Chi-Square Test Calculator
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Chi-Square Statistic (χ²): ${chiSquareSum.toFixed(4)} Degrees of Freedom (df): ${degreesOfFreedom} To interpret these results, compare your calculated Chi-Square statistic to a critical value from a Chi-Square distribution table. You will need your degrees of freedom and your chosen significance level (e.g., 0.05). If your calculated χ² is greater than the critical value, you reject the null hypothesis. `; } } // Initialize with two category rows on page load window.onload = function() { addCategoryRow(); addCategoryRow(); };Understanding the Chi-Square Test
The Chi-Square (χ²) test is a fundamental statistical tool used to examine the relationship between categorical variables. It's particularly useful for determining if there's a significant difference between observed frequencies (what you actually see in your data) and expected frequencies (what you would expect to see if there were no relationship or difference).
When to Use the Chi-Square Test
There are two primary applications for the Chi-Square test:
- Goodness-of-Fit Test: This test determines if an observed frequency distribution differs significantly from an expected distribution. For example, you might use it to see if the number of votes received by different political candidates in a sample matches the proportions predicted by a previous poll.
- Test of Independence: This test assesses whether two categorical variables are independent of each other. For instance, you could use it to determine if there's a relationship between gender and preference for a certain type of product.
The Chi-Square Formula
The Chi-Square statistic is calculated using the following formula:
χ² = Σ [(Oᵢ – Eᵢ)² / Eᵢ]
- Σ (Sigma) means "sum of"
- Oᵢ represents the observed frequency for each category
- Eᵢ represents the expected frequency for each category
In essence, the formula calculates the squared difference between observed and expected frequencies for each category, divides it by the expected frequency, and then sums these values across all categories. A larger Chi-Square value indicates a greater discrepancy between observed and expected frequencies.
Degrees of Freedom (df)
Degrees of Freedom (df) are crucial for interpreting the Chi-Square statistic. For a goodness-of-fit test (which this calculator primarily addresses by summing individual categories), the degrees of freedom are calculated as:
df = (Number of Categories – 1)
For a test of independence (contingency tables), the formula is: df = (Number of Rows – 1) * (Number of Columns – 1).
The degrees of freedom represent the number of independent pieces of information used to calculate the statistic. They are necessary to look up the critical value in a Chi-Square distribution table.
Interpreting the Results
Once you have your Chi-Square statistic and degrees of freedom, you compare your calculated χ² value to a critical value from a Chi-Square distribution table. This comparison is made at a chosen significance level (alpha, commonly 0.05 or 0.01).
- If your calculated χ² is greater than the critical value, you reject the null hypothesis. This suggests that there is a statistically significant difference between your observed and expected frequencies, or that the two categorical variables are not independent.
- If your calculated χ² is less than or equal to the critical value, you fail to reject the null hypothesis. This suggests that any observed differences are likely due to random chance.
Example Scenario
Imagine a researcher wants to test if a new teaching method has an effect on student performance categories (Pass, Fail). Historically, 80% of students pass and 20% fail. After implementing the new method, out of 100 students, 90 passed and 10 failed.
Observed Frequencies:
- Pass: 90
- Fail: 10
Expected Frequencies (based on historical data for 100 students):
- Pass: 100 * 0.80 = 80
- Fail: 100 * 0.20 = 20
Calculation:
- For Pass: (90 – 80)² / 80 = 10² / 80 = 100 / 80 = 1.25
- For Fail: (10 – 20)² / 20 = (-10)² / 20 = 100 / 20 = 5.00
Total Chi-Square (χ²) = 1.25 + 5.00 = 6.25
Degrees of Freedom (df) = Number of Categories – 1 = 2 – 1 = 1
Using a Chi-Square distribution table with 1 degree of freedom and a significance level of 0.05, the critical value is approximately 3.841. Since our calculated χ² (6.25) is greater than the critical value (3.841), we would reject the null hypothesis. This suggests that the new teaching method likely has a significant effect on student performance, as the observed results differ significantly from historical expectations.