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Chi-Square Table: Critical Values, Made Interactive
A chi-square table turns your degrees of freedom and significance level into the critical value a test statistic must beat to count as significant. Set the inputs below to light up the exact cell, read the value off a shaded curve, switch to a two-sided variance interval, or flip to reverse mode to turn a chi-square statistic back into a p-value. Every number is generated live and matches R's qchisq().
I want to find a critical chi-square value
Upper tail is the usual chi-square test (goodness-of-fit, independence): the tool looks up qchisq(1 − α, df). Two-sided returns both variance-interval bounds, qchisq(α/2, df) and qchisq(1 − α/2, df).
χ²* = 18.307
Critical value for df = 10.
Full chi-square distribution table
The complete critical-value table below is rendered as plain HTML, so it prints, copies and is fully indexable. Each row is a value of degrees of freedom; each column is an upper-tail area α (the cumulative probability below the value is under it). The cell is the chi-square value with area α to its right, so small α gives a large critical value. Your current selection highlights the exact cell.
| df | 0.995 | 0.99 | 0.975 | 0.95 | 0.9 | 0.1 | 0.05 | 0.025 | 0.01 | 0.005 |
|---|---|---|---|---|---|---|---|---|---|---|
| cumulative P(χ² ≤ crit) | 0.005 | 0.01 | 0.025 | 0.05 | 0.1 | 0.9 | 0.95 | 0.975 | 0.99 | 0.995 |
| 1 | 0.0000393 | 0.000157 | 0.000982 | 0.0039 | 0.0158 | 2.706 | 3.841 | 5.024 | 6.635 | 7.879 |
| 2 | 0.0100 | 0.0201 | 0.0506 | 0.1026 | 0.2107 | 4.605 | 5.991 | 7.378 | 9.210 | 10.597 |
| 3 | 0.0717 | 0.1148 | 0.2158 | 0.3518 | 0.5844 | 6.251 | 7.815 | 9.348 | 11.345 | 12.838 |
| 4 | 0.2070 | 0.2971 | 0.4844 | 0.7107 | 1.064 | 7.779 | 9.488 | 11.143 | 13.277 | 14.860 |
| 5 | 0.4117 | 0.5543 | 0.8312 | 1.145 | 1.610 | 9.236 | 11.070 | 12.833 | 15.086 | 16.750 |
| 6 | 0.6757 | 0.8721 | 1.237 | 1.635 | 2.204 | 10.645 | 12.592 | 14.449 | 16.812 | 18.548 |
| 7 | 0.9893 | 1.239 | 1.690 | 2.167 | 2.833 | 12.017 | 14.067 | 16.013 | 18.475 | 20.278 |
| 8 | 1.344 | 1.646 | 2.180 | 2.733 | 3.490 | 13.362 | 15.507 | 17.535 | 20.090 | 21.955 |
| 9 | 1.735 | 2.088 | 2.700 | 3.325 | 4.168 | 14.684 | 16.919 | 19.023 | 21.666 | 23.589 |
| 10 | 2.156 | 2.558 | 3.247 | 3.940 | 4.865 | 15.987 | 18.307 | 20.483 | 23.209 | 25.188 |
| 11 | 2.603 | 3.053 | 3.816 | 4.575 | 5.578 | 17.275 | 19.675 | 21.920 | 24.725 | 26.757 |
| 12 | 3.074 | 3.571 | 4.404 | 5.226 | 6.304 | 18.549 | 21.026 | 23.337 | 26.217 | 28.300 |
| 13 | 3.565 | 4.107 | 5.009 | 5.892 | 7.042 | 19.812 | 22.362 | 24.736 | 27.688 | 29.819 |
| 14 | 4.075 | 4.660 | 5.629 | 6.571 | 7.790 | 21.064 | 23.685 | 26.119 | 29.141 | 31.319 |
| 15 | 4.601 | 5.229 | 6.262 | 7.261 | 8.547 | 22.307 | 24.996 | 27.488 | 30.578 | 32.801 |
| 16 | 5.142 | 5.812 | 6.908 | 7.962 | 9.312 | 23.542 | 26.296 | 28.845 | 32.000 | 34.267 |
| 17 | 5.697 | 6.408 | 7.564 | 8.672 | 10.085 | 24.769 | 27.587 | 30.191 | 33.409 | 35.718 |
| 18 | 6.265 | 7.015 | 8.231 | 9.390 | 10.865 | 25.989 | 28.869 | 31.526 | 34.805 | 37.156 |
| 19 | 6.844 | 7.633 | 8.907 | 10.117 | 11.651 | 27.204 | 30.144 | 32.852 | 36.191 | 38.582 |
| 20 | 7.434 | 8.260 | 9.591 | 10.851 | 12.443 | 28.412 | 31.410 | 34.170 | 37.566 | 39.997 |
| 21 | 8.034 | 8.897 | 10.283 | 11.591 | 13.240 | 29.615 | 32.671 | 35.479 | 38.932 | 41.401 |
| 22 | 8.643 | 9.542 | 10.982 | 12.338 | 14.041 | 30.813 | 33.924 | 36.781 | 40.289 | 42.796 |
| 23 | 9.260 | 10.196 | 11.689 | 13.091 | 14.848 | 32.007 | 35.172 | 38.076 | 41.638 | 44.181 |
| 24 | 9.886 | 10.856 | 12.401 | 13.848 | 15.659 | 33.196 | 36.415 | 39.364 | 42.980 | 45.559 |
| 25 | 10.520 | 11.524 | 13.120 | 14.611 | 16.473 | 34.382 | 37.652 | 40.646 | 44.314 | 46.928 |
| 26 | 11.160 | 12.198 | 13.844 | 15.379 | 17.292 | 35.563 | 38.885 | 41.923 | 45.642 | 48.290 |
| 27 | 11.808 | 12.879 | 14.573 | 16.151 | 18.114 | 36.741 | 40.113 | 43.195 | 46.963 | 49.645 |
| 28 | 12.461 | 13.565 | 15.308 | 16.928 | 18.939 | 37.916 | 41.337 | 44.461 | 48.278 | 50.993 |
| 29 | 13.121 | 14.256 | 16.047 | 17.708 | 19.768 | 39.087 | 42.557 | 45.722 | 49.588 | 52.336 |
| 30 | 13.787 | 14.953 | 16.791 | 18.493 | 20.599 | 40.256 | 43.773 | 46.979 | 50.892 | 53.672 |
| 35 | 17.192 | 18.509 | 20.569 | 22.465 | 24.797 | 46.059 | 49.802 | 53.203 | 57.342 | 60.275 |
| 40 | 20.707 | 22.164 | 24.433 | 26.509 | 29.051 | 51.805 | 55.758 | 59.342 | 63.691 | 66.766 |
| 45 | 24.311 | 25.901 | 28.366 | 30.612 | 33.350 | 57.505 | 61.656 | 65.410 | 69.957 | 73.166 |
| 50 | 27.991 | 29.707 | 32.357 | 34.764 | 37.689 | 63.167 | 67.505 | 71.420 | 76.154 | 79.490 |
| 60 | 35.534 | 37.485 | 40.482 | 43.188 | 46.459 | 74.397 | 79.082 | 83.298 | 88.379 | 91.952 |
| 70 | 43.275 | 45.442 | 48.758 | 51.739 | 55.329 | 85.527 | 90.531 | 95.023 | 100.425 | 104.215 |
| 80 | 51.172 | 53.540 | 57.153 | 60.391 | 64.278 | 96.578 | 101.879 | 106.629 | 112.329 | 116.321 |
| 90 | 59.196 | 61.754 | 65.647 | 69.126 | 73.291 | 107.565 | 113.145 | 118.136 | 124.116 | 128.299 |
| 100 | 67.328 | 70.065 | 74.222 | 77.929 | 82.358 | 118.498 | 124.342 | 129.561 | 135.807 | 140.169 |
Columns 0.90 and above give the small (lower-tail) cutoffs used for the lower bound of a variance interval; columns 0.10 and below give the large upper-tail cutoffs used for chi-square tests. R: qchisq(1 − α, df) reproduces any cell.
How this value is computed
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When you reach for a chi-square table
Three everyday jobs, all answered by the same distribution of critical values.
Compare observed counts to what a model predicts. The test statistic is upper-tail: if it exceeds the critical value for your df and α, reject the null. A 2 by 2 table has df = 1, so the cutoff is 3.841 at α = 0.05.
The interval for a population variance is two-sided, so it needs both a lower and an upper chi-square value. Switch to two-sided mode to read qchisq(α/2, df) and qchisq(1 − α/2, df) at once.
Already have a chi-square value? Reverse mode gives the upper-tail p-value pchisq(x, df, lower.tail = FALSE) and compares it to your α so you can read the decision straight off.
Common critical chi-square values at α = 0.05
| Situation | df | R call | Critical value |
|---|---|---|---|
| 2 × 2 table of independence | 1 | qchisq(0.95, 1) | 3.841 |
| Goodness-of-fit, 3 categories | 2 | qchisq(0.95, 2) | 5.991 |
| Goodness-of-fit, 4 categories | 3 | qchisq(0.95, 3) | 7.815 |
| 3 × 3 table of independence | 4 | qchisq(0.95, 4) | 9.488 |
| Larger contingency table | 10 | qchisq(0.95, 10) | 18.307 |
All of these are upper-tail cutoffs: the chi-square statistic must be larger than the value to be significant. As df grows, the distribution shifts right and the critical value climbs, because a bigger table has more cells that can differ by chance.
Chi-square table questions
How do I read a chi-square table?
Find the row for your degrees of freedom and the column for your upper-tail significance level, then read the cell where they meet: that is the critical chi-square value. If your observed statistic is larger than the cell, the result is significant at that α. At df = 1 and α = 0.05 the cutoff is 3.841.
What are the degrees of freedom for a chi-square test?
For a goodness-of-fit test, df is the number of categories minus one, minus any parameters you estimated from the data. For a test of independence in an r by c table, df is (r − 1)(c − 1), so a 2 by 2 table has df = 1 and a 3 by 4 table has df = 6. This tool accepts any df, whole or decimal.
What is the critical chi-square value for alpha = 0.05?
It depends on the degrees of freedom. At df = 1 it is 3.841, at df = 2 it is 5.991, at df = 3 it is 7.815, and at df = 10 it is 18.307. In R this is qchisq(0.95, df), the value with 0.05 of the area in the upper tail.
Why does a variance confidence interval need two chi-square values?
A confidence interval for a variance is two-sided, so it splits alpha across both tails. You need the lower critical value qchisq(alpha/2, df) and the upper critical value qchisq(1 - alpha/2, df). The two-sided mode of this tool reports both bounds at once.
Does this chi-square table match R?
Yes. Every printed value and every interactive result comes from the same routine, checked against R's qchisq() and pchisq() across more than 2000 cases to better than seven significant figures.
Why does the critical value get bigger as df increases?
The chi-square distribution is the sum of squared standard normals, so its mean equals its degrees of freedom. More df means the whole distribution sits farther right, which pushes the upper-tail cutoff higher. A large contingency table therefore needs a larger statistic to reach significance.
Why does a variance interval use two chi-square values?
A confidence interval for a variance is two-sided, so it divides α between the two tails. The lower bound uses qchisq(α/2, df) and the upper bound uses qchisq(1 − α/2, df). The two-sided mode reports both at once, which is what you plug into the variance interval formula.
Does this match R and printed textbook tables?
Yes. The printed table and the live result share one routine, checked against R's qchisq() and pchisq() across more than 2000 cases to better than seven significant figures. Small last-digit gaps versus an old textbook table are rounding in the book, not here.