χ²
Your calculated chi-square value
df
Number of categories minus 1
P-Value
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Understanding Chi-Square Tests
The chi-square (χ²) test is one of the most widely used statistical tests for analyzing categorical data. It measures how much observed frequencies deviate from expected frequencies.
χ² = Σ [(Observed − Expected)² ÷ Expected]
Sum this ratio across all categories. Then use degrees of freedom (df = categories − 1) to find the p-value.
Two Main Uses
- Goodness-of-fit test: Does a sample match a theoretical distribution? (e.g., is a die fair?)
- Test of independence: Are two categorical variables independent? (e.g., is gender independent of product preference?)
💡 Degrees of freedom for a goodness-of-fit test = (number of categories − 1). For a contingency table = (rows − 1) × (columns − 1).
Chi-Square Critical Values Reference
Common critical values at α = 0.05 (reject H₀ if χ² exceeds these):
- df = 1: 3.841 | df = 2: 5.991 | df = 3: 7.815
- df = 4: 9.488 | df = 5: 11.070 | df = 6: 12.592
- df = 7: 14.067 | df = 8: 15.507 | df = 10: 18.307
At α = 0.01: df=1 → 6.635; df=3 → 11.345; df=5 → 15.086. For df > 10, use the calculator above for exact p-values.
Frequently Asked Questions
The p-value is the probability of observing a chi-square statistic as large as (or larger than) yours if the null hypothesis were true. A small p-value (typically < 0.05) means the result is statistically significant and you reject the null hypothesis.
Degrees of freedom (df) represent the number of values free to vary in the calculation. For a goodness-of-fit test with k categories, df = k − 1. For a contingency table with r rows and c columns, df = (r−1) × (c−1). The df parameter shapes the chi-square distribution used to find the p-value.
The chi-square statistic measures how far observed data deviates from expected data — it is a raw number. The p-value converts that number into a probability using the chi-square distribution with the given degrees of freedom. The p-value tells you whether the deviation is statistically meaningful.
There is no universally "good" chi-square value — it depends on degrees of freedom and your significance threshold. What matters is the resulting p-value. Critical chi-square values at α = 0.05: df=1 → 3.841; df=2 → 5.991; df=3 → 7.815; df=4 → 9.488; df=5 → 11.070.
Chi-square tests are unreliable when expected cell frequencies are below 5. In those cases, use Fisher's Exact Test instead. As a rule of thumb: all expected frequencies should be ≥ 5, and no more than 20% of cells should have expected frequencies below 5. Increasing sample size is the best fix for small expected cell counts.
The chi-square distribution is a family of probability distributions parametrized by degrees of freedom. It is always right-skewed and takes only positive values. As df increases, it becomes more symmetric and bell-shaped. The p-value is the area under the chi-square curve to the right of your test statistic — the tail probability.
Sources & Methodology
This calculator's logic and benchmarks are based on the following authoritative sources, reviewed regularly for accuracy.
NIST/SEMATECH e-Handbook of Statistical Methods
Chi-square distribution and hypothesis testing — nist.gov
Khan Academy — Chi-Square Tests
Goodness-of-fit and independence test instruction — khanacademy.org
Pearson (1900) — Chi-Square Original Paper
Original chi-square test publication — philosophical basis
Methodology: Chi-Square P-Value Calculations calculations use industry-standard formulas verified against the sources above. Results are estimates — consult a qualified professional for significant financial, legal, or structural decisions.
Last reviewed: March 2026
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