How To Calculate The Expected Frequency

Expected frequency is a cornerstone concept in statistics, particularly within the realm of hypothesis testing and categorical data analysis. It represents the frequency we expect to see in a cell of a contingency table if there is no association between the variables being studied. Understanding how to calculate expected frequency is crucial for conducting chi-square tests and drawing meaningful conclusions from data. Let's explore the process in detail.

Understanding Expected Frequency

The concept of expected frequency revolves around the null hypothesis, which posits that there is no relationship between the categorical variables under investigation. In simpler terms, it assumes that the observed frequencies are merely due to chance. The expected frequency, therefore, represents the theoretical frequency that would occur if the null hypothesis were true.

Calculating the expected frequency helps us determine whether the observed frequencies deviate significantly from what we would expect by chance alone. If the difference between the observed and expected frequencies is large enough, we reject the null hypothesis and conclude that there is a statistically significant association between the variables.

The Formula for Calculating Expected Frequency

The formula for calculating the expected frequency in a contingency table is straightforward:

Expected Frequency (E) = (Row Total * Column Total) / Grand Total

Where:

• Row Total is the sum of all frequencies in the row containing the cell of interest.
• Column Total is the sum of all frequencies in the column containing the cell of interest.
• Grand Total is the total number of observations in the entire contingency table.

This formula essentially distributes the overall observations proportionally based on the marginal distributions of the row and column variables.

Step-by-Step Calculation with Examples

Let's break down the calculation process with several examples to illustrate its application.

Example 1: A Simple 2x2 Contingency Table

Imagine we are investigating the relationship between gender (Male, Female) and preference for a certain brand of coffee (Brand A, Brand B). Our observed data is summarized in the following contingency table:

Step 1: Calculate the Expected Frequency for Each Cell

• Cell (Male, Brand A): E = (Row Total for Male * Column Total for Brand A) / Grand Total = (100 * 90) / 200 = 45
• Cell (Male, Brand B): E = (Row Total for Male * Column Total for Brand B) / Grand Total = (100 * 110) / 200 = 55
• Cell (Female, Brand A): E = (Row Total for Female * Column Total for Brand A) / Grand Total = (100 * 90) / 200 = 45
• Cell (Female, Brand B): E = (Row Total for Female * Column Total for Brand B) / Grand Total = (100 * 110) / 200 = 55

Step 2: Construct the Expected Frequency Table

Now, we can create a table showing the expected frequencies:

Interpretation:

These expected frequencies represent what we would expect to see in each cell if there were no relationship between gender and coffee brand preference. For instance, we'd expect 45 males to prefer Brand A and 55 males to prefer Brand B, solely based on the overall distribution of preferences and the number of males in the sample.

Example 2: A Larger Contingency Table (3x3)

Let's consider a more complex scenario where we are examining the relationship between education level (High School, Bachelor's, Master's) and employment status (Employed, Unemployed, Self-Employed). The observed data is:

Step 1: Calculate the Expected Frequency for Each Cell

• Cell (High School, Employed): E = (120 * 350) / 500 = 84
• Cell (High School, Unemployed): E = (120 * 60) / 500 = 14.4
• Cell (High School, Self-Employed): E = (120 * 90) / 500 = 21.6
• Cell (Bachelor's, Employed): E = (200 * 350) / 500 = 140
• Cell (Bachelor's, Unemployed): E = (200 * 60) / 500 = 24
• Cell (Bachelor's, Self-Employed): E = (200 * 90) / 500 = 36
• Cell (Master's, Employed): E = (180 * 350) / 500 = 126
• Cell (Master's, Unemployed): E = (180 * 60) / 500 = 21.6
• Cell (Master's, Self-Employed): E = (180 * 90) / 500 = 32.4

Step 2: Construct the Expected Frequency Table

Interpretation:

Again, these expected frequencies show what we anticipate if there is no association between education level and employment status. For example, we would expect 84 individuals with a high school education to be employed, 14.4 to be unemployed, and 21.6 to be self-employed, based on the overall distribution of employment statuses and the number of people with a high school education in our sample.

Example 3: Examining the Impact of Sample Size

To emphasize the importance of sample size, let's revisit the coffee brand preference example but with a smaller sample. Suppose our observed data is:

Step 1: Calculate the Expected Frequency for Each Cell

• Cell (Male, Brand A): E = (20 * 18) / 40 = 9
• Cell (Male, Brand B): E = (20 * 22) / 40 = 11
• Cell (Female, Brand A): E = (20 * 18) / 40 = 9
• Cell (Female, Brand B): E = (20 * 22) / 40 = 11

Step 2: Construct the Expected Frequency Table

While the relative differences between observed and expected frequencies might appear similar to the first example, the absolute differences are smaller. With a smaller sample size, the chi-square test statistic will likely be smaller, potentially leading to a failure to reject the null hypothesis, even if a real association exists. This highlights the importance of having a sufficiently large sample size to detect statistically significant relationships.

Calculating the Chi-Square Statistic

Once you have calculated the expected frequencies, the next step is to calculate the chi-square statistic. This statistic quantifies the overall discrepancy between the observed and expected frequencies. The formula for the chi-square statistic is:

χ² = Σ [(O - E)² / E]

Where:

• χ² represents the chi-square statistic.
• Σ denotes the summation across all cells in the contingency table.
• O represents the observed frequency in a cell.
• E represents the expected frequency in the same cell.

Let's calculate the chi-square statistic for our first coffee brand preference example:

• Cell (Male, Brand A): (60 - 45)² / 45 = 5
• Cell (Male, Brand B): (40 - 55)² / 55 = 4.09
• Cell (Female, Brand A): (30 - 45)² / 45 = 5
• Cell (Female, Brand B): (70 - 55)² / 55 = 4.09

χ² = 5 + 4.09 + 5 + 4.09 = 18.18

This chi-square statistic, along with the degrees of freedom, is then used to determine the p-value, which indicates the probability of observing such a large discrepancy between observed and expected frequencies if the null hypothesis were true.

Degrees of Freedom

The degrees of freedom (df) are a crucial component of the chi-square test. They represent the number of independent pieces of information available to estimate a parameter. For a contingency table, the degrees of freedom are calculated as:

df = (Number of Rows - 1) * (Number of Columns - 1)

In our 2x2 coffee brand preference example, df = (2-1) * (2-1) = 1. In the 3x3 education and employment example, df = (3-1) * (3-1) = 4.

The degrees of freedom are used in conjunction with the chi-square statistic to determine the p-value. A larger chi-square statistic with the same degrees of freedom will result in a smaller p-value.

Interpreting the Results

The p-value obtained from the chi-square test is compared to a predetermined significance level (alpha), typically set at 0.05.

• If p-value ≤ alpha: We reject the null hypothesis. This suggests that there is a statistically significant association between the variables. The observed frequencies deviate significantly from what we would expect by chance alone.
• If p-value > alpha: We fail to reject the null hypothesis. This suggests that there is not enough evidence to conclude that there is a statistically significant association between the variables. The observed frequencies are reasonably consistent with what we would expect by chance.

Important Considerations:

• Expected Frequency Rule: A common rule of thumb is that all expected frequencies should be at least 5. If some expected frequencies are less than 5, the chi-square approximation may not be accurate. In such cases, consider combining categories or using alternative tests like Fisher's exact test.
• Causation vs. Association: A statistically significant association does not imply causation. It only suggests that the variables are related. There may be other confounding variables influencing the relationship.
• Sample Size: As demonstrated earlier, a sufficiently large sample size is crucial for detecting statistically significant associations. Small sample sizes can lead to a failure to reject the null hypothesis, even if a real association exists.
• Yate's Correction for Continuity: For 2x2 contingency tables, Yate's correction for continuity is sometimes applied to adjust the chi-square statistic. This correction reduces the magnitude of the chi-square statistic, particularly when expected frequencies are small. However, its use is debated among statisticians.

Common Mistakes to Avoid

• Incorrectly Calculating Expected Frequencies: Double-check your calculations to ensure you are using the correct formula and values.
• Ignoring the Expected Frequency Rule: Be mindful of the expected frequency rule and take appropriate action if some expected frequencies are too small.
• Misinterpreting the Results: Remember that a statistically significant association does not imply causation.
• Using the Chi-Square Test with Non-Categorical Data: The chi-square test is specifically designed for categorical data. Do not use it with continuous variables.
• Forgetting Degrees of Freedom: The degrees of freedom are essential for determining the p-value.

Alternatives to the Chi-Square Test

While the chi-square test is a widely used method for analyzing categorical data, other options are available depending on the specific research question and data characteristics:

• Fisher's Exact Test: This test is particularly useful when dealing with small sample sizes or when expected frequencies are less than 5. It provides an exact p-value, rather than relying on the chi-square approximation.
• G-Test (Likelihood Ratio Test): The G-test is an alternative to the chi-square test that is often preferred when dealing with small sample sizes.
• McNemar's Test: This test is used for analyzing paired categorical data, where the same subjects are measured at two different time points or under two different conditions.
• Cochran's Q Test: This test is an extension of McNemar's test for situations where you have more than two related samples.

Conclusion

Calculating the expected frequency is a fundamental step in performing chi-square tests and analyzing categorical data. By understanding the formula, following the step-by-step calculation process, and interpreting the results correctly, you can draw meaningful conclusions about the relationships between categorical variables. Remember to consider the expected frequency rule, the importance of sample size, and the potential need for alternative tests when appropriate. Mastering this concept will empower you to effectively analyze data and make informed decisions in various fields, from social sciences to healthcare to business.