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Chi-Square Critical Value Calculator
The chi-square critical value is the cutoff your χ² statistic must exceed to be significant at a chosen level α with given degrees of freedom — for example 3.841 at α = 0.05 with df = 1. This page computes critical values from α and df, and p-values from χ² and df, in both directions.
Reviewed by the crosstabs.com methods team · Last updated
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Use the left panel to find the critical value for a significance level and degrees of freedom, or the right panel to convert a chi-square statistic you already have into its p-value.
Critical value from α and df
Critical value χ²₍0.05₎
3.841
Reject H₀ at α = 0.05 when χ²(1) exceeds this value.
p-value from χ² and df
Right-tail p-value
p = .050
P(χ²(1) ≥ 3.84) — not significant at α = 0.05.
Critical value or p-value — which do you need?
They are two views of the same decision. The critical value approach fixes α in advance (usually 0.05) and rejects the null hypothesis when χ² exceeds the cutoff. The p-valueapproach converts the observed χ² into the probability of a value at least that large under the null, and rejects when p < α. Both always agree; modern software (and journals) generally report the p-value.
Note these are upper-tail values: chi-square tests of independence and goodness of fit are one-tailed on the right, because any departure from the null inflates χ².
Formula
Definition
P(χ²(df) ≥ critical value) = α
- α
- = the significance level — the accepted probability of a false positive
- df
- = degrees of freedom: (r − 1)(c − 1) for a test of independence, k − 1 for goodness of fit
- critical value
- = the (1 − α) quantile of the chi-square distribution; this calculator inverts the p-value function numerically
Worked example
Worked example
You run a chi-square test of independence on a 2×2 table and get χ² = 5.20 with df = 1.
The critical value at α = 0.05 with df = 1 is 3.841. Since 5.20 > 3.841, the result is significant at the 5% level. Equivalently, the p-value of χ² = 5.20 with df = 1 is p = .023, which is below 0.05 — the same conclusion.
At the stricter α = 0.01 the critical value is 6.635; 5.20 does not reach it, so the result is not significant at the 1% level.
How to interpret it
Rule of thumb
Reject the null hypothesis when your χ² statistic exceeds the critical value for your α and df — equivalently, when the p-value falls below α. Significance says the departure is real; report an effect size such as Cramér's V to say how large it is.
Chi-square critical value table (df 1–10)
Common upper-tail critical values. For other α levels or df up to 100, use the calculator above.
| df | α = 0.05 | α = 0.01 |
|---|---|---|
| 1 | 3.841 | 6.635 |
| 2 | 5.991 | 9.210 |
| 3 | 7.815 | 11.345 |
| 4 | 9.488 | 13.277 |
| 5 | 11.070 | 15.086 |
| 6 | 12.592 | 16.812 |
| 7 | 14.067 | 18.475 |
| 8 | 15.507 | 20.090 |
| 9 | 16.919 | 21.666 |
| 10 | 18.307 | 23.209 |
Frequently asked questions
- What is the chi-square critical value for α = 0.05 and df = 1?
- 3.841. If your chi-square statistic on 1 degree of freedom exceeds 3.841, the result is significant at the 5% level. At α = 0.01 the df = 1 cutoff is 6.635.
- How do I get a p-value from a chi-square statistic?
- Enter the χ² value and its degrees of freedom in the right panel above. The p-value is the upper-tail probability P(χ²(df) ≥ your statistic), computed from the chi-square distribution — the same calculation as Excel's CHISQ.DIST.RT or R's pchisq(x, df, lower.tail = FALSE).
- Is the chi-square test one-tailed or two-tailed?
- Chi-square tests of independence and goodness of fit use only the upper (right) tail, because any deviation from the null hypothesis — in either direction — makes χ² larger. So the critical value and p-value here are upper-tail values.
- How are degrees of freedom determined?
- For a test of independence on an r × c contingency table, df = (r − 1)(c − 1) — so a 2×2 table has df = 1. For a goodness of fit test with k categories, df = k − 1.
References & further reading
- Pearson, K. (1900). On the criterion that a given system of deviations… Philosophical Magazine, 50(302), 157–175.
- Chi-squared distribution — Wikipedia
- Wilson, E. B. & Hilferty, M. M. (1931). The distribution of chi-square. PNAS, 17(12), 684–688.
Try it on your own data — free, no signup
Upload a CSV or XLSX. Everything runs in your browser; your file never leaves your device.
Open the workspace →