Introduction to Contingency Tables: Videos & Practice Problems

Contingency tables display frequencies for two categorical variables, allowing for the analysis of marginal, joint, and conditional probabilities. Marginal probability assesses the likelihood of a single category, while joint probability evaluates the intersection of two events. Conditional probability determines the likelihood of an event given that another has occurred. Understanding these concepts is crucial for interpreting data effectively, particularly in fields like business analytics and statistics, where categorical data analysis is common.

Introduction to Contingency Tables

When analyzing data involving two categorical variables, contingency tables serve as a valuable tool to display frequencies. These tables allow us to explore relationships between variables, such as whether students drive a car and their grade level. Each cell in a contingency table represents the frequency of responses that fall into specific categories, which can be read by identifying the corresponding row and column. For instance, a cell located in the junior row and the "no" column indicates the number of juniors who do not drive a car.

Contingency tables often include totals for each row and column, providing a quick reference for category totals. For example, if the total for the junior row is 50, this indicates that there are 50 juniors surveyed. The grand total, typically found in the last row and column, represents the overall number of responses, which in this case is 100.

Understanding probabilities within this context involves three key types: marginal, joint, and conditional probabilities. Marginal probability refers to the likelihood of a single category occurring, calculated by dividing the total of that category by the grand total. For example, to find the probability that a student drives a car, one would take the total number of students who drive a car (60) and divide it by the grand total (100), resulting in a probability of 0.6 or 60%.

Joint probability, on the other hand, assesses the likelihood of two events occurring simultaneously. This is determined by taking the frequency from the relevant cell and dividing it by the grand total. For instance, to find the probability that a student is both a senior and drives a car, one would look at the intersection of the senior row and the drives a car column, which might yield a frequency of 40. Dividing this by the grand total gives a joint probability of 0.4 or 40%.

Lastly, conditional probability evaluates the likelihood of one event occurring given that another event has already occurred. This is calculated by taking the frequency from the relevant cell and dividing it by the total of the row or column that corresponds to the known event. For example, to find the probability that a student drives a car given that they are a senior, one would use the frequency of seniors who drive a car (40) and divide it by the total number of seniors (50), resulting in a conditional probability of 0.8 or 80%.

By mastering these concepts, students can effectively interpret contingency tables and apply their understanding of marginal, joint, and conditional probabilities to real-world scenarios.

The table below shows the results from a drug trial for a new ADHD medication. Use the table to find the probability that a person's symptoms improved, given that they received the placebo and identify the type of probability found. 

• A. 0.1; Marginal probability
• B. 0.1; Conditional probability
• C. 0.2; Conditional probability
• D. 0.2; Marginal probability

Answer: (C)

The table below shows the results from a drug trial for a new ADHD medication. Use the table to find the probability that a person's symptoms didn't improve and they received the non-placebo and identify the type of probability found. 

• A. 0.4; Joint probability
• B. 0.4; Conditional probability
• C. 0.2; Joint probability
• D. 0.2; Conditional probability

Answer: (C)

The table below shows the results from a drug trial for a new ADHD medication. Use the table to find the probability that a person's symptoms improved and identify the type of probability found.

• A. 0.8; Marginal Probability
• B. 0.8; Joint Probability
• C. 0.4; Joint Probability
• D. 0.4; Marginal Probability

Answer: (D)

Introduction to Contingency Tables Example 1

In this exercise, we will construct a contingency table based on survey data regarding hair and eye color among 50 individuals. The grand total of surveyed individuals is 50, which will be reflected in both the total row and total column of the table.

First, we note that 28% of the surveyed individuals have blue eyes. Calculating this, we find that 28% of 50 equals 14, indicating that 14 people have blue eyes. This value is placed in the total cell of the blue-eyed column.

Similarly, we are informed that 28% of the individuals are blonde, which also results in 14 people. This number is recorded in the total cell of the blonde-haired row.

Next, we learn that 20% of the surveyed individuals are both blonde and blue-eyed. Calculating 20% of 50 gives us 10, which we place in the cell where the blonde hair and blue eyes intersect.

It is also stated that no individuals have both black hair and hazel eyes, so we enter a zero in the corresponding cell for this combination.

Continuing, we find that 40% of the individuals have brown hair. This translates to 20 people, which we place in the total cell of the brown-haired row. Since we have accounted for 34 individuals with either brown or blonde hair, we deduce that the remaining individuals must have black hair, totaling 16 people.

Next, we are told that 60% of the surveyed individuals have brown eyes, which amounts to 30 people. This value is recorded in the total row of the brown-eyed column. By subtracting the accounted individuals with brown and blue eyes from the total, we find that 6 individuals must have hazel eyes.

We also learn that one out of seven blue-eyed individuals has black hair. With 14 blue-eyed individuals, this means there are 2 individuals with both blue eyes and black hair, which we place in the intersecting cell.

From the previous data, we know that among the 14 blue-eyed individuals, 10 have blonde hair, leaving 2 who must have brown hair. Additionally, we find that 50% of the 6 individuals with hazel eyes have blonde hair, resulting in 3 hazel-eyed individuals with blonde hair. The remaining 3 hazel-eyed individuals must have brown hair.

To complete the table, we can fill in the remaining cells using the totals. For instance, since 14 individuals have blonde hair and we have accounted for 13 of them with blue or hazel eyes, the remaining individual must have brown eyes. Similarly, for the 20 individuals with brown hair, we have accounted for 5 with hazel or blue eyes, leaving 15 who must have brown eyes. Finally, for the 16 individuals with black hair, we have accounted for 2 with blue eyes and none with hazel eyes, meaning the remaining 14 must have brown eyes.

This systematic approach allows us to accurately fill in the contingency table, reflecting the relationships between hair color and eye color among the surveyed individuals.

Introduction to Contingency Tables Example 2

In analyzing the dietary preferences of wedding guests, we can derive both the conditional distribution for vegetarians and the marginal distribution of diet types. The conditional distribution focuses on the probabilities of certain characteristics given a specific condition, in this case, being a vegetarian.

To find the conditional probabilities, we first look at the number of vegetarians who have allergies versus those who do not. Out of 13 vegetarians, 9 do not have allergies. Therefore, the probability of no allergies given that a person is vegetarian is calculated as:

$$ P(\text{No Allergies} | \text{Vegetarian}) = \frac{9}{13} \approx 0.69 \text{ or } 69\% $$

Conversely, the probability of having allergies given that a person is vegetarian is determined by the 4 vegetarians who do have allergies:

$$ P(\text{Allergies} | \text{Vegetarian}) = \frac{4}{13} \approx 0.31 \text{ or } 31\% $$

This indicates that approximately 69% of vegetarians do not have allergies, while 31% do.

Next, we explore the marginal distribution of diet types, which provides insights into the overall percentages of each dietary category among all guests. With a total of 85 guests, we can calculate the probabilities for each diet type:

1. The probability of being a vegetarian is:

$$ P(\text{Vegetarian}) = \frac{13}{85} \approx 0.15 \text{ or } 15\% $$

2. The probability of being vegan is:

$$ P(\text{Vegan}) = \frac{7}{85} \approx 0.08 \text{ or } 8\% $$

3. The probability of being neither vegetarian nor vegan is:

$$ P(\text{Neither}) = \frac{65}{85} \approx 0.76 \text{ or } 76\% $$

Although the percentages do not sum to exactly 100% due to rounding, the data reveals that approximately 15% of the guests are vegetarian, 8% are vegan, and 76% are neither. This analysis provides a clear understanding of the dietary preferences among the wedding guests.

Here’s what students ask on this topic:

What is a contingency table, and how is it used in statistics?

A contingency table is a type of table used in statistics to display the frequencies of two categorical variables. It organizes data into rows and columns, where each cell represents the frequency of a specific combination of categories. For example, if we survey students about their grade level (junior or senior) and whether they drive a car (yes or no), the table will show how many students fall into each combination of these categories. Contingency tables are essential for analyzing relationships between variables and calculating probabilities, such as marginal, joint, and conditional probabilities. They help summarize data and provide insights into patterns or associations between the variables.

How do you calculate marginal probability using a contingency table?

Marginal probability is the probability of a single category occurring, regardless of the other variable. To calculate it using a contingency table, follow these steps:
1. Identify the total frequency for the category of interest (row or column total).
2. Divide this total by the grand total (sum of all frequencies in the table).
For example, if the total number of students who drive a car is 60 and the grand total is 100, the marginal probability is calculated as: $\frac{60}{100}$, which simplifies to 0.6 or 60%.

What is the difference between joint probability and conditional probability?

Joint probability is the probability of two events occurring simultaneously, such as being a senior and driving a car. It is calculated by dividing the frequency of the intersection of the two events by the grand total. For example, if 40 seniors drive a car out of 100 students, the joint probability is: $\frac{40}{100}$, or 0.4 (40%).
Conditional probability, on the other hand, is the probability of one event occurring given that another event has already occurred. For example, the probability of driving a car given that the student is a senior is calculated by dividing the relevant cell frequency (40) by the total for the given condition (50 seniors): $\frac{40}{50}$, or 0.8 (80%).

How can you identify a conditional probability problem in a word problem?

Conditional probability problems often include keywords or phrases that indicate one event has already occurred. Look for terms like "given," "assuming," or "if we know." For example, a question might ask, "What is the probability that a student drives a car given they are a senior?" This phrasing suggests that the condition (being a senior) is already established, and you are calculating the probability of the second event (driving a car) within that subset. Recognizing these cues helps distinguish conditional probability from other types, such as joint or marginal probability.

What are the steps to calculate joint probability using a contingency table?

To calculate joint probability using a contingency table, follow these steps:
1. Identify the cell frequency that represents the intersection of the two events of interest.
2. Divide this cell frequency by the grand total (sum of all frequencies in the table).
For example, if you want to find the probability of a student being a senior and driving a car, locate the cell where the "senior" row and "drives a car" column intersect. If this cell frequency is 40 and the grand total is 100, the joint probability is: $\frac{40}{100}$, or 0.4 (40%).

What is the purpose of the total row and total column in a contingency table?

The total row and total column in a contingency table summarize the data by providing the category totals for each variable. The total row sums up the frequencies across all columns for each row category, while the total column sums up the frequencies across all rows for each column category. These totals are essential for calculating probabilities, such as marginal probabilities, where the category total is divided by the grand total. Additionally, the grand total (intersection of the total row and total column) represents the sum of all frequencies in the table, which is used as the denominator in many probability calculations.