Constructing Graphical and Tabular Displays of Data: Pie Charts and Two-Way Tables

Chapter 3: Constructing Graphical and Tabular Displays of Data (3.2)

Pie Charts

Pie charts are a common graphical tool used to display the distribution of a categorical variable. Each slice of the pie represents a category, and the size of the slice is proportional to the relative frequency (usually expressed as a percentage) of that category.

• Definition: A pie chart is a circular graph divided into slices to illustrate numerical proportions of categories in a dataset.

• Relative Frequency: The proportion of the total represented by each category, often shown as a percentage.

• Interpretation: The area of each slice corresponds to the proportion of observations in that category.

Example: Children's Dream Jobs

A survey of 273 children asked what job they would want to do. The results are summarized below:

Job	Percent
Spy	16
Veterinarian	13
Professional athlete	12
Movie star	10
Video game designer	8
Doctor	6
Other	35

• Constructing a Pie Chart: Use the percentages to create slices for each job category. The sum of all slices should be 100%.

• Finding Proportions:

  • Proportion for 'Spy': $0.16$ (16%)

  • Proportion NOT 'Spy': $1 - 0.16 = 0.84$ (84%)

  • Proportion for 'Professional athlete' OR 'Movie star': $0.12 + 0.10 = 0.22$ (22%)

• Statistical Inference: Results from a sample (e.g., 6% want to be a doctor) cannot be generalized to the entire population without considering sampling and nonsampling errors.

Two-Way Tables

Two-way tables (also called contingency tables) are used to summarize the relationship between two categorical variables. Each cell in the table shows the frequency or count for a specific combination of categories.

• Definition: A two-way table displays the frequency distribution of variables that have two or more categories.

• Rows and Columns: Typically, one variable is represented by rows and the other by columns.

• Totals: Marginal totals (row and column totals) show the total counts for each category.

Example: Novel Reading by Gender

Gender	Did Not Read Novel	Read Novel	Total
Female	6	19	25
Male	6	11	17
Total	12	30	42

• Total who read a novel: 30 students

• Proportion who did not read a novel: $\frac{12}{42} \approx 0.286$ (28.6%)

• Proportion of women who read a novel: $\frac{19}{25} = 0.76$ (76%)

• Proportion of men who read a novel: $\frac{11}{17} \approx 0.647$ (64.7%)

Key Points for Two-Way Tables

• To find proportions for a subgroup, use only the relevant row or column total as the denominator.

• Two-way tables are useful for comparing groups and identifying relationships between categorical variables.

Additional info:

• Pie charts are best for displaying parts of a whole for a single categorical variable, while two-way tables are ideal for examining the interaction between two categorical variables.

• Always check that the sum of the relative frequencies in a pie chart equals 1 (or 100%).

• When interpreting survey results, consider the possibility of sampling error (random variation due to the sample) and nonsampling error (biases or mistakes in data collection).