Critical value calculator
Pick a distribution and a significance level to get the boundary of the rejection region — with the decision rule written out, so there is no ambiguity about which side of it counts.
Enter the degrees of freedom
The critical value is the boundary of the rejection region — the statistic you would need to reach in order to call a result significant at your α.
Critical value
Critical value
Decision rule
What a critical value is
Every test statistic has a distribution it would follow if the null hypothesis were true. The critical value is the point on that distribution with exactly α of the probability beyond it. Land past it and your result is rarer than the threshold you set, so you reject the null.
That is all a critical value is: a quantile, chosen so the area in the tail equals the false-positive rate you are willing to accept. Everything else — the tables in the back of textbooks, the 1.96 people memorise — is bookkeeping around that one idea.
How to find a critical value
Whichever distribution you are working in, the procedure is the same four steps. The calculator above does all of them at once, but it is worth knowing what it is doing.
- Pick your significance level. α = 0.05 is the usual choice, and it has to be settled before you look at the data.
- Decide one-tailed or two-tailed. Two-tailed splits α across both tails, so you look up α/2 in each. Chi-square and F are always right-tailed.
- Get the degrees of freedom. Z needs none. Everything else does, and this is where the mistake usually happens — see the table further down.
- Take the quantile. The critical value is the point with exactly α (or α/2) of the probability beyond it.
Then compare: if your statistic is further from zero than the critical value, you reject the null hypothesis. For a two-tailed test the comparison is on the absolute value, since either tail counts.
The critical value formula
There is no arithmetic formula for a critical value in the way there is for a mean. The critical value is the inverse of the distribution function — the quantile — so the formula is a statement about which quantile you want:
| Test | Critical value |
|---|---|
| Z, one-tailed | z* = Φ⁻¹(1 − α) |
| Z, two-tailed | z* = Φ⁻¹(1 − α/2) |
| t, two-tailed | t* = F⁻¹t,df(1 − α/2) |
| Chi-square | χ²* = F⁻¹χ²,df(1 − α) |
| F | F* = F⁻¹F,df₁,df₂(1 − α) |
| Pearson r | r* = t* / √(t*² + n − 2), with df = n − 2 |
Φ⁻¹ is the inverse standard normal CDF — the function Excel calls NORM.S.INV and R calls qnorm. It has no closed form, which is the historical reason critical values were printed in tables at all: the quantile has to be computed numerically, and before computers that meant computing it once and publishing the answer. The last row is the useful one, because it shows that the correlation threshold is just a t critical value rearranged.
Finding a t critical value
The t-distribution is where most people arrive, because t-tests are where most people arrive. Select "t" in the t critical value calculator above, enter your degrees of freedom and α, and you get the boundary for one- and two-tailed tests together.
Two things make t different from Z. The distribution has heavier tails, so every critical value is larger than the corresponding z — and the gap grows as the sample shrinks. And the value moves with df, so a t critical value is never a single number you can memorise the way 1.96 is. At α = 0.05 two-tailed it runs from 12.706 at df = 1 down through 2.228 at df = 10 and 2.042 at df = 30, approaching 1.960 as df grows.
Critical value chart
The two panels beside this text are a critical value chart for the Z and t distributions, covering the α values in ordinary use. Unlike the chart in the back of a textbook, they are generated from this site's own quantile functions when the page is built, and every cell in them is checked against SciPy by the test suite — so a transcription error cannot survive in them.
For chi-square and F, and for a wider grid of degrees of freedom than fits in a sidebar, see the full p-value and critical value tables. If your df or α is not on the chart, use the calculator rather than interpolating between rows — interpolation on a t table is only approximate, and the error is worst exactly where the table is sparsest.
Critical value or p-value?
They always give the same verdict, so the choice is about what you want to see.
| Critical value | P-value | |
|---|---|---|
| Computed from | Your α, before the data | Your data |
| The comparison | Is my statistic past the line? | Is my p below α? |
| Tells you | Reject or do not reject | Reject or not, and by how much |
| Best for | Planning; drawing the rejection region | Reporting a result |
The critical-value route is older, and it exists because computing an exact tail area by hand was impractical — so statisticians precomputed the boundary for a handful of conventional α values and printed them. With a computer there is no reason to discard the extra information, which is why journals now expect an exact p-value. Use the p-value calculator for reporting; use this page for planning, for coursework that asks for the critical value explicitly, and for building an interval.
Why the two-tailed value is larger
A two-tailed test has to allocate α across two rejection regions, so each gets α/2 and both boundaries move outward. At α = 0.05 the one-tailed z is 1.645 and the two-tailed z is 1.960. The gap is not a rounding artefact — it is the cost of being able to detect an effect in either direction.
Which makes one-tailed testing tempting, and that is precisely the problem. It buys sensitivity in one direction by giving up the other entirely: a large effect in the unexpected direction produces a p-value above 0.5 and no finding at all. That trade is only legitimate if you committed to the direction before seeing data. Our guide to one-tailed vs two-tailed tests goes through when it genuinely applies.
Critical values build confidence intervals
The same number does double duty. A 95% confidence interval is the point estimate plus and minus the two-tailed critical value at α = 0.05 times the standard error: estimate ± t* × SE. The 1.96 in the familiar mean ± 1.96 × SE is a critical value.
This is why the interval and the test always agree. If a 95% interval excludes zero, the statistic must have cleared the critical value, and the p-value must be below 0.05 — three statements of one fact. The interval is the most useful of the three, because it carries the units of your measurement.
Getting the degrees of freedom right
For anything except Z, the critical value moves with the degrees of freedom, so this is where the error usually happens rather than in the lookup itself.
| Test | Degrees of freedom |
|---|---|
| One-sample t-test | n − 1 |
| Paired t-test | pairs − 1 |
| Two-sample t, pooled | n₁ + n₂ − 2 |
| Two-sample t, Welch | Welch–Satterthwaite, usually fractional |
| Chi-square, independence | (rows − 1) × (columns − 1) |
| One-way ANOVA | df₁ = groups − 1; df₂ = N − groups |
| Pearson correlation | n − 2 (enter n above) |
Small df punishes you hard. At df = 4 the two-tailed 5% boundary is 2.776; at df = 30 it is 2.042. A t of 2.4 is significant in the second case and not in the first, from the same statistic.
Frequently asked questions
What is a critical value?
It is the boundary of the rejection region: the value your test statistic must pass for the result to count as significant at your chosen significance level. Formally it is a quantile of the distribution the statistic follows when the null hypothesis is true — the point with exactly alpha of the probability beyond it. Anything further out is rarer than your threshold allows, so you reject the null.
What is the critical value for a 95% confidence level?
For a two-tailed test on the standard normal distribution it is 1.960, which is where the familiar 1.96 comes from. For a t-distribution it depends on the degrees of freedom: 2.776 at df = 4, 2.228 at df = 10, 2.042 at df = 30, and 1.984 at df = 100, converging on 1.96 as df grows. For a right-tailed chi-square test with df = 1 it is 3.841.
How is a critical value different from a p-value?
They are two routes to the same decision. The critical value comes from your significance level before you look at the data; the p-value comes from your data. Comparing your statistic to the critical value and comparing your p-value to alpha always agree. The p-value carries more information, though — it tells you how far past the line you landed, which the critical-value comparison discards.
Why is the two-tailed critical value larger?
Because alpha has to be split between the two tails. At alpha = 0.05 a one-tailed test puts the whole 5% in one tail, giving z = 1.645. A two-tailed test puts 2.5% in each, pushing the boundary out to z = 1.960. That is the price of being able to detect an effect in either direction.
Do chi-square and F tests have a lower critical value?
Not in ordinary use. Both statistics grow with any departure from the null — chi-square sums squared deviations, F is a ratio of explained to unexplained variance — so all the evidence against the null sits in the upper tail. A very small chi-square means your data look like the null predicted, which is not grounds for rejecting it. The calculator locks these to right-tailed for that reason.
How do I find the critical value for a t-test?
Select "t" in the t critical value calculator above, enter your degrees of freedom — n − 1 for a one-sample or paired test, n₁ + n₂ − 2 for a pooled two-sample test — and choose your alpha and tail count. Unlike z, a t critical value is not a single memorable number: at alpha = 0.05 two-tailed it is 12.706 at df = 1, 2.228 at df = 10 and 2.042 at df = 30, converging on 1.960 as the sample grows.
Is there a formula for the critical value?
Not an arithmetic one. The critical value is a quantile — the inverse of the distribution function — so the formula is z* = Φ⁻¹(1 − α) for a one-tailed z test, Φ⁻¹(1 − α/2) two-tailed, and the equivalent inverse for t, chi-square and F at the relevant degrees of freedom. None of these inverses has a closed form, which is precisely why critical values were historically printed in tables instead of being derived on the spot.
Where can I find a critical value chart?
The Z and t charts on this page cover the alpha values in ordinary use, and the p-value tables page carries a wider grid plus chi-square and F. Both are generated from this site’s own quantile functions at build time and every cell is verified against SciPy by the test suite, so they cannot carry a transcription error. If your degrees of freedom are not listed, use the calculator rather than interpolating between rows.
What critical value do I need for a correlation?
It depends almost entirely on sample size. At alpha = 0.05 two-tailed you need r = 0.878 with n = 5, r = 0.632 with n = 10, r = 0.444 with n = 20, and only r = 0.197 with n = 100. Select "r" above and enter your n. Note that this is a threshold for detectability, not for importance — large samples make trivial correlations significant.