What Is Chi-Square P-Value?
The p-value for a chi-square test is the probability of obtaining a statistic at least as large as yours if the null hypothesis were true. Chi-square tests are usually right-tailed, so the p-value is the area under the χ² curve to the right of your statistic.
Enter the chi-square value and the degrees of freedom. The calculator returns the right-tail p-value, the left-tail probability, the critical value for your significance level and where your statistic sits relative to the mean and spread of the distribution. If you only have raw counts, run the chi-square calculator first to get the statistic.
Below you will find the formula, a variables table, a worked example and answers to common questions such as how to read a chi-square table and what a small p-value means.
How to Use the Chi-Square P-Value Calculator
- Enter the chi-square statistic and the degrees of freedom.
- Leave the tail at right-tailed for tests of independence and goodness of fit.
- Pick α and read the decision. Compare your χ² with the critical value as a cross-check.
- Use the reference table for exam or lookup purposes.
Chi-Square P-Value Formula
Right-tail p-value
Chi-square density
Mean and variance
Chi-Square P-Value Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| χ² | Chi-square statistic | From your test |
| df | Degrees of freedom | Also the mean of the distribution |
| p | Right-tail p-value | P(χ² ≥ observed) |
| α | Significance level | Sets the critical value |
Chi-Square P-Value Worked Example
Using the example values (press “Load example” above to try them):
- Distribution: χ² with 3 degrees of freedom; statistic x = 7.81.
- Right-tail p-value = 1 − F(x) = 0.0501061.
- Critical value for α = 0.05: χ²(0.95, 3) = 7.81473.
- x is below the critical value, matching the p-value decision.
Result: p-value (right-tail) = 0.0501061; Decision at α = 0.05 = Fail to reject H₀; Critical value (right tail, α = 0.05) = 7.81473.
Which tail should I use?
Chi-square tests for counts are essentially always right-tailed. A very small χ² (left tail) means the data fit suspiciously well, which is occasionally used to question data quality.
| Situation | Tail | Reason |
|---|---|---|
| Goodness-of-fit or independence test | Right | Large χ² means the data disagree with the null |
| Test of a single variance, H₁: σ² > σ₀² | Right | Large χ² supports a larger variance |
| Test of a single variance, H₁: σ² < σ₀² | Left | Small χ² supports a smaller variance |
| Test of a single variance, H₁: σ² ≠ σ₀² | Two-sided | Either tail counts |
Understanding the shape of the distribution
The chi-square distribution is the distribution of a sum of squared standard normal variables, with df giving the number of terms. It is skewed to the right and starts at zero. With df = 1 or 2 it is monotone decreasing; as df grows it moves right and becomes more symmetric, approaching a normal curve with mean df and variance 2·df.
Using a chi-square table
Printed tables give critical values only for a few significance levels. Find the row for your df and compare your χ² to the column for your α. If your statistic exceeds the tabulated value, p is below α. This calculator gives the exact p-value and the same critical values for any df.
Where Chi-Square P-Value Is Used
Finishing a chi-square test
Turning a hand-calculated statistic into a p-value.
Replacing a printed table
Getting an exact p-value instead of a range from a table.
Finding critical values
Looking up the cut-off for a chosen α and degrees of freedom.
Understanding the curve
Seeing the shaded tail area that the p-value represents.
Common Chi-Square P-Value Mistakes to Avoid
- Degrees of freedom depend on the test; check the formula for yours.
- Report the exact p-value when you can, not just “p < 0.05”.
- The p-value is not the probability that the null hypothesis is true.
Chi-Square P-Value Terms and Definitions
- Critical value
- The χ² value that cuts off an area α in the right tail.
- Right-tail probability
- The area under the density to the right of the statistic.
Chi-Square P-Value FAQs
How do I find the p-value from a chi-square statistic?
Compute the right-tail area of the chi-square distribution with your degrees of freedom beyond the statistic. This calculator does it directly.
What is the critical value for chi-square with 1 degree of freedom at 0.05?
About 3.841. It is the value that cuts off 5% in the right tail.
Why is the chi-square test one-tailed?
Because the statistic is a sum of squares and only large values indicate a mismatch between observed and expected counts.
What does a p-value of 0.03 mean?
If the null hypothesis were true, you would see a statistic this extreme or more about 3% of the time.
Can the degrees of freedom be a decimal?
Yes for the distribution itself, though standard tests use whole numbers.
What does it mean if my p-value is exactly 1?
The statistic is zero or extremely small, meaning observed and expected counts match almost perfectly.
Is the chi-square p-value one-tailed or two-tailed?
It is normally the right tail: the probability of a statistic at least as large as yours. Large χ² values signal a poor fit, so the upper tail is the one that matters.
What is the difference between the chi-square statistic and the p-value?
The statistic measures how far observed counts are from expected counts. The p-value converts that distance, together with the degrees of freedom, into a probability under the null hypothesis.