X2 Score (Chi-Square) Calculators

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The chi-square (χ², X²) statistic tests whether observed categorical data differ significantly from expected values. Two main applications: (1) Goodness-of-fit test: does a sample distribution match an expected distribution (e.g., do observed Mendelian phenotype ratios match the expected 3:1)? (2) Test of independence: are two categorical variables independent (e.g., is smoking status independent of disease outcome)? The chi-square statistic: χ² = Σ(O − E)² / E, where O = observed count and E = expected count. Significance is assessed by comparing χ² to the chi-square distribution with the appropriate degrees of freedom.

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Chi-Square Formula

χ² = Σ (O − E)² / E

Summed over all categories or cells. Large χ² → large deviation between observed and expected → evidence against H₀.

Goodness of Fit Test

Example: observed phenotype counts 312, 104, 108, 36 vs. expected 9:3:3:1 ratio from 560 total. Expected = 315, 105, 105, 35. χ² = (312−315)²/315 + (104−105)²/105 + (108−105)²/105 + (36−35)²/35 = 0.029 + 0.010 + 0.086 + 0.029 = 0.154. df = 4−1 = 3. Critical value χ²(0.05, 3) = 7.815. 0.154 << 7.815 → fail to reject H₀ → data consistent with 9:3:3:1 ratio.

Test of Independence (Contingency Table)

Test whether two categorical variables are independent. E = (row total × column total) / grand total. df = (rows−1) × (columns−1).

Degrees of Freedom

  • Goodness of fit: df = k − 1 (k = categories) minus additional constraints
  • Test of independence: df = (r−1)(c−1) (r = rows, c = columns)

Assumptions

All expected frequencies ≥ 5 (use Fisher's exact test for 2×2 tables with small E). Random sample. Observations independent. Categorical data only.

Glossary

Chi-Square Statistic (χ²)
χ² = Σ(O−E)²/E; measures deviation of observed from expected categorical frequencies; compared to chi-square distribution with df = k−1 (goodness of fit) or (r−1)(c−1) (independence).
Goodness-of-Fit Test
Chi-square test comparing observed category frequencies to expected frequencies from a hypothesized distribution; df = categories − 1; used for Mendelian ratio testing and distribution comparisons.
Test of Independence
Chi-square test for a contingency table examining whether two categorical variables are associated; df = (rows−1)(cols−1); E = row total × column total / grand total.

Frequently Asked Questions

The chi-square (χ²) test evaluates whether observed frequencies of categorical data differ significantly from expected frequencies. Two applications: (1) Goodness-of-fit — does the observed distribution match the expected? E.g., are observed phenotype ratios consistent with a 3:1 Mendelian ratio? (2) Test of independence — are two categorical variables associated? E.g., does vaccine status relate to disease outcome? Formula: χ² = Σ(O−E)²/E, summed over all categories. H₀: observed matches expected (goodness of fit) or variables are independent (independence test). Reject H₀ when χ² > critical value at the chosen α level.

Step 1: State H₀ (the expected ratio/distribution). Step 2: Calculate expected counts: E = total × expected proportion for each category. Step 3: Calculate χ² = Σ(O−E)²/E for each category. Step 4: df = number of categories − 1 (−additional constraints if any). Step 5: Compare χ² to critical value from chi-square table at chosen α (usually 0.05). If χ² > critical value: reject H₀ (observed differs significantly from expected). Example: df = 3, α = 0.05: critical χ² = 7.815. If observed χ² = 2.4: p > 0.05; data consistent with H₀.

2×2 or larger contingency tables: each cell has observed frequency O. Expected frequency E = (row total × column total) / grand total. χ² = Σ(O−E)²/E over all cells. df = (r−1)(c−1). For a 2×2 table (df=1): χ² > 3.841 (α=0.05) → reject H₀ of independence. Example: 2×2 vaccine vs. disease table: row totals = 200, 200 (vaccinated/unvaccinated); column totals = 60 disease, 340 no disease; grand total = 400. E₁₁ = 200×60/400 = 30; etc. Calculate χ² from observed vs. expected for each cell.

Chi-square assumptions: (1) Random sample — observations must be randomly selected from the population. (2) Independence — each observation belongs to exactly one category; multiple observations per subject violate this. (3) Adequate expected frequencies — all expected values should be ≥ 5. When any E < 5 in a 2×2 table: use Fisher's exact test instead. For larger tables: if fewer than 20% of cells have E < 5 and no cell has E < 1, the approximation is still reasonable. (4) Categorical data — chi-square requires counts (frequencies), not means or continuous measurements. Combining cells with very small expected values is sometimes used to meet the assumption, but this changes the df and test interpretation.