How to Read the Chi-Square Table
Degrees of freedom sit in the left column. Significance levels run across the top. The number where they cross is your critical value.
Karl Pearson published the original chi-square test in 1900, and the table format has barely changed since — rows for df, columns for α, and a grid of thresholds that tell you whether your data actually disagree with the null hypothesis or just look like they do. Say you ran a goodness-of-fit test with 5 categories, so df = 4. You want α = 0.05. Row 4, column 0.05, and the cell reads 9.488 — meaning your calculated χ² needs to beat that number before the result counts as significant at the 5% level. Modern software computes exact p-values, but the table is still the fastest way to build intuition for what "significant" actually requires at different degrees of freedom.
Common Critical Values
The three significance levels below cover nearly all published chi-square work. For degrees of freedom above 30, use the interactive table at the top of the page.
| df | α = 0.10 | α = 0.05 | α = 0.01 | Typical Use |
|---|---|---|---|---|
| 1 | 2.706 | 3.841 | 6.635 | 2×2 contingency table |
| 2 | 4.605 | 5.991 | 9.210 | 3-category goodness of fit |
| 3 | 6.251 | 7.815 | 11.345 | 2×4 table or 4-category test |
| 4 | 7.779 | 9.488 | 13.277 | 5-category test or 3×3 table |
| 5 | 9.236 | 11.070 | 15.086 | 6-category test |
| 6 | 10.645 | 12.592 | 16.812 | 3×4 contingency table |
| 7 | 12.017 | 14.067 | 18.475 | 8-category test |
| 8 | 13.362 | 15.507 | 20.090 | 3×5 or 5×3 table |
| 9 | 14.684 | 16.919 | 21.666 | 10-category test |
| 10 | 15.987 | 18.307 | 23.209 | 4×4 table or 11-category test |
| 12 | 18.549 | 21.026 | 26.217 | 4×5 contingency table |
| 15 | 22.307 | 24.996 | 30.578 | 5×5 contingency table |
| 20 | 28.412 | 31.410 | 37.566 | Large survey with many categories |
| 25 | 34.382 | 37.652 | 44.314 | Variance test with n = 26 |
| 30 | 40.256 | 43.773 | 50.892 | Variance test with n = 31 |
Critical values grow with df because a higher-dimensional distribution needs a larger χ² to reach the same tail probability. At df = 1, a value of 3.841 sits at the 95th percentile. At df = 30, that same threshold requires 43.773. The gap between α = 0.05 and α = 0.01 also narrows as df increases — 73% at df = 1, down to 16% at df = 30 — because the distribution converges toward normal shape at large degrees of freedom.
When to Use the Chi-Square Distribution
Three different tests share this one table, and the only thing that changes between them is how you count degrees of freedom.
A goodness-of-fit test checks whether observed counts match a hypothesized distribution. Five categories gives you df = 4, because one count is locked once you know the other four and the total.
A test of independence asks whether two categorical variables in a contingency table are related. A 3×4 table gives df = 6, computed as (rows − 1) × (columns − 1).
A variance test compares sample variance to a hypothesized population value, with df = n − 1.
The distribution itself is always right-skewed, but by df = 30 it starts looking roughly normal — which is why most printed tables stop well before 100.
Assumptions Behind the Test
Cochran's 1954 paper set the rule that still gates most chi-square work today: every cell in your table needs an expected frequency of at least 5, or the chi-square approximation stops being trustworthy and you should switch to Fisher's exact test instead. Later simulation work loosened that threshold a bit, but most review boards and textbooks still enforce the original cutoff.
The other two assumptions are easier to check. The data must be counts, not continuous measurements. And each observation must be independent — one person cannot show up in two cells at the same time.
Frequently Asked Questions
How do I use the chi-square table to find a critical value?
The whole process takes about five seconds once you see the layout. Find your df in the left column, slide across to the α column you need, and the number at the intersection is your critical value.
If the χ² from your test beats that number, reject the null. If it doesn't, you lack evidence at that significance level. No software required — pen and table are enough during an exam.
What are degrees of freedom in a chi-square test?
Getting df wrong is the single fastest way to flip your conclusion, because it shifts you to the wrong row of the table entirely.
For a goodness-of-fit test with k categories, df = k − 1. For an r × c contingency table, df = (r − 1)(c − 1). The logic is the same both times: once you know all but one value, the last one is locked by the totals.
What significance level should I use?
The short answer: α = 0.05 for most coursework and published research. That means a 5% chance you reject a true null.
Medical trials drop to 0.01 or 0.001 — a false positive there could mean approving a drug that does nothing. Pilot studies go the other way with 0.10 to catch weak signals early. Pick whichever matches the cost of being wrong.
Why does the chi-square table only go up to df = 100?
A 10×10 contingency table only gives df = 81, and tables that large are rare in practice. Beyond df ≈ 30 the distribution is close enough to normal that an approximation works fine.
If you need a df not listed here, type it into the quick lookup field above the table — it computes the exact critical value on the spot for any degree of freedom.
What is the difference between the chi-square table and the chi-square test?
The test is where the math happens — you crunch (observed − expected)² / expected for every cell and add them up into one χ² number.
The table is the ruler you hold that number up against. It tells you whether the χ² you got is big enough to mean something at your chosen significance level, given your degrees of freedom.