Statistics How-To
How to Read a Chi-Square Table
A chi-square table lists critical values. Find the row for your degrees of freedom and the column for your significance level (the area in the right tail), and read the number where they meet. If your chi-square statistic is larger than that number, the result is significant at that level. With 2 degrees of freedom and α = 0.05 the critical value is 5.991.
The table
Each cell is the chi-square value with the right-tail area shown in the column heading above it.
| df | α = 0.10 | α = 0.05 | α = 0.025 | α = 0.01 | α = 0.005 |
|---|---|---|---|---|---|
| 1 | 2.706 | 3.841 | 5.024 | 6.635 | 7.879 |
| 2 | 4.605 | 5.991 | 7.378 | 9.210 | 10.597 |
| 3 | 6.251 | 7.815 | 9.348 | 11.345 | 12.838 |
| 4 | 7.779 | 9.488 | 11.143 | 13.277 | 14.860 |
| 5 | 9.236 | 11.070 | 12.833 | 15.086 | 16.750 |
| 6 | 10.645 | 12.592 | 14.449 | 16.812 | 18.548 |
| 10 | 15.987 | 18.307 | 20.483 | 23.209 | 25.188 |
| 20 | 28.412 | 31.410 | 34.170 | 37.566 | 39.997 |
| 30 | 40.256 | 43.773 | 46.979 | 50.892 | 53.672 |
The full table, for 1 to 100 degrees of freedom and ten tail areas, is on the chi-square table page.
Reading it in four steps
- Compute your statistic. Add up (observed − expected)² / expected over all cells, as in the chi-square test in Excel guide.
- Find the degrees of freedom. (rows − 1)(columns − 1) for a test of independence, or the number of categories − 1 for goodness of fit; see degrees of freedom explained.
- Choose the significance level. The column headed 0.05 is the right-tail area for a 5% test.
- Compare. Statistic greater than the critical value means reject the null hypothesis; less than or equal means fail to reject.
Three worked examples
| Situation | Statistic and df | Reading the table | Conclusion |
|---|---|---|---|
| A 2 × 3 table of group by preference | 16.67, 2 df | Larger than 10.597, the α = 0.005 value for 2 df | p < 0.005 (exact 0.00024); significant |
| A die rolled 60 times, faces 5, 8, 9, 8, 10, 20 | 13.4, 5 df | Between 12.833 (α = 0.025) and 15.086 (α = 0.01) | p between 0.01 and 0.025 (exact 0.0199); significant at 5% |
| A weak association in a 2 × 3 table | 4.1, 2 df | Smaller than 4.605, the α = 0.10 value for 2 df | p > 0.10 (exact 0.1287); not significant |
A table cannot give the exact p-value, only the bracket, but the bracket is enough to decide at the usual levels. For the exact value use a p-value calculator.
Software gives the same numbers
| Tool | Critical value for α = 0.05, 2 df | p-value for a statistic of 13.4 with 5 df |
|---|---|---|
| Excel | =CHISQ.INV.RT(0.05,2) gives 5.9915 | =CHISQ.DIST.RT(13.4,5) gives 0.0199 |
| Google Sheets | =CHIINV(0.05,2) gives 5.9915 | =CHIDIST(13.4,5) gives 0.0199 |
| R | qchisq(0.95, df = 2) gives 5.991465 | pchisq(13.4, df = 5, lower.tail = FALSE) gives 0.0199 |
| Python (SciPy) | chi2.ppf(0.95, 2) gives 5.991465 | chi2.sf(13.4, 5) gives 0.0199 |
Mistakes that pick the wrong column
- Using a left-tail table. Some books print the area to the left. The 0.05 column of such a table is the value with 95% to its right, which is a very different number. Check the heading, or the sketch of the curve next to the table.
- Doubling α for a two-sided test. Chi-square tests for independence and goodness of fit are right-tailed by construction, so use the α column as it stands.
- Using n instead of the degrees of freedom. A 3 × 4 table has 6 degrees of freedom, whatever the number of observations.
- Treating the critical value as the effect. Beating the critical value shows an association exists, not that it is strong; report an effect size such as Cramér's V as well. Read the chi-square test explained for the reasoning.
Try the Chi-Square Table
Look up chi-square critical values for 1 to 100 degrees of freedom across ten tail areas.
Try the P-Value Calculator
Turn a chi-square statistic and its degrees of freedom into an exact p-value.
Frequently Asked Questions
What does the number in a chi-square table mean?
It is the critical value: the chi-square statistic that leaves the column's probability in the right tail of the distribution for that row's degrees of freedom. For 2 df and the 0.05 column it is 5.991, meaning that 5% of the chi-square distribution with 2 df lies above 5.991. A statistic larger than that is significant at the 5% level.
What is the chi-square critical value for α = 0.05 with 1 degree of freedom?
3.841. Chi-square with 1 df is the square of a standard normal variable, so 3.841 is 1.96 squared. A 2 × 2 table has 1 degree of freedom, so a statistic above 3.841 in such a table is significant at the 5% level.
What do I do if my degrees of freedom are not in the table?
Use the next smaller row, which is conservative, or compute the value exactly: =CHISQ.INV.RT(0.05,25) in Excel gives 37.652 for 25 degrees of freedom, and qchisq(0.95, 25) gives the same in R. Printed tables usually jump from 30 to 40 and beyond, so software is the reliable route for those rows.
Why does the table only show the right tail?
Chi-square tests measure how far the observed counts are from the expected ones, and a large distance is the evidence against the null hypothesis, so the rejection region is in the right tail. The left tail matters only for special cases, such as asking whether a fit is suspiciously perfect, or in confidence intervals for a variance.
How do I get an exact p-value instead of a range?
A table can only bracket it. For an exact value use software: =CHISQ.DIST.RT(13.4,5) in Excel returns 0.0199, and 1 - pchisq(13.4, 5) returns the same in R. The p-value calculator on this site does the same for any statistic and degrees of freedom.
Is the chi-square table the same for all chi-square tests?
Yes. The table depends only on the degrees of freedom and the tail area. What differs between tests is how the statistic and the degrees of freedom are found: (rows − 1)(columns − 1) for independence, categories − 1 for goodness of fit, and n − 1 for a variance test.