Chi-Square Critical Value Calculator
Find left-tail, right-tail, or two-tailed chi-square critical values from degrees of freedom and significance level.
What is a chi-square critical value?
A chi-square critical value is a cutoff point on the chi-square distribution used in hypothesis testing. If your test statistic falls beyond that cutoff, your result is considered statistically significant for the chosen significance level.
The exact value depends on two things: the degrees of freedom (df) and the significance level (α). Because chi-square distributions are right-skewed and change shape with df, the critical value is not fixed like a z-score.
When this calculator is useful
- Chi-square goodness-of-fit tests
- Chi-square test of independence in contingency tables
- Chi-square test of homogeneity
- Confidence intervals for population variance (for normally distributed data)
How to use this calculator
1) Enter degrees of freedom
Degrees of freedom are usually tied to your table dimensions or number of categories:
- Goodness-of-fit: df = categories − 1 − estimated parameters
- Independence test: df = (rows − 1)(columns − 1)
2) Enter α (significance level)
Common values are 0.10, 0.05, and 0.01. Smaller α means stricter evidence is required to reject the null hypothesis.
3) Choose tail type
- Right-tail: the standard setup for most chi-square hypothesis tests.
- Left-tail: less common, but useful in some variance-related settings.
- Two-tailed: returns both lower and upper critical values, each using α/2.
Interpretation guide
For a right-tail test, reject the null hypothesis if:
For two-tailed decisions (such as variance interval bounds), compare your statistic to both ends:
Example
Suppose you run a chi-square test of independence with a 3×4 contingency table:
- df = (3 − 1)(4 − 1) = 6
- α = 0.05
- Tail = right-tail
Enter those values and calculate. If your observed test statistic exceeds the computed critical value, you reject the null hypothesis of independence at the 5% level.
Behind the scenes (calculation method)
This page computes critical values by numerically inverting the chi-square cumulative distribution function:
where k is the degrees of freedom, γ is the lower incomplete gamma function, and Γ is the gamma function. The script uses stable numerical methods (series expansion, continued fraction, and bisection search) for reliable values.
Common mistakes to avoid
- Using the wrong df formula for your test design
- Confusing right-tail α with two-tailed α
- Typing α as 5 instead of 0.05
- Rounding too early before making the decision
FAQ
Is this the same as a p-value calculator?
No. This tool gives critical cutoffs. A p-value calculator starts with your observed χ² statistic and returns probability.
Can I use this for confidence intervals of variance?
Yes. For normal-population variance intervals, two chi-square quantiles are required; choose the two-tailed option.
Why does the critical value change with df?
Because the chi-square distribution’s shape depends on df. Low df is highly skewed; higher df becomes more symmetric.