Chi-Squared Calculator
Input your observed and expected frequencies to calculate the Chi-Squared statistic.
Chi-Squared Statistic (χ²)
Understanding the Chi-Squared (χ²) Test and Calculator
The Chi-Squared (χ²) test is a fundamental statistical method used to determine if there is a significant difference between expected frequencies and observed frequencies in one or more categories. It is widely employed in fields such as genetics, marketing research, social sciences, and quality control to analyze categorical data.
What does the Chi-Squared Statistic tell us?
The Chi-Squared statistic quantifies the discrepancy between what we expect to see and what we actually observe in our data. A larger χ² value indicates a greater difference between the observed and expected frequencies, suggesting that the observed pattern is less likely to have occurred by chance and that there might be a real association or effect.
How the Chi-Squared Calculator Works
This calculator implements the formula for the Chi-Squared goodness-of-fit test. The formula is:
χ² = Σ [ (Oᵢ - Eᵢ)² / Eᵢ ]
Where:
χ²is the Chi-Squared statistic.Σ(Sigma) represents the sum across all categories.Oᵢis the observed frequency for category i.Eᵢis the expected frequency for category i.
The calculator takes comma-separated lists of observed and expected frequencies. It then calculates the difference for each pair of observed and expected values, squares that difference, and divides it by the expected value. Finally, it sums up these values for all categories to produce the final Chi-Squared statistic.
When to Use This Calculator
- Goodness-of-Fit Test: To determine if a sample distribution matches a hypothesized population distribution (e.g., does a die land on each face with equal probability?).
- Test of Independence: To assess whether two categorical variables are independent or if there's a significant association between them (e.g., is there a relationship between gender and political preference?). For this calculator, you would typically input observed counts from a contingency table and the expected counts derived under the assumption of independence.
- Test of Homogeneity: To determine if the distribution of a categorical variable is the same across different populations (e.g., do different age groups have the same preference for a product?).
Interpreting the Result
The calculated Chi-Squared statistic alone is not enough; it needs to be interpreted in the context of degrees of freedom and a chosen significance level (alpha, α). You would typically compare the calculated χ² value to a critical value from a Chi-Squared distribution table or calculate a p-value.
- Degrees of Freedom (df): For a goodness-of-fit test, df = (number of categories – 1). For a test of independence or homogeneity with a contingency table of r rows and c columns, df = (r – 1) * (c – 1).
- Significance Level (α): Commonly set at 0.05.
- Decision: If the calculated χ² value is greater than the critical value (or if the p-value is less than α), you reject the null hypothesis. This means there is a statistically significant difference or association.
Example Calculation
Let's say we are testing if a four-sided die is fair. We roll it 100 times and observe the following frequencies:
- Side 1: Observed = 20
- Side 2: Observed = 30
- Side 3: Observed = 25
- Side 4: Observed = 25
If the die were fair (null hypothesis), we would expect each side to occur 100 / 4 = 25 times.
Observed Frequencies: 20, 30, 25, 25
Expected Frequencies: 25, 25, 25, 25
Using the calculator with these inputs:
- Category 1:
(20 - 25)² / 25 = (-5)² / 25 = 25 / 25 = 1.0 - Category 2:
(30 - 25)² / 25 = (5)² / 25 = 25 / 25 = 1.0 - Category 3:
(25 - 25)² / 25 = (0)² / 25 = 0 / 25 = 0.0 - Category 4:
(25 - 25)² / 25 = (0)² / 25 = 0 / 25 = 0.0
Total Chi-Squared Statistic: 1.0 + 1.0 + 0.0 + 0.0 = 2.0
For this example, the degrees of freedom would be 4 (categories) – 1 = 3. If our significance level was 0.05, the critical value for χ² with 3 df is 7.815. Since 2.0 is less than 7.815, we would not reject the null hypothesis and conclude there is no statistically significant evidence that the die is unfair.