Contingency Tables: Videos & Practice Problems

A contingency table is a tool used to analyze the relationship between two categorical variables, displaying observed frequencies across different categories. To find expected frequencies, assume independence between variables and use the formula: $\frac{\mathrm{Row\; Total}\mathrm{Column\; Total}}{\mathrm{Total\; Sample\; Size}}.$ This approach helps in understanding data distributions and making informed decisions based on statistical analysis.

Contingency Tables & Expected Frequencies

Understanding the relationship between two variables can provide valuable insights, especially in educational contexts. For instance, when examining students' heights in relation to their grade levels, a contingency table becomes an essential tool. A contingency table is a type of frequency distribution that displays the frequencies of different categories across two variables, allowing for a clearer analysis of their relationship.

To navigate a contingency table, one must identify the intersection of the relevant row and column. For example, if you want to find out how many students are between 5'1" and 5'6", you would locate the corresponding row for grade levels and the column for height, then read the observed frequency at their intersection. This straightforward method allows for quick answers to specific questions about the data.

In addition to observed frequencies, expected frequencies are crucial for understanding the data's underlying patterns. Expected frequencies are calculated under the assumption that the two variables are independent. The formula for determining expected frequencies is:

E = \frac{(Row \ Total) \times (Column \ Total)}{Grand \ Total}

Here, E represents the expected frequency for a specific cell in the table. The row total corresponds to the total number of students in a particular grade, while the column total reflects the number of students within a specific height range. The grand total is the overall number of students surveyed.

For example, if there are 77 graders and 72 students between 5'1" and 5'6", the expected frequency for that cell would be calculated as follows:

E = \frac{77 \times 72}{161} \approx 34.43

This means that, assuming independence, we would expect approximately 34.43 students to fall into that category, even though the observed frequency is 40. This discrepancy can indicate a relationship between the variables that warrants further investigation.

By applying this method to each cell in the contingency table, one can derive expected frequencies for all combinations of the two variables, providing a comprehensive view of the data. This approach not only enhances understanding but also prepares students for more complex statistical analyses in the future.

The table below shows the results from a drug trial for a new ADHD medication. Find the frequencies that we would expect if improvement of symptoms was independent of whether a participant received the placebo.

• A. E_1,1 = 25.0; E_2,1 = 25.0; E_1,2 = 25.0; E_2,2 = 25.0
• B. E_1,1 = 28.6; E_2,1 = 26.4; E_1,2 = 23.4; E_2,2 = 21.6
• C. E_1,1 = 26.4; E_2,1 = 21.6; E_1,2 = 28.6; E_2,2 = 23.4
• D. E_1,1 = 18; E_2,1 = 30; E_1,2 = 37; E_2,2 = 15

Answer: (C)

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 statistical tool used to analyze the relationship between two categorical variables. It organizes data into rows and columns, where each cell represents the frequency of occurrences for a specific combination of categories. For example, it can show the number of students in different grade levels and height ranges. Contingency tables are used to calculate observed frequencies and expected frequencies under the assumption of independence. Expected frequencies are calculated using the formula:

$\frac{\left(\mathrm{Row\; Total}\right)\left(\mathrm{Column\; Total}\right)}{\mathrm{Total\; Sample\; Size}}$

These tables are essential for understanding relationships between variables and performing statistical tests like the chi-square test.

How do you calculate expected frequencies in a contingency table?

To calculate expected frequencies in a contingency table, you assume the variables are independent. The formula for expected frequency in a specific cell is:

$\frac{\left(\mathrm{Row\; Total}\right)\left(\mathrm{Column\; Total}\right)}{\mathrm{Total\; Sample\; Size}}$

Here’s how it works: Multiply the total number of observations in the row by the total number of observations in the column, then divide by the grand total of all observations in the table. For example, if the row total is 77, the column total is 72, and the grand total is 161, the expected frequency is:

$\frac{\left(77\right)\left(72\right)}{161}$

This calculation helps determine whether the observed frequencies differ significantly from what is expected under independence.

What is the difference between observed and expected frequencies in a contingency table?

Observed frequencies are the actual counts recorded in the data for each cell of a contingency table. They represent the real-world occurrences of specific combinations of categories. Expected frequencies, on the other hand, are theoretical values calculated under the assumption that the two variables are independent. The formula for expected frequency is:

$\frac{\left(\mathrm{Row\; Total}\right)\left(\mathrm{Column\; Total}\right)}{\mathrm{Total\; Sample\; Size}}$

For example, if the observed frequency in a cell is 40 but the expected frequency is 34.43, it suggests a potential relationship between the variables. Comparing observed and expected frequencies is crucial for statistical tests like the chi-square test, which evaluates the independence of variables.

How do you interpret the results of a contingency table?

Interpreting a contingency table involves analyzing the observed frequencies and comparing them to the expected frequencies. If the observed frequencies closely match the expected frequencies, it suggests that the variables may be independent. However, significant differences between observed and expected frequencies indicate a potential relationship between the variables. Statistical tests, such as the chi-square test, are often used to quantify this relationship. For example, if the chi-square test yields a p-value less than the significance level (e.g., 0.05), you reject the null hypothesis of independence and conclude that the variables are related. Additionally, subtotals and grand totals in the table provide insights into overall trends and distributions within the data.

What is the formula for calculating joint probability in a contingency table?

Joint probability in a contingency table represents the likelihood of two events occurring together. If the events are independent, the joint probability is calculated by multiplying their individual probabilities. The formula is:

$P(AB)=P\left(A\right)P\left(B\right)$

For example, if event A is selecting a student in grade 8 and event B is selecting a student between 5'1" and 5'6", their probabilities are calculated as:

$\frac{77}{161}\frac{72}{161}$

The joint probability is the product of these fractions. This concept is essential for understanding relationships between variables in contingency tables.