Chi-Square Calculator

Frequently Asked Questions

Why must I use counts, not percentages?

The chi-square formula divides by the expected count E, which acts as a variance estimate. If you substitute proportions or percentages, the denominator no longer has the right units and the resulting statistic does not follow a chi-square distribution.

What should I do when expected counts are less than 5?

Combine adjacent categories to raise expected counts above 5, or use Fisher's exact test (for 2×2 tables), or a Monte Carlo permutation test (for larger tables). The chi-square approximation is unreliable with small expectations.

Can chi-square detect the direction of an association?

No. It only tests whether the observed distribution departs from the null. To understand direction and magnitude, examine standardized residuals and compute a measure like Cramér's V or the phi coefficient.

What is the difference between goodness-of-fit and test of independence?

Goodness-of-fit tests whether a single variable matches a specified distribution. Test of independence tests whether two variables in a contingency table are related. Computationally, both use the same χ² = Σ(O−E)²÷E formula, differing only in how expected values are determined.

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This calculator provides estimates for informational purposes only. Results are based on assumptions and may not reflect actual outcomes. Consult qualified professionals in relevant fields before making important decisions based on these results.

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How This Calculator Works

The chi-square statistic measures how far observed counts deviate from the counts a null model predicts: χ² = Σ (O − E)² ÷ E. In a goodness-of-fit test you supply observed and expected frequencies directly; df = k − 1, where k is the number of categories. In a test of independence you supply a contingency table; the expected count for cell (i,j) is (row total_i × column total_j) ÷ grand total, and df = (rows − 1) × (columns − 1).

The upper-tail p-value P(χ²_df > observed) is computed from the χ² CDF, which equals the regularized lower incomplete gamma function P(df÷2, χ²÷2). This tool evaluates that function with a series-plus-continued-fraction routine accurate to roughly four decimal places.

Worked Example: Goodness-of-Fit

Observed counts: 18, 22, 20, 25, 15. Expected counts: 20, 20, 20, 20, 20 (a uniform distribution with 5 categories). Contributions: (−2)²÷20 + 2²÷20 + 0 + 5²÷20 + (−5)²÷20 = 0.2 + 0.2 + 0 + 1.25 + 1.25 = 2.90. With df = 4, the upper-tail p-value is about 0.575 - far above α = 0.05, so we fail to reject H₀. These deviations are consistent with ordinary sampling noise from a fair die.

When to Use Chi-Square

Goodness-of-fit: does a categorical variable follow a hypothesized distribution - uniform, Mendelian ratio, market-share benchmark?

Independence: are two categorical variables associated - treatment × outcome, region × preference, gender × vote choice?

Homogeneity: mechanically identical to independence - comparing the distribution of one categorical variable across several populations.

Assumptions and Common Mistakes

The test requires counts (raw frequencies), not proportions, percentages, or rates. Always convert back to raw numbers first. The chi-square approximation degrades when expected cell counts are small: the common rule is every expected count ≥ 5 with none below 1. For 2 × 2 tables with small expected counts, use Fisher's exact test or a Yates continuity correction.

Observations must be independent: repeated measures on the same subject violate this. A significant result indicates association, never causation or direction.

Effect Size and Interpretation

A large χ² relative to its df means the data are unlikely under the null of no difference or no association. However, χ² scales with sample size, so huge datasets will flag trivially small departures as significant. To quantify the strength of association, pair the test with an effect-size measure: Cramér's V = √(χ² ÷ (N × min(r−1, c−1))) ranges from 0 to 1, with 0.1 considered small, 0.3 medium, and 0.5 large.

Always inspect standardized residuals (O − E) ÷ √E to see which cells drive the result. A significant omnibus χ² without examining residuals is an incomplete analysis.

Chi-Square Calculator

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