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how to find expected value in chi square test

how to find expected value in chi square test

2 min read 30-12-2024
how to find expected value in chi square test

The chi-square (χ²) test is a powerful statistical tool used to determine if there's a significant association between two categorical variables. Understanding how to calculate the expected values is crucial for conducting this test accurately. This article will guide you through the process.

Understanding Expected Values in the Chi-Square Test

Before diving into calculations, let's clarify what expected values represent. In a chi-square test, we compare observed frequencies (the actual counts in your data) with expected frequencies. Expected values represent the frequencies you'd expect to see in each category if there were no association between the variables. A large discrepancy between observed and expected values suggests a significant relationship.

Calculating Expected Values: A Step-by-Step Guide

The formula for calculating expected values in a chi-square test is straightforward:

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

Let's break this down with a clear example:

Scenario: Imagine you're investigating whether there's a relationship between gender and preference for coffee or tea. You collect the following data:

Coffee Tea Total
Male 30 20 50
Female 25 35 60
Total 55 55 110

Step 1: Identify Row and Column Totals:

The table already provides these. For example, the row total for "Male" is 50, and the column total for "Coffee" is 55.

Step 2: Determine the Grand Total:

This is the total number of observations in your entire dataset (110 in this case).

Step 3: Calculate Expected Values for Each Cell:

Let's calculate the expected value for the cell representing "Male" who prefer "Coffee":

  • Row Total (Male): 50
  • Column Total (Coffee): 55
  • Grand Total: 110

Expected Value (Male prefers Coffee) = (50 * 55) / 110 = 25

Let's do one more:

Expected Value (Female prefers Tea) = (60 * 55) / 110 = 30

We repeat this calculation for each cell in the table:

Coffee (Observed) Tea (Observed) Total Coffee (Expected) Tea (Expected)
Male 30 20 50 25 25
Female 25 35 60 30 30
Total 55 55 110 55 55

Interpreting Expected Values

Notice that if there were no association between gender and drink preference, we'd expect roughly equal numbers of males and females to prefer coffee and tea. The differences between observed and expected values will be used in the chi-square calculation to assess the significance of any relationship. A larger difference suggests a stronger association.

Beyond the Basics: More Complex Scenarios

The principles remain the same for more complex contingency tables (tables with more rows and columns). You'll simply apply the same formula to each cell, calculating the expected frequency based on its corresponding row and column totals.

Conducting the Chi-Square Test

Once you've calculated the expected values, you can proceed with the chi-square test itself. This involves calculating the chi-square statistic, determining the degrees of freedom, and comparing your results to a chi-square distribution table to obtain a p-value. A low p-value (typically below 0.05) indicates a statistically significant association between the variables. Many statistical software packages and online calculators can automate this process.

Conclusion

Calculating expected values is a fundamental step in performing a chi-square test. By understanding this process and correctly calculating these values, you can accurately assess the association between categorical variables in your data. Remember to always consult statistical resources and software for precise calculations and interpretations, especially in complex scenarios.

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