Chi-Square Test Calculator

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Mode Test of Independence Goodness-of-Fit

Rows × Columns

× Update

For independence tables

Categories

Update

For goodness-of-fit

Estimated Parameters (m)

Contingency Table (Observed)

Row labels (comma separated)

Column labels (comma separated)

Enter non-negative counts. Row/column totals computed automatically.

Category labels (comma separated)

Observed Counts

Expected Counts

Set Uniform Scale to Observed Total

Provide expected counts or set uniform expectations.

Calculate Clear

Formulas Used

Chi-square statistic: χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ, summing over cells or categories.

• Independence: Eᵢⱼ = (rowᵢ total × colⱼ total) / N, df = (r−1)(c−1), optional Yates correction for 2×2.
• Goodness-of-fit: user-provided expectations, df = k − 1 − m, where m is parameters estimated from data.
• p-value: P(Χ² ≥ χ² | df) from the chi-square distribution via regularized gamma functions.
• Residuals: Pearson (O−E)/√E; adjusted for independence (O−E)/√(E(1−rᵢ)(1−cⱼ)).
• Effect sizes: φ = √(χ²/N) for 2×2; V = √(χ²/(N·min(r−1,c−1))); C = √(χ²/(χ²+N)).

How to Use This Calculator

1. Select the test type and adjust alpha, precision, and options.
2. Set table size or categories, then enter observed counts.
3. For independence, optionally enter row/column labels.
4. For goodness-of-fit, enter expected counts or choose uniform.
5. Click Calculate to view χ², df, p-value, and details.
6. Export results as CSV or PDF, or save/load your JSON setup.

FAQs

When should I use the test of independence?

Use it to check association between two categorical variables in a contingency table. Counts, not percentages. Adequate expected counts (ideally ≥5). Null states variables are independent; alternative states association exists.

When should I use goodness‑of‑fit?

Use it to compare one categorical distribution to a specified target distribution. Provide expected counts or proportions scaled to the observed total. Degrees of freedom are categories minus one minus parameters estimated from data.

What if expected counts are small?

If many expected counts are below five, combine sparse categories if substantively reasonable. For 2×2 tables, you may enable Yates continuity correction or consider Fisher’s exact test. Report limitations clearly and interpret results cautiously.

How do I interpret residuals?

Pearson residuals show standardized cell deviations from expectation. Adjusted residuals account for row and column proportions; values beyond about ±2 often flag influential cells. Use them to locate categories contributing most to the overall chi‑square statistic.

Which effect size should I report?

For general r×c tables, report Cramér’s V. For 2×2 tables, Phi is common; you may also show the contingency coefficient. Interpret magnitudes using discipline‑specific conventions rather than fixed universal thresholds.

Can I paste percentages instead of counts?

Please enter raw counts. If you only have percentages, multiply each percentage by a common base total to obtain integer or fractional counts, ensuring the scaled expected counts align with the observed total.