Chi-Square Test Calculator
Enter the observed and expected frequencies for each category. Click "Add Category" to include more categories in your analysis.
Understanding the Chi-Square Test
The Chi-Square (χ²) test is a non-parametric statistical test used to determine if there is a significant association between two categorical variables or if an observed distribution differs significantly from an expected distribution. It's commonly applied in two main scenarios:
- Goodness-of-Fit Test: To determine if an observed frequency distribution matches an expected distribution. For example, testing if the number of red, green, and blue candies in a bag matches the manufacturer's stated proportions.
- Test of Independence: To determine if there is a significant relationship between two categorical variables. For example, testing if gender is independent of political party affiliation.
The Chi-Square Formula
The Chi-Square statistic is calculated using the following formula:
χ² = Σ [(Oᵢ – Eᵢ)² / Eᵢ]
Where:
- Σ (Sigma) represents the sum across all categories.
- Oᵢ is the observed frequency (the actual count) for the i-th category.
- Eᵢ is the expected frequency (the count you would expect if the null hypothesis were true) for the i-th category.
Degrees of Freedom (df)
Degrees of freedom are a crucial component for interpreting the Chi-Square value. For a goodness-of-fit test, the degrees of freedom are calculated as:
df = (Number of Categories) – 1
For a test of independence (using a contingency table), it's:
df = (Number of Rows – 1) * (Number of Columns – 1)
This calculator focuses on the goodness-of-fit scenario where you provide observed and expected frequencies for different categories.
Interpreting the Chi-Square Result
Once you have the Chi-Square value and the degrees of freedom, you compare your calculated χ² to a critical value from a Chi-Square distribution table at a chosen significance level (e.g., 0.05). Alternatively, statistical software provides a p-value.
- If your calculated χ² is greater than the critical value (or p-value < significance level), you reject the null hypothesis. This suggests a statistically significant difference between the observed and expected frequencies, meaning the observed distribution does not fit the expected distribution, or there is an association between variables.
- If your calculated χ² is less than the critical value (or p-value ≥ significance level), you fail to reject the null hypothesis. This suggests that any observed differences are likely due to random chance, and there is no significant difference or association.
Example Calculation
Imagine a genetics experiment where you expect a 3:1 ratio of dominant to recessive phenotypes in offspring. You observe 70 dominant and 30 recessive offspring from a total of 100.
- Category 1 (Dominant):
- Observed (O₁): 70
- Expected (E₁): (3/4) * 100 = 75
- Contribution: (70 – 75)² / 75 = (-5)² / 75 = 25 / 75 = 0.3333
- Category 2 (Recessive):
- Observed (O₂): 30
- Expected (E₂): (1/4) * 100 = 25
- Contribution: (30 – 25)² / 25 = (5)² / 25 = 25 / 25 = 1.0000
Total Chi-Square (χ²): 0.3333 + 1.0000 = 1.3333
Degrees of Freedom (df): Number of Categories – 1 = 2 – 1 = 1
With 1 degree of freedom, a common critical value for α = 0.05 is 3.841. Since our calculated χ² (1.3333) is less than 3.841, we fail to reject the null hypothesis. This suggests that the observed frequencies are consistent with the expected 3:1 ratio.