Critical Value Tables
t, z, chi-square and F critical values at any degrees of freedom, plus exact p-values from an observed statistic.
About this tool
Critical Value Tables
t, z, chi-square and F critical values at any degrees of freedom, plus exact p-values from an observed statistic.
A critical value table gives the cut-off a test statistic must pass for a result to sit in the rejection region at a chosen significance level α, and this page computes those cut-offs live for the Student t, standard normal z, chi-square and F distributions. It is built for anyone reading a printed table and finding their degrees of freedom missing from it — students, analysts and researchers whose df is 37, or 214, or the fractional 18.7 that a Welch–Satterthwaite correction produced. Every figure comes from the distribution functions themselves: tail probabilities from the regularised incomplete beta I_x(a, b) and the regularised incomplete gamma Q(a, x) evaluated by continued fraction, inverted by bracketed bisection, so t(0.05, two-tailed, df = 10) returns 2.228138852 rather than the rounded 2.228 a book prints.
How to use Critical Value Tables
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Open Critical Value Tables on AltFTool — it loads instantly in your browser.
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Choose Table mode for a full sheet of critical values, or Solve mode to work from one observed statistic.
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Pick the distribution: Student t, standard normal z, chi-square, or F.
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Set the tail — one-tailed or two-tailed for t and z, upper or lower tail for chi-square. F tables are upper-tail.
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In Table mode, type the degrees of freedom you want as rows, using commas and ranges such as 1-30, 40, 60, 120, inf. For F, set α and give the numerator df across the top and the denominator df down the side.
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In Solve mode, enter the observed test statistic, the significance level α, and the degrees of freedom, then read the exact p-value and the matching critical value.
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Use the copy button to take the table or the result away as tab-separated text.
Why use Critical Value Tables
Degrees of freedom are not capped
Type any positive df, whole or fractional, up to 200,000 or ∞ — the row is computed, not looked up.
Exact p-values, not table bounds
Enter the observed statistic and the page returns the tail probability itself, so you can report p = 0.0231 instead of p < 0.05.
Four distributions, one page
t, z, chi-square and F share the same tail machinery, so the numbers are consistent across all of them.
Verifiable against identities
χ²(0.05, df = 1) = 3.841458821 equals z(0.05, two-tailed)² = 1.959963985², and F(α; 1, v) equals t(α, two-tailed; v)² — both hold here to better than 1e-12 relative.
Never goes stale
These are properties of the distributions, not published data. There is no source to update and no revision date to watch.
Common use cases
- Look up a t critical value at a degrees of freedom that printed tables skip — they typically jump from 30 straight to 40, 60, 120 and ∞
- Turn an observed test statistic into an exact p-value instead of reporting only 'p < 0.05' because the table ran out of columns
- Get the chi-square cut-off for a goodness-of-fit or independence test at any df, in either the upper or the lower tail
- Read an F critical value for an ANOVA with an unusual numerator and denominator df combination
- Check a fractional degrees of freedom from a Welch–Satterthwaite correction, which no printed table can list
Frequently asked questions
What is the critical value of t at 0.05 with 10 degrees of freedom?
For a two-tailed test at α = 0.05 with 10 degrees of freedom the t critical value is 2.228138852, which printed tables round to 2.228. For a one-tailed test at the same α and df it is 1.812461123. The two-tailed figure is the point that leaves α/2 = 0.025 in each tail, which is why it is the larger of the two.
Why do printed t tables stop at 30 degrees of freedom?
Because past about df = 30 the t distribution is close enough to the standard normal that a book saves space by jumping to 40, 60, 120 and ∞ and letting readers interpolate. The gap is real but small: at α = 0.05 two-tailed, t is 2.042 at df = 30, 2.021 at df = 40 and 1.960 at df = ∞. This page computes the missing rows exactly, so df = 37 gives 2.026 without any interpolation.
What is the difference between a critical value and a p-value?
A critical value is fixed before you see the data — it is the statistic value that cuts off probability α in the tail, such as 1.959963985 for a two-tailed z test at α = 0.05. A p-value is computed after, and is the tail probability beyond the statistic you actually observed. The two always agree at the boundary: a statistic exactly equal to the critical value has p exactly equal to α.
How accurate are the values on this page?
Tail probabilities are evaluated by continued fractions for the regularised incomplete beta and incomplete gamma functions, iterated to a relative tolerance of 1e-15, and critical values invert them by bisection to a relative bracket width of 1e-13. Against published five-decimal tables every printed digit matches — 3.841 for χ²(0.05, df = 1), 18.307 for χ²(0.05, df = 10), 3.708 for F(0.05, df₁ = 3, df₂ = 10) — and against exact cross-distribution identities the agreement is better than 1e-12 relative.
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