How To Calculate Expected Frequency In Chi Square

Unlocking the power of the Chi-Square test starts with understanding expected frequency, a cornerstone concept that allows us to compare observed data against theoretical expectations, ultimately revealing whether a relationship exists between categorical variables.

Understanding Expected Frequency in Chi-Square Tests

The Chi-Square test is a statistical tool used to determine if there is a significant association between two categorical variables. In essence, it examines whether the observed distribution of data differs significantly from what we would expect if there were no relationship between the variables. At the heart of this test lies the concept of expected frequency: the number of observations we would anticipate in each category if the variables were independent.

Why is Expected Frequency Important?

Expected frequency serves as the baseline against which we compare our observed frequencies (the actual counts in our data). By calculating and comparing these two values, we can quantify the discrepancy between what we see and what we would expect by chance alone. A large discrepancy suggests a potential relationship between the variables, leading us to reject the null hypothesis of independence. Without calculating expected frequencies, the Chi-Square test wouldn't be possible. They are fundamental to determining the Chi-Square statistic and subsequently, the p-value, which informs our decision about statistical significance.

Calculating Expected Frequency: A Step-by-Step Guide

Calculating expected frequencies is a straightforward process. Here's a breakdown of the steps, complete with examples to solidify your understanding.

1. Construct a Contingency Table

The first step is to organize your data into a contingency table, also known as a cross-tabulation. This table displays the frequency distribution of two categorical variables. Rows represent one variable, columns represent the other, and each cell contains the count of observations that fall into that specific combination of categories.

Example: Let's say we want to investigate whether there's a relationship between smoking habits and the development of lung cancer. We collect data from 500 individuals and create the following contingency table:

• Observed Frequency (O): The values within the table (60, 140, 30, 270) are our observed frequencies.
• Row Totals: The totals for each row (200, 300).
• Column Totals: The totals for each column (90, 410).
• Grand Total: The total number of observations (500).

2. Apply the Formula

The formula for calculating the expected frequency (E) for each cell in the contingency table is:

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

In essence, the expected frequency represents the number of observations we would expect in a particular cell if the two variables were completely independent.

3. Calculate Expected Frequencies for Each Cell

Using the formula, we calculate the expected frequency for each cell in our smoking and lung cancer example:

• Smoker & Lung Cancer: E = (200 * 90) / 500 = 36
• Smoker & No Lung Cancer: E = (200 * 410) / 500 = 164
• Non-Smoker & Lung Cancer: E = (300 * 90) / 500 = 54
• Non-Smoker & No Lung Cancer: E = (300 * 410) / 500 = 246

We can now create a table of expected frequencies:

4. Verify Your Calculations

A helpful check to ensure your calculations are correct is to verify that the row totals and column totals of the expected frequency table match the row totals and column totals of the original contingency table.

• 36 + 164 = 200 (matches the Smoker row total)
• 54 + 246 = 300 (matches the Non-Smoker row total)
• 36 + 54 = 90 (matches the Lung Cancer column total)
• 164 + 246 = 410 (matches the No Lung Cancer column total)

If these totals match, you can be confident in your calculations.

Example: Another Scenario

Let's consider another example. A researcher wants to investigate if there is a relationship between educational attainment and political affiliation. They survey 400 people and gather the following data:

Let's calculate the expected frequencies:

• High School & Democrat: E = (90 * 150) / 400 = 33.75
• High School & Republican: E = (90 * 120) / 400 = 27
• High School & Independent: E = (90 * 130) / 400 = 29.25
• Bachelor's Degree & Democrat: E = (140 * 150) / 400 = 52.5
• Bachelor's Degree & Republican: E = (140 * 120) / 400 = 42
• Bachelor's Degree & Independent: E = (140 * 130) / 400 = 45.5
• Graduate Degree & Democrat: E = (170 * 150) / 400 = 63.75
• Graduate Degree & Republican: E = (170 * 120) / 400 = 51
• Graduate Degree & Independent: E = (170 * 130) / 400 = 55.25

Resulting in the following expected frequency table:

The Chi-Square Formula and Interpretation

Once you have calculated the expected frequencies, you can proceed to calculate the Chi-Square statistic (χ²). The formula for the Chi-Square statistic is:

χ² = Σ [(O - E)² / E]

Where:

• Σ represents the summation across all cells in the contingency table.
• O is the observed frequency for a cell.
• E is the expected frequency for the same cell.

Steps to Calculate the Chi-Square Statistic:

1. For each cell: Subtract the expected frequency (E) from the observed frequency (O).
2. Square the difference: Square the result from step 1.
3. Divide by the expected frequency: Divide the squared difference from step 2 by the expected frequency (E).
4. Sum the results: Sum the results from step 3 for all cells in the contingency table. This gives you the Chi-Square statistic (χ²).

Using the Lung Cancer Example:

Let's calculate the Chi-Square statistic for our smoking and lung cancer example:

• Smoker & Lung Cancer: [(60 - 36)² / 36] = 16
• Smoker & No Lung Cancer: [(140 - 164)² / 164] = 3.54
• Non-Smoker & Lung Cancer: [(30 - 54)² / 54] = 10.67
• Non-Smoker & No Lung Cancer: [(270 - 246)² / 246] = 2.34

χ² = 16 + 3.54 + 10.67 + 2.34 = 32.55

Interpreting the Chi-Square Statistic:

The Chi-Square statistic tells us the overall discrepancy between the observed and expected frequencies. A larger Chi-Square statistic indicates a greater difference between the observed and expected values, suggesting a stronger association between the variables.

Determining Statistical Significance:

To determine if the Chi-Square statistic is statistically significant, we need to compare it to a critical value from the Chi-Square distribution or calculate the p-value. This requires knowing the degrees of freedom (df), which is calculated as:

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

In our lung cancer example, df = (2 - 1) * (2 - 1) = 1

We then compare our calculated Chi-Square statistic (32.55) with the critical value for a Chi-Square distribution with 1 degree of freedom at a chosen significance level (alpha, typically 0.05). Alternatively, we can use statistical software or an online calculator to determine the p-value associated with our Chi-Square statistic.

• If the Chi-Square statistic is greater than the critical value (or the p-value is less than alpha): We reject the null hypothesis and conclude that there is a statistically significant association between the two variables. In our example, 32.55 is significantly larger than the critical value of 3.84 (at alpha = 0.05), and the p-value would be very small (much less than 0.05). Therefore, we would reject the null hypothesis and conclude that there is a significant association between smoking and lung cancer.
• If the Chi-Square statistic is less than the critical value (or the p-value is greater than alpha): We fail to reject the null hypothesis and conclude that there is no statistically significant association between the two variables.

Important Considerations and Assumptions

While the Chi-Square test is a powerful tool, it's important to be aware of its limitations and assumptions:

• Independence of Observations: The data points must be independent of each other. This means that one observation should not influence another.
• Expected Frequencies: A general rule of thumb is that all expected frequencies should be 5 or greater. If an expected frequency is less than 5, the Chi-Square test may not be accurate. In such cases, consider combining categories or using an alternative test like Fisher's exact test. This is particularly important in smaller sample sizes.
• Categorical Data: The Chi-Square test is designed for categorical data. It cannot be used with continuous variables directly. If you have continuous data, you may need to categorize it first.
• Does Not Imply Causation: A significant association does not imply causation. Even if you find a statistically significant relationship between two variables, it doesn't necessarily mean that one variable causes the other. There may be other confounding variables at play.

Common Mistakes to Avoid

• Incorrectly Calculating Expected Frequencies: Double-check your calculations to ensure accuracy. A small error in calculating expected frequencies can significantly impact the Chi-Square statistic and your conclusions.
• Violating the Assumption of Independence: Ensure that your data points are truly independent. If there's a dependency between observations, the Chi-Square test may not be appropriate.
• Ignoring Low Expected Frequencies: Be mindful of the expected frequency rule. If you have cells with expected frequencies less than 5, consider alternative approaches.
• Misinterpreting Statistical Significance: Remember that statistical significance does not equal practical significance. A statistically significant result may not be meaningful in the real world. Consider the magnitude of the effect size and the context of your research.
• Drawing Causal Conclusions: Avoid drawing causal conclusions based solely on a Chi-Square test. Association does not equal causation.

Advanced Applications and Variations

While the basic Chi-Square test is widely used, there are several variations and advanced applications:

• Chi-Square Test for Goodness-of-Fit: This test assesses how well a sample distribution matches a hypothesized distribution.
• Yates' Correction for Continuity: This correction is sometimes applied to the Chi-Square test when dealing with 2x2 contingency tables, especially when sample sizes are small. It helps to reduce the overestimation of the Chi-Square statistic.
• Fisher's Exact Test: This test is an alternative to the Chi-Square test for 2x2 contingency tables, particularly when expected frequencies are low. It provides a more accurate p-value in such cases.
• Cochran-Mantel-Haenszel Test: This test is used to assess the association between two categorical variables while controlling for a third confounding variable.

FAQ

• What does expected frequency mean in chi-square?

  Expected frequency represents the number of observations you would anticipate in a cell of a contingency table if the two categorical variables being analyzed were completely independent of each other. It's the "baseline" against which you compare your observed frequencies.

• How do you find the expected value?

  In the context of the Chi-Square test, the expected value (or expected frequency) for each cell in the contingency table is calculated using the formula: E = (Row Total * Column Total) / Grand Total.

• What happens if expected counts are too low?

  If expected counts are too low (typically less than 5 in any cell), the Chi-Square test might produce inaccurate results. You could consider combining categories to increase the expected counts, or use an alternative test like Fisher's exact test (especially for 2x2 tables).

• What is the difference between observed and expected frequency?

  Observed frequency is the actual count of data points you have in each category combination in your sample. Expected frequency is the theoretical count you would expect to see in each category combination if there were no relationship between the variables. The Chi-Square test compares these two to see if the difference is statistically significant.

• Can the expected frequency be a decimal?

  Yes, the expected frequency can be a decimal. It represents the average number of observations you would expect over many repeated samples if the variables were independent. The observed frequencies, on the other hand, must always be whole numbers.

Conclusion

Mastering the calculation of expected frequencies is crucial for effectively utilizing the Chi-Square test. By understanding the underlying principles and following the steps outlined above, you can confidently analyze categorical data, uncover meaningful relationships, and draw informed conclusions. Remember to always consider the assumptions and limitations of the test and to interpret your results in the context of your research question. The Chi-Square test, when applied correctly, provides a valuable tool for exploring relationships within categorical data and advancing our understanding of the world around us.