Displaying and Describing Categorical Data: Titanic Case Study and Contingency Tables

Study Guide - Smart Notes

Tailored notes based on your materials, expanded with key definitions, examples, and context.

Displaying and Describing Categorical Data

Introduction to Categorical Data Visualization

Categorical data analysis is essential for understanding how groups or categories compare within a dataset. In statistics, visual displays and tables are used to summarize and interpret such data, revealing patterns and relationships that may not be obvious from raw numbers alone.

Visualizing Categorical Data: The Titanic Example

The Area Principle and Misleading Graphs

When displaying categorical data, it is crucial to ensure that the area representing each category is proportional to its value. Violating this principle can lead to misinterpretation.

Area Principle: The area occupied by a part of a graph should correspond to the magnitude of the value it represents.
Example: The Titanic ship graphic (see below) attempts to show the number of people in each class (Crew, Third, Second, First) by the size of ship icons. However, the area of each ship exaggerates the differences, making it appear as though the majority were crew, when in fact only about 40% were.

Misleading Titanic ship graphic violating the area principle

Contingency Tables: Summarizing Two Categorical Variables

Definition and Purpose

A contingency table (or two-way table) displays the frequency distribution of variables that are categorical. It allows us to examine the relationship between two categorical variables by showing the counts for each combination of categories.

Example: Titanic Class and Survival

The table below summarizes the number of Titanic passengers and crew by class and survival status:

	First	Second	Third	Crew	Total
Alive	203	118	178	212	711
Dead	122	167	528	673	1490
Total	325	285	706	885	2201

Contingency table of Titanic class and survival Contingency table highlighting crew deaths

Marginal and Joint Distributions

Marginal Distribution: The totals for each category (row or column), showing the overall distribution of a single variable.
Joint Distribution: The proportion of cases falling into each combination of categories (cell), often expressed as a percentage of the total.

Example of joint distribution (percent of total):

	First	Second	Third	Crew	All
Alive	9.22	5.36	8.09	9.63	32.30
Dead	5.54	7.59	23.99	30.58	67.70
All	14.77	12.95	32.08	40.21	100.00

Joint and marginal distribution table

Conditional Distributions

A conditional distribution shows the distribution of one variable for individuals who satisfy a condition on another variable. For example, the distribution of class among survivors, or the survival rate within each class.

To compute the conditional distribution of class among survivors, divide the number of survivors in each class by the total number of survivors.

Example (percent of survivors by class):

	First	Second	Third	Crew	Total
Alive	28.6%	16.6%	25.0%	29.8%	100%

Conditional distribution of class among survivors

Example (percent of deaths by class):

	First	Second	Third	Crew	Total
Dead	8.2%	11.2%	35.4%	45.2%	100%

Conditional distribution of class among those who died

Using Statistical Software for Contingency Tables

Statistical software (such as Minitab) can be used to generate contingency tables and calculate row, column, and total percentages. The following screenshots show how to set up cross-tabulation and select the appropriate display options:

To display row percentages (conditional on survival status):
To display column percentages (conditional on class):

Interpreting Conditional Percentages

Conditional percentages help us understand the relationship between variables. For example, the survival rate within each class:

	First	Second	Third	Crew	All
Alive	62.46	41.40	25.21	23.95	32.30
Dead	37.54	58.60	74.79	76.05	67.70
All	100.00	100.00	100.00	100.00	100.00

Conditional distribution: survival status within class

Summary Table: Types of Distributions in Contingency Tables

Type	What it Shows	How to Calculate
Marginal Distribution	Totals for each row or column	Sum counts across rows or columns, divide by grand total
Joint Distribution	Proportion in each cell	Cell count divided by grand total
Conditional Distribution	Distribution of one variable given a category of the other	Cell count divided by row or column total

Key Points and Best Practices

Always check that the area principle is respected in visual displays.
Use contingency tables to explore relationships between two categorical variables.
Calculate and interpret marginal, joint, and conditional distributions to gain insights into the data.
Use statistical software to efficiently compute and visualize these distributions.
Be cautious of misleading graphics and always verify the underlying numbers.

Example Application

In the Titanic dataset, contingency tables and conditional distributions reveal that survival rates varied greatly by class, with first-class passengers having a much higher chance of survival than third-class passengers or crew. This kind of analysis is foundational for understanding associations in categorical data.