Summarizing Data in Tables and Graphs: Introductory Statistics Study Notes

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 and displaying qualitative data using tables and graphs.

2.1.1 Organize Qualitative Data in Tables

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

• Relative Frequency: The proportion (or percent) of observations within a category, calculated as: $\text{Relative Frequency} = \frac{\text{Frequency of the category}}{\text{Sum of all frequencies}}$

Example: Frequency distribution of M&M colors in a sample bag.

The table above shows the frequency of each color in the sample.

Relative frequency can be added to the table to show the proportion of each color:

2.1.2 Construct Bar Graphs

A bar graph visually represents the frequency or relative frequency of categories. Each bar's height corresponds to the frequency or relative frequency, and all bars have equal width. Bar graphs can be used for both frequency and relative frequency data.

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

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

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

Example: Frequency bar graph for M&M colors.

Example: Relative frequency bar graph for M&M colors.

Example: Side-by-side bar graph comparing marital status in 1990 and 2006.

Example: Horizontal bar graph for educational attainment.

2.1.3 Construct Pie Charts

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

Example: Pie chart of marital status in 2006.

2.2 Organizing Quantitative Data: The Popular Displays

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

2.2.1 Organize Discrete Data in Tables

Discrete data with few unique values can be organized in frequency and relative frequency tables.

Example: Number of cars per household (sample of 50 households).

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 unique value or class, and bars touch each other to indicate continuity.

Example: Histogram for number of cars per household (frequency and relative frequency).

2.2.3 Organize Continuous Data in Tables

Continuous data are grouped into intervals called classes. Each class has a lower and upper class limit, and the class width is the difference between consecutive lower class limits.

• Guidelines for Choosing Classes: Use between 5 and 20 classes. Choose a convenient lower class limit, often just below the smallest observation.

• Class Width Formula: $\text{Class Width} = \frac{\text{Largest Data Value} - \text{Smallest Data Value}}{\text{Number of Classes}}$

Example: Time between eruptions at Old Faithful Geyser (in seconds).

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. The choice of class width affects the appearance and interpretability of the histogram.

Example: Histogram for time between eruptions (class width = 10 and 5).

2.2.5 Draw Stem-and-Leaf Plots

A stem-and-leaf plot displays quantitative data by splitting each value into a "stem" (all but the final digit) and a "leaf" (the final digit). This plot preserves the original data values and shows the distribution's shape.

• Split Stems: Used when data are bunched, to better reveal the distribution.

• Advantage: Raw data can be retrieved from the plot, unlike histograms.

Example: Unemployment rates by state (June 2008).

2.2.6 Draw Dot Plots

A dot plot places each observation as a dot above its value on a number line. It is useful for small data sets and shows clusters, gaps, and outliers.

Example: Dot plot for number of cars per household.

2.2.7 Identify the Shape of a Distribution

The shape of a distribution can be described as:

• Uniform: All values occur with roughly equal frequency.

• Bell-shaped (Symmetric): Highest frequency in the middle, tails off on both sides.

• Skewed Right: Tail on the right is longer; most data are on the left.

• Skewed Left: Tail on the left is longer; most data are on the right.

Example: Histogram for time between eruptions at Old Faithful (identify the shape).

2.2.8 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: Closing prices of the Dow Jones Industrial Average (1990–2007).

2.3 Graphical Misrepresentations of Data

Graphs can be misleading or deceptive if not constructed properly. This section discusses how to avoid common pitfalls in graphical representation.

2.3.1 Describe What Can Make a Graph Misleading or Deceptive

• Distortion: Manipulating axes or scales to exaggerate or minimize differences.

• Clutter: Excessive gridlines, backgrounds, or 3D effects can distract or mislead.

• Omitting Labels or Units: Makes interpretation difficult or impossible.

• Relative Graphs Without Data: Avoid graphs that lack clear data or scales.

Example: Misleading pie chart exaggerating a category.

Example: Properly constructed pie chart for comparison.

• Guidelines for Good Graphics:

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

  • Avoid distortion and minimize white space.

  • Avoid clutter and unnecessary design elements.

  • Let the data speak for themselves; do not force attention to specific areas.