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.
Critical Values for Large Degrees of Freedom (df > 100)
Printed tables stop at df = 30 or df = 100 because paper runs out, not because the distribution does. Real lookups go far beyond that — variance tests on large samples, log-likelihood model comparisons, contingency tables built from production data. The interactive table above lists exact rows up to df = 1000, and the quick lookup field computes the value for any df up to 10,000.
At df = 337: 380.809 (α = 0.05) and 400.319 (α = 0.01). At df = 1423: 1511.872 (α = 0.05). All from exact CDF inversion.
The classical workaround for large df is the Wilson–Hilferty approximation, published in 1931: χ²α ≈ df · (1 − 2/(9df) + zα·√(2/(9df)))³.
We checked it against exact CDF values and it never drifts past two decimal places once df exceeds 100. That margin of error is what kept the formula in printed textbooks for the better part of a century — close enough to be trustworthy, fast enough to run by hand. These days the only situation where it matters is when someone has a normal table and a pencil and nothing else, which does still happen in exam halls where calculators are banned.
A Worked Example, Start to Finish
A critical value means more once you watch one settle an actual question. Picture a market researcher who samples 400 shoppers and records which of five brands each one prefers. The null hypothesis is the dull one: all five brands are equally popular, so you would expect 80 picks each.
The counts come back uneven — 90, 60, 104, 95, and 51. The chi-square statistic measures how far that spread sits from the flat 80-per-brand expectation, summing (observed − expected)² / expected across all five categories to 26.775. One brand does most of the damage: 51 picks against an expected 80 contributes 10.51 on its own, nearly 40% of the whole statistic.
Five categories fix the degrees of freedom at df = 4, since the final count is locked once you know the other four and the total of 400. That sends you to row 4 of the table. At α = 0.05 the critical value is 9.488, and 26.775 clears it with room to spare — so the brands are not equally popular, and you reject the null at the 5% level.
It clears the stricter cutoffs too: 13.277 at α = 0.01, and even 18.467 at α = 0.001. That last comparison is why software reports this result as p < 0.001 rather than a marginal call — the exact p-value works out near 0.00002. To run the arithmetic on your own counts instead of reading thresholds off a grid, the chi-square test calculator does every step and reports the statistic directly.
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 do printed chi-square tables stop at df = 100?
Because df rarely exceeds 100 in real contingency tables — a 10×10 table gives df = 81 — and past df ≈ 30 the Wilson-Hilferty approximation (1931) is accurate enough that extra printed rows were not worth the paper.
No such constraint here: the interactive table lists exact rows up to df = 1000, and the quick lookup field computes the exact critical value on the spot for any df up to 10,000.
How do I find the chi-square critical value for df = 337 (or any large df)?
Type 337 into the quick lookup above the table — it returns 380.809 for α = 0.05 and 400.319 for α = 0.01, computed exactly rather than approximated.
Lookups like this usually come from variance tests or model comparisons on large samples, where df lands on values no printed table covers. Every df up to 10,000 works the same way.
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.
Is the chi-square test one-tailed or two-tailed?
For goodness-of-fit and independence tests you only ever check the right tail, so the lookup is effectively one-tailed — which catches people off guard, because the hypotheses themselves sound two-sided.
The reason is mechanical. Squaring (observed − expected) throws away the direction of every difference, so a gap on either side pushes χ² upward, never down. A large statistic just means "far from expected," and the only critical value you compare against is the upper one in the table.