Frequency Distribution: A table listing each category with its relative frequency.

• Example: A physical therapist records the body part requiring rehabilitation for 30 patients and organizes the data into a frequency and relative frequency table.

2.1.2 Construct Bar Graphs

• Bar Graph: Categories are labeled on one axis, and frequencies or relative frequencies on the other. Rectangles of equal width represent each category, with height corresponding to frequency or relative frequency.

• Pareto Chart: A bar graph with bars ordered in decreasing order of frequency or relative frequency.

• Side-by-Side Bar Graphs: Used to compare two or more data sets, 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 bar graphs.

2.1.3 Construct Pie Charts

• Pie Chart: A circle divided into sectors, each representing a category. The area of each sector is proportional to the category's frequency.

• Example: Educational attainment data for U.S. residents aged 25+ in 2017 displayed as a pie chart.

2.2 Organizing Quantitative Data: The Popular Displays

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

2.2.1 Organize Discrete Data in Tables

• Discrete Data: Data with countable values (e.g., number of customers in a time interval).

• Frequency and Relative Frequency Distributions: Tables summarizing how often each value occurs and its proportion of the total.

2.2.2 Construct Histograms of Discrete Data

• Histogram: Rectangles represent each class of data. The height is the frequency or relative frequency, and the rectangles touch each other to indicate continuity.

2.2.3 Organize Continuous Data in Tables

• Classes: Intervals into which data are grouped for large or continuous data sets.

• Class Limits: The smallest (lower) and largest (upper) values within a class.

• Class Width: The difference between consecutive lower class limits.

• Guidelines: Choose a convenient lower class limit (often the smallest observation or slightly lower) and a class width that results in 5–20 classes.

• Example: Parking fine data grouped into classes with a chosen class width.

2.2.4 Construct Histograms of Continuous Data

• Histogram for Continuous Data: Similar to discrete, but classes are intervals. The height of each rectangle is the frequency or relative frequency for that interval.

2.2.5 Draw Dot Plots

• Dot Plot: Each observation is plotted as a dot above its value on a number line. Multiple dots stack vertically for repeated values.

• Example: Number of arrivals at a restaurant shown as a dot plot.

2.2.6 Identify the Shape of a Distribution

• Uniform Distribution: Frequencies are evenly spread across values.

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

• Skewed Right: Tail on the right is longer than on the left.

• Skewed Left: Tail on the left is longer than on the right.

• Example: Analyzing the shape of a histogram for parking fines.

2.2.7 Draw Time Series Graphs

• Time Series Data: Values measured at different points in time.

• Time-Series Plot: Time is on the horizontal axis, variable values on the vertical axis, with points connected by line segments.

• Example: Partisan Conflict Index over time.

2.3 Graphical Misrepresentations of Data

Graphs can be misleading or deceptive if not constructed properly. This section discusses common pitfalls and guidelines for accurate graphical representation.

2.3.1 Describe What Can Make a Graph Misleading or Deceptive

• Misleading Graph: Unintentionally creates an incorrect impression.

• Deceptive Graph: Purposely creates an incorrect impression.

• Common Issues:

  • Improper scale (e.g., inconsistent increments, misplaced origin)

  • Truncated or non-zero baselines

  • Distorted proportions (e.g., exaggerated pie chart sectors)

• 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.