A critical value is the boundary of the rejection region: the point on a sampling distribution beyond which a result is treated as too extreme to be attributed to chance at your chosen significance level. This calculator computes that boundary for the four distributions that carry almost all of introductory and applied hypothesis testing — the standard normal, Student's t, chi-square and F — at any alpha you type, in either a one-tailed or a two-tailed arrangement.
Arb Digital built this page because the printed tables that most courses still rely on are truncated. They list a handful of alpha values, they stop at 30 degrees of freedom and jump to infinity, and they force you to interpolate the moment your design does not match a row. This tool evaluates the underlying distribution functions numerically, so a t critical value at 137 degrees of freedom and an alpha of 0.037 is as easy to get as the textbook 0.05 at df 10. Nothing here is read from a stored table.
What This Critical Value Calculator Does
It runs the inverse cumulative distribution function. You give it a tail area; it returns the statistic that leaves exactly that much probability beyond it. For z and t the distributions are symmetric, so a two-tailed test at alpha produces a matched pair of cut-offs at plus and minus the same magnitude. For chi-square and F the distributions are skewed and bounded below at zero, so the lower and upper cut-offs are genuinely different numbers and the calculator reports both.
This is the mirror image of our p-value calculator, and that is the one-sentence boundary worth keeping straight: the p-value calculator starts from a test statistic you have already computed and returns the probability attached to it, while this page starts from the probability you are willing to accept and returns the statistic. Same distributions, opposite direction of travel. Use whichever end of the problem you actually hold.
How to Use It
- Choose the distribution. z for a known population standard deviation or a large sample, t for an estimated standard deviation, chi-square for variance and goodness-of-fit tests, F for comparing two variances or running an ANOVA.
- Set the number of tails. Two-tailed if the alternative hypothesis simply says "different", one-tailed if it specifies a direction and you committed to that direction before seeing the data.
- Type the alpha level. Any value between zero and one works. It does not have to be 0.05.
- Enter the degrees of freedom. One value for t and chi-square, two for F. The z field ignores both because the standard normal has no degrees of freedom.
- Compare your statistic to the cut-off. If the absolute value of your test statistic exceeds the critical value shown, the result is significant at that alpha.
How Critical Values Are Calculated
Every critical value is the solution to one equation: find the value x such that the area under the density curve beyond x equals the tail probability you specified. Formally, x is the inverse of the cumulative distribution function evaluated at 1 minus that tail area. For the standard normal the calculator uses a rational approximation to the inverse normal CDF accurate to roughly one part in a billion. For the other three it evaluates the CDF directly — the regularised incomplete beta function for t and F, the regularised incomplete gamma function for chi-square — and then inverts it by bisection until the answer stops moving.
Work through the default. With a t distribution, two tails and alpha 0.05, each tail holds 0.025. The calculator solves for the t value that leaves 2.5% of the area above it at 10 degrees of freedom, which is 2.228. Anything beyond plus or minus 2.228 is significant. Drop to a one-tailed test at the same alpha and the whole 5% sits in one tail, so the cut-off falls to 1.812 — a lower bar, which is exactly why the choice of tails has to be made before the data is seen. Both figures match the standard published t table row for df 10, and the NIST/SEMATECH e-Handbook page on the t distribution sets out the density these numbers come from.
Chi-square and F work the same way but without symmetry. At 5 degrees of freedom and alpha 0.05 upper-tailed, the chi-square critical value is 11.070; the corresponding lower-tail cut-off at the same area is 1.145. They are not mirror images because the distribution is skewed right and cannot go below zero. The e-Handbook's chi-square distribution page and its F distribution page give the density functions behind both.
One Tail or Two: The Decision That Changes the Answer
The tails switch is the single most consequential control on this page, and it is the one most often set after the fact. A two-tailed test at alpha 0.05 puts 2.5% in each end. A one-tailed test puts the full 5% in one end, which pulls the cut-off closer to zero and makes significance easier to reach. For the standard normal the difference is 1.960 versus 1.645 — a gap wide enough that a substantial band of results are significant one-tailed and not significant two-tailed.
That gap is exactly why the choice must be locked in before the data arrives. Deciding to go one-tailed after seeing which way the effect points is not a one-tailed test at all; it is a two-tailed test with the alpha quietly doubled, and it inflates the false positive rate from 5% to 10%. The legitimate case for one tail is narrow: a result in the opposite direction would lead to the same decision as no result at all. A quality inspector who only ever acts when defects rise, and does nothing whatsoever if they fall, has a genuine one-tailed problem. Most research questions do not.
Why the t Cut-Off Moves With Sample Size but z Never Does
The standard normal critical value at alpha 0.05 two-tailed is 1.960 regardless of how much data you have. The t critical value at the same alpha is 12.71 at 1 degree of freedom, 2.228 at 10, 2.042 at 30, 1.984 at 100 and 1.962 at 1,000. It converges on the z value from above and never quite reaches it.
The reason is that t carries an extra source of uncertainty. When you estimate the population standard deviation from the sample rather than knowing it, the estimate itself wobbles, and small samples wobble badly. The t distribution has heavier tails to pay for that, which pushes the cut-off further out and demands a larger observed effect before you call it real. Beyond about 30 degrees of freedom the two are close enough that the distinction rarely changes a decision, which is where the familiar rule of thumb comes from — but the rule is a convenience, not a threshold, and there is no reason to fall back on z once this calculator will give you the exact t value at any df. Our t-test calculator uses the same distribution to produce the test statistic you would compare against these cut-offs.
The Relationship Between Critical Values and Confidence Intervals
A two-tailed critical value and a confidence interval are the same quantity wearing different clothes. The 95% confidence interval for a mean is the sample mean plus and minus the two-tailed critical value at alpha 0.05 multiplied by the standard error. That is why the grid on this page reports the confidence level alongside the cut-off: 1 minus alpha, expressed as a percentage.
The practical consequence is that the two approaches always agree. If a 95% interval excludes the null value, the two-tailed test rejects at alpha 0.05, and vice versa — they cannot disagree, because they are built from the same number. The interval is usually the more informative presentation because it shows the size and precision of the effect rather than a bare accept-or-reject verdict. Feed the multiplier from this page into our confidence interval calculator to see the interval form of the same result.
Multiple Comparisons Move the Cut-Off You Should Use
The critical value on this page controls the error rate of one test. Run twenty independent tests at alpha 0.05 and the probability that at least one of them produces a false positive is about 64%, not 5%. Nothing is wrong with the arithmetic; the alpha you typed simply is not the error rate of the family of tests you actually ran.
The usual repair is to shrink alpha before computing the critical value. A Bonferroni correction divides the target alpha by the number of comparisons: twelve tests at a family-wide 0.05 means typing 0.00417 into the alpha field, which pushes the two-tailed z cut-off from 1.960 out to 2.865. That is a real and often uncomfortable increase, which is the point — it prices the extra chances you gave yourself. Type the corrected alpha into this calculator rather than mentally adjusting the standard cut-off afterwards, because the relationship between alpha and the critical value is not linear and intuition consistently under-corrects. Where several groups are being compared at once, running a single ANOVA with an F critical value is often cleaner than a pile of pairwise tests.
Reading Chi-Square and F Cut-Offs Correctly
Chi-square and F tests are usually one-tailed in the upper direction even when the alternative hypothesis is non-directional, and this trips people up. A goodness-of-fit test measures how far observed counts sit from expected counts; any mismatch, in any direction, makes the statistic larger. There is no lower end to worry about, so the entire alpha goes in the upper tail even though the question was two-sided in spirit.
The exception is a genuine two-sided test on a variance, where a variance far below expectation is as interesting as one far above. That is when the lower critical value in the grid earns its place. The F distribution has a further quirk worth knowing: its lower critical value at df1 and df2 equals the reciprocal of the upper critical value with the degrees of freedom swapped. Older tables printed only upper values and expected you to perform that flip by hand. This calculator computes both directly, so the reciprocal trick is a check rather than a chore. For the count-based version of the same test, our chi-square calculator produces the statistic itself.
Arb Digital sets significance thresholds before a campaign test starts, not after the numbers land, so results survive scrutiny.
Browse All Free Tools Talk To Our TeamCommon Mistakes to Avoid
- Switching to one tail after seeing the data — that doubles the real false positive rate while appearing to keep alpha at 0.05.
- Halving alpha yourself and then choosing two tails — the calculator already splits alpha across the tails, so entering 0.025 for a two-tailed 5% test produces the cut-off for a 2.5% test.
- Using n instead of degrees of freedom — a one-sample t test on 25 observations has 24 degrees of freedom, and the difference matters most in exactly the small samples where people are most casual about it.
- Assuming a chi-square or F cut-off is symmetric — both distributions are skewed and bounded at zero, so the lower value is never the negative of the upper one.
- Ignoring multiple comparisons — the alpha for one test is not the alpha for the whole family, and the critical value has to move to reflect that.
Related Free Tools From Arb Digital
Convert a raw observation into standard units with the z-score calculator, get the probability attached to a statistic with the p-value calculator, work out how many observations a design needs with the sample size calculator, look up areas under the bell curve with the normal distribution calculator, or summarise your data first with the descriptive statistics calculator. The free online tools hub lists every statistics calculator we publish.
Frequently Asked Questions
It is the cut-off point on a sampling distribution that separates results considered consistent with the null hypothesis from results extreme enough to reject it at your chosen significance level.
For the standard normal distribution it is 1.960 for a two-tailed test. For a t distribution it depends on the degrees of freedom, ranging from 12.71 at one degree of freedom down towards 1.960 as the sample grows.
Use two tails unless a result in the opposite direction would lead to exactly the same decision as no result at all. The choice must be made before you look at the data, because switching afterwards doubles the false positive rate.
A critical value goes from a chosen alpha to a cut-off statistic. A p-value goes from an observed statistic to a probability. They are inverses of each other and always agree about significance.
Because the t distribution accounts for the extra uncertainty in estimating the standard deviation from the sample. Smaller samples give a less reliable estimate, so the tails are heavier and the cut-off sits further from zero.
The chi-square distribution is skewed to the right and cannot take negative values, so the areas in its two tails are reached at points that are not mirror images of one another.
Yes. Any value strictly between zero and one is accepted, because the calculator inverts the distribution function rather than reading a fixed table of standard alpha levels.
If you are running several tests and care about the chance of any false positive across all of them, yes. Divide your target alpha by the number of comparisons and enter that smaller value here.
This page explains a statistical calculation for educational purposes only. It is not medical, legal, or financial advice, and study design decisions should be reviewed by a qualified statistician.
