Chi-Square Calculator

Calculate Chi-Square (rho�^2) statistic for goodness-of-fit tests. Determine if observed frequencies differ significantly from expected frequencies. Includes p-value and detailed statistical interpretation.

Observed frequency for category 1

Expected frequency for category 1

Observed frequency for category 2

Expected frequency for category 2

Observed frequency for category 3

Expected frequency for category 3

Observed frequency for category 4

Expected frequency for category 4

Observed frequency for category 5

Expected frequency for category 5

Observed frequency for category 6

Expected frequency for category 6

Chi-Square Statistic: rho�^2 = Σ[(O - E)^2 / E] Where: O = Observed frequency E = Expected frequency Σ = Sum across all categories Degrees of Freedom: df = number of categories - 1 Decision Rule: If rho�^2 > critical value → Reject H₀ (significant) If rho�^2 <= critical value → Fail to reject H₀ (not significant) Or: If p-value < alpha → Reject H₀ Assumptions: • Independent observations • Expected frequency >= 5 in each category • Categorical data • Random sampling
Example 1 (Dice Fairness): Test if a die is fair (expected: each face 1/6 probability) Rolled 120 times: Observed: [25, 18, 22, 19, 21, 15] Expected: [20, 20, 20, 20, 20, 20] Calculations: Category 1: (25-20)^2/20 = 1.25 Category 2: (18-20)^2/20 = 0.20 Category 3: (22-20)^2/20 = 0.20 Category 4: (19-20)^2/20 = 0.05 Category 5: (21-20)^2/20 = 0.05 Category 6: (15-20)^2/20 = 1.25 rho�^2 = 1.25 + 0.20 + 0.20 + 0.05 + 0.05 + 1.25 = 3.00 df = 6 - 1 = 5 Critical value (alpha=0.05) = 11.070 Result: rho�^2 (3.00) < 11.070 → NOT significant Conclusion: No evidence die is unfair Example 2 (Survey Analysis): Expected equal preference (25% each) Sample size: 200 Observed: [65, 45, 50, 40] Expected: [50, 50, 50, 50] rho�^2 = (15^2/50) + (-5^2/50) + (0^2/50) + (-10^2/50) = 4.5 + 0.5 + 0 + 2.0 = 7.0 df = 3, Critical (alpha=0.05) = 7.815 Result: NOT significant (7.0 < 7.815)

What is the Chi-Square test?

The Chi-Square (rho�^2) test is a statistical test used to determine if there is a significant association between categorical variables or if observed frequencies differ from expected frequencies. It measures the difference between observed and expected data, with larger rho�^2 values indicating greater discrepancy.

How do you calculate the Chi-Square statistic?

Formula: rho�^2 = Σ[(O - E)^2 / E], where O = observed frequency, E = expected frequency, and Σ means sum across all categories. For each category: subtract expected from observed, square the result, divide by expected, then sum all values. Example: O=30, E=25 gives (30-25)^2/25 = 1.0 for that cell.

What are degrees of freedom in Chi-Square tests?

Degrees of freedom (df) = number of categories - 1 for goodness-of-fit tests, or df = (rows - 1) * (columns - 1) for independence tests. Example: A 2*3 contingency table has df = (2-1)*(3-1) = 2. Higher df requires larger rho�^2 values for significance.

How do you interpret Chi-Square results?

Compare rho�^2 statistic to critical value at chosen significance level (alpha, usually 0.05). If rho�^2 > critical value, reject null hypothesis (significant association exists). Alternatively, if p-value < alpha, reject null hypothesis. Example: rho�^2=7.82, df=3, critical value (alpha=0.05)=7.815 → significant result.

What is the difference between Chi-Square goodness-of-fit and independence tests?

Goodness-of-fit tests whether observed frequencies match expected distribution (one variable). Independence tests whether two categorical variables are related (contingency table). Example: Goodness-of-fit: Do dice rolls match expected 1/6? Independence: Is smoking related to lung cancer?

What assumptions must be met for Chi-Square tests?

Requirements: (1) Independent observations, (2) Expected frequency >= 5 in each category (some allow >= 1), (3) Categorical data, (4) Random sampling. If expected frequencies < 5, consider Fisher's exact test or combine categories. Violating assumptions can lead to incorrect conclusions.

What is a p-value in Chi-Square testing?

The p-value is the probability of obtaining a rho�^2 statistic as extreme as observed, assuming null hypothesis is true. Lower p-values indicate stronger evidence against null hypothesis. Standard: p < 0.05 = significant, p < 0.01 = highly significant, p >= 0.05 = not significant. Example: p=0.03 suggests significant relationship.

What are real-world applications of Chi-Square tests?

Medical research (treatment effectiveness), genetics (Mendel's laws), marketing (customer preference vs demographics), quality control (defect rates), social sciences (survey analysis), A/B testing (conversion rates), education (grading distributions), epidemiology (disease association with factors). Widely used for categorical data analysis.

What is the critical value in Chi-Square tests?

The critical value is the threshold from Chi-Square distribution tables based on degrees of freedom and significance level (alpha). If calculated rho�^2 exceeds critical value, result is significant. Example: df=2, alpha=0.05 gives critical value 5.991. rho�^2=6.5 > 5.991 → reject null hypothesis.

Can Chi-Square tests determine causation?

No, Chi-Square tests only detect associations or relationships between variables, not causation. A significant result means variables are related, but doesn't prove one causes the other. Correlation ≠ causation. Example: Ice cream sales and drowning are associated (both increase in summer) but ice cream doesn't cause drowning.