Chi-Square Goodness-of-Fit

χ² = Σ(O−E)²/E. Test if observed frequencies match expected.

Inputs

Result

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How to use this calculator

  • Enter observed counts + expected counts.

About this calculator

Chi-square goodness-of-fit test: do observed counts match expected? χ² = Σ(O−E)²/E. Larger = bigger discrepancy. Compare to χ² distribution at df = (categories − 1). Common use: testing if data follows hypothesized distribution (uniform, normal, Poisson). Sample size requirement: each expected ≥ 5 for good approximation. Source: Pearson (1900); NIST Engineering Statistics Handbook.

Frequently asked

Goodness-of-fit vs. independence?+
Goodness-of-fit: 1 categorical variable vs. expected. Independence: 2 categorical variables tested for relationship.
Sample size?+
Each expected count ≥ 5 for good approximation. Below: use Fisher's exact test.
Effect size?+
Cramer's V = √(χ² / N(k−1)). 0.1 small, 0.3 medium, 0.5 large.
Critical values?+
df=1: 3.84 (α=.05), 6.63 (α=.01). df=2: 5.99 / 9.21. df=3: 7.81 / 11.34.
Multi-way contingency?+
For r×c table: χ² = ΣΣ(O−E)²/E with df = (r−1)(c−1).

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