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Wize University Statistics Textbook > Displaying & Summarizing Categorical Data

Contingency Tables

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Contingency Table

A contingency table, also known as cross tabulation or crosstab, is a matrix format that displays the frequency distribution of variables. It is a two-way table that displays the frequency counts of two categorical variables.

Example

There are two ways to treat Idiocy, an epidemic disease: medication or natural remedies. The contingency table below summarizes the success rate of each treatment. Which treatment is better?

To get a better idea about the rate of success for each treatment, we need to convert this table from counts to percentages.

Do we convert to percentages of the row totals or the column totals?
It depends on the context.

Row Totals


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Column Totals


Here, the column totals are much more useful than the row totals because you can compare success rates:
  • 83.3% of those who received medication were successfully treated.
  • Only 14.6% of those who received natural remedies were successfully treated.
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Simpson’s Paradox

Simpson's Paradox is the effect that occurs in which there appears to be a certain trend in multiple groups; however, this trend disappears or is the complete opposite when the data of these groups are combined or aggregated. This happens when you have a lurking variable, which is a variable that is not included in an experiment or observation but it does truly affect the variables of interest.

Watch Out!
Aggregating can be dangerous!



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Example:

Suppose there are two instructors for COMM 291: Dr. Simpson and Dr. Griffin. Based on the grades from the pervious term, 78 out of 240 students got A’s in Dr. Simpson’s classes; 36 out of 150 students got A’s in Dr. Griffin’s classes. In percentages:


Does this mean you have a better chance of getting an A in Dr. Simpson’s class? Is Dr. Griffin a hard or bad instructor?

Not so fast! You should be aware that they each teach three class sections, which we have aggregated.

Let’s break them down into sections:


  • You see that Dr. Simpson teaches all morning classes; Dr. Griffin teaches one morning class and two evening classes.
  • You also see that more students get A’s in morning classes, including the morning class that Dr. Griffin teaches.
  • In fact, more than half of his morning section got A’s – more than any COMM 291 section.
What is the lurking variable in this example? The sections: morning and evening.

Practice: Contingency Tables

We asked 240 randomly selected people in the province asked how satisfied they are with their premier. Results:



The survey was conducted via stratified sampling based on three age groups: 40 millennials, 100 adults, and 100 seniors.


(a) What percent of people surveyed is at least “Somewhat Satisfied” with the premier? (Enter answer in decimal form e.g. 0.356)


(b) What percent of seniors surveyed is at least “Somewhat Satisfied” with the premier? (Enter answer in decimal form e.g. 0.16)


(c) Almost all the seniors and more than half of adults surveyed are at least “Somewhat Dissatisfied” with the premier. Why does the data contradict with what we observe in part (a)? [Hint: Answer is a two-word term.]


Extra Practice
Contingency Table
Jeremy runs an online store for phone accessories. He has records of who paid full price and who used discount codes when they checkout and constructed the following contingency table:
Full Price Discount Total
Male 54 26 80
Contingency Table
Jeremy runs an online store for phone accessories. He has records of who paid full price and who used discount codes when they checkout and constructed the following contingency table:
Full Price Discount Total
Male 54 26 80
Contingency Table
We want to see if the proportion of male who know how to swim differ than those of females. Here is a contingency table:
The value of the test-statistic is
Contingency Table
We want to see if the proportion of male who know how to swim differ than those of females. Here is a contingency table:
The estimate of the proportion of all males that can swim is
Contingency Table
Jeremy runs an online store for phone accessories. He has records of who paid full price and who used discount codes when they checkout and constructed the following contingency table:
Full Price Discount Total
Male 54 26 80
Contingency tables
Jeremy runs an online store for phone accessories. He has records of who paid full price and who used discount codes when they checkout and constructed the following contingency table:
Full Price Discount Total
Male 54 26 80
Contingency Table
Jeremy runs an online store for phone accessories. He has records of who paid full price and who used discount codes when they checkout and constructed the following contingency table:
Full Price Discount Total
Male 54 26 80
Contingency Tables
A group of people were asked about the type of soda pop drink they had the last time they had BBQ. The data is entered in the following table.
What percent of people had neither Coca-Cola nor Pepsi?
Contingency Tables
A sample of 80 adults that drink coffee are asked; where do you usually go to get your morning coffee, Tim Hortons, McDonalds, Starbucks, or other? The results are recorded below:
Marginal Distribution
Here's the data collected on male and female's preferences on ice cream flavours.
FemaleMaleStrawberry5235Chocolate2421Vanilla1216\begin{array}{|l|l|l|l|} \hline &\text{Female}&\text{Male}\\ \hline \text{Strawberry}&52&35\\ \hline \text{Chocolate}&24&21\\ \hline \text{Vanilla}&12&16\\ \hline \end{array}
What is the marginal distribution of ice cream flavours?
Contingency Tables
The following contingency table displays the number of first and second year students that have math, science, literature, or history as their majors.
MajorFirst year studentSecond year studentTotalMath20,82213,76434,586Science18,44313,01931,462Literature80,96550,977131,942History42,69023,24165,931Total162,920101,001263,921\begin{array}{|l|l|l|l|} \hline \text{Major}&\text{First year student}&\text{Second year student}&\text{Total}\\ \hline \text{Math}&20,822&13,764&34,586\\ \hline \text{Science}&18,443&13,019&31,462\\ \hline \text{Literature}&80,965&50,977&131,942\\ \hline \text{History}&42,690&23,241&65,931\\ \hline \text{Total}&162,920&101,001&263,921\\ \hline \end{array}
Calculate the following percentages:
Contingency table
A group of people were asked about the type of soda pop drink they had the last time they had BBQ. The data is entered in the following table.
What percent of those that had Pepsi were female?
Contingency tables
A group of people were asked about the type of soda pop drink they had the last time they had BBQ. The data is entered in the following table.
What percent of people had neither Coca-Cola nor Pepsi?