Chi-Squared Test Calculator
Use this calculator to determine the Chi-Squared (χ²) statistic and degrees of freedom for a goodness-of-fit test. Enter your observed and expected frequencies for each category below. The calculator will sum up the contributions from all valid pairs.
Frequency Data (Enter at least two pairs):
Results:
Understanding the Chi-Squared Test
The Chi-Squared (χ²) test is a non-parametric statistical test used to determine if there is a significant difference between the observed frequencies and the expected frequencies in one or more categories. It's particularly useful for analyzing categorical data.
When to Use the Chi-Squared Test
There are two primary types of Chi-Squared tests:
- 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 red, green, and blue candies in a bag matches the manufacturer's stated proportions. This calculator is designed for the goodness-of-fit test.
- Test of Independence: This test determines if there is a significant association between two categorical variables. For example, you might use it to see if there's a relationship between gender and political party preference.
Null and Alternative Hypotheses
For a goodness-of-fit test, the hypotheses are typically:
- Null Hypothesis (H₀): There is no significant difference between the observed frequencies and the expected frequencies. Any observed differences are due to random chance.
- Alternative Hypothesis (H₁): There is a significant difference between the observed frequencies and the expected frequencies. The observed differences are not due to random chance.
The Chi-Squared Formula
The Chi-Squared statistic is calculated using the following formula:
χ² = Σ [(O – E)² / E]
Where:
- Σ (Sigma) means "sum of"
- O represents the Observed Frequency (the actual counts from your data)
- E represents the Expected Frequency (the counts you would expect if the null hypothesis were true)
The formula essentially measures the discrepancy between what you observed and what you expected. Larger discrepancies lead to a larger Chi-Squared value.
Degrees of Freedom (df)
Degrees of freedom refer to the number of independent values that can vary in an analysis without violating any constraints. For a goodness-of-fit test, the degrees of freedom are calculated as:
df = k – 1
Where k is the number of categories or groups being compared.
Interpreting the Results
Once you have calculated the Chi-Squared statistic and the degrees of freedom, you need to compare your calculated χ² value to a critical value from a Chi-Squared distribution table. This table provides critical values for different degrees of freedom and significance levels (alpha, α).
- Significance Level (α): This is the probability of rejecting the null hypothesis when it is actually true (Type I error). Common alpha levels are 0.05 (5%) or 0.01 (1%).
- If Calculated χ² > Critical Value: You reject the null hypothesis. This suggests that there is a statistically significant difference between your observed and expected frequencies.
- If Calculated χ² ≤ Critical Value: You fail to reject the null hypothesis. This suggests that there is no statistically significant difference, and any observed differences could be due to random chance.
Alternatively, statistical software often provides a p-value. If the p-value is less than your chosen significance level (α), you reject the null hypothesis.
Example Scenario
Imagine a genetics experiment where you expect a 9:3:3:1 phenotypic ratio (16 total categories). You observe the following counts in 160 offspring:
- Category A: Observed = 95, Expected = (9/16)*160 = 90
- Category B: Observed = 25, Expected = (3/16)*160 = 30
- Category C: Observed = 35, Expected = (3/16)*160 = 30
- Category D: Observed = 5, Expected = (1/16)*160 = 10
Using the calculator, you would input these observed and expected frequencies. The calculator would then compute the Chi-Squared statistic and the degrees of freedom (which would be 4 – 1 = 3 in this case). You would then compare the calculated χ² to a critical value from a table at your chosen alpha level (e.g., 0.05) with 3 degrees of freedom to draw a conclusion about whether the observed ratio significantly deviates from the expected 9:3:3:1 ratio.