What this calculator does
A critical value is the cutoff on a test statistic's distribution that marks where a hypothesis test switches from "fail to reject" to "reject the null hypothesis." Anything more extreme than the critical value falls in the rejection region; anything less extreme does not. It is worked out purely from the significance level, the tail direction and, for the t-distribution, the degrees of freedom, before a single piece of sample data is collected.
This calculator handles both common cases: the critical t value, used when the population standard deviation is unknown and estimated from the sample (the usual case for a t-test), and the critical z value, used when the population standard deviation is known or the sample is large enough that the normal approximation applies. The t-distribution has heavier tails than the normal distribution at small degrees of freedom, so a critical t value is always at least as large in magnitude as the matching critical z value, converging to it as degrees of freedom grow.
The formula
For a two-tailed test at significance level α, the critical value is the point where the cumulative probability reaches 1 − α/2, since α is split evenly between both tails. For a one-tailed test the whole of α sits in one tail, so the cutoff is at cumulative probability 1 − α. The z case reads that cutoff straight off the standard normal distribution; the t case reads it off the Student's t-distribution for the stated degrees of freedom, which has no closed-form inverse and is solved here numerically.
| Term | Meaning |
|---|---|
| Critical value | The cutoff on a test statistic's distribution that separates "fail to reject" from "reject the null hypothesis." |
| Significance level (α) | The probability of rejecting a true null hypothesis, chosen before the test, typically 5% or 1%. |
| Degrees of freedom | For the t-distribution, a value tied to sample size that controls how heavy its tails are; smaller values mean fatter tails and a larger critical t value. |
| Two-tailed vs one-tailed | A two-tailed test splits α across both ends of the distribution; a one-tailed test puts all of α in the single direction being tested. |
The inputs explained
| Field | What to enter |
|---|---|
| Distribution | Choose t if the population standard deviation is unknown and was estimated from the sample (the usual case); choose z if it is known, or the sample is large. |
| Significance level (α) (%) | The significance level for the test, most commonly 5% or 1%. |
| Tail | Two-tailed for a "different from" hypothesis; one-tailed for a "greater than" or "less than" hypothesis. |
| Degrees of freedom (t-distribution only) | Sample size minus the number of estimated parameters, ignored when the z-distribution is selected. |
When to use it
Setting a rejection rule before running a t-test
Working out the critical t value in advance, at df = 10 and α = 5%, and only afterwards comparing the sample t-statistic against it, keeps the test's decision rule separate from and unbiased by the result itself.
Checking a published t-table figure
Textbook t-tables only list common combinations of degrees of freedom and significance level. This calculator reproduces those same figures for any degrees of freedom, not just the ones a printed table happened to include.
Comparing the t and z cutoffs at the same significance level
At small sample sizes the critical t value is noticeably larger than the matching critical z value, which is why using the normal distribution's cutoff on a small sample makes a test look more significant than it really is.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How the critical t value changes with degrees of freedom
A fixed 5% two-tailed significance level, across a range of degrees of freedom.
How the critical value changes with significance level
A fixed 10 degrees of freedom, across a range of common significance levels.
Questions
Why is the two-tailed critical value larger than the one-tailed critical value at the same α?
A two-tailed test spreads the significance level across both tails, so each individual tail only gets α/2 of the probability, pushing the cutoff further out. A one-tailed test puts the full α into a single tail, so the cutoff sits closer to the centre of the distribution for the same overall significance level.
When should I use the critical t value instead of the critical z value?
Use the t-distribution whenever the population standard deviation is unknown and has been estimated from the sample itself, which describes most real hypothesis tests on sample data. The z-distribution is appropriate only when the true population standard deviation is known, or as an approximation once the sample is large (as a rough guide, above about 30).
How is this different from a p-value?
A critical value is a fixed cutoff on the test statistic's scale, set before the data is examined. A p-value is worked out afterwards, from the actual sample test statistic. The two are two routes to the same decision: reject the null hypothesis if the sample statistic passes the critical value, which is exactly equivalent to rejecting when the p-value falls below α.
Why does the critical t value get smaller as degrees of freedom increase?
Degrees of freedom controls how heavy the tails of the t-distribution are. A small sample carries more uncertainty about the true population standard deviation, which fattens the tails and pushes the critical value out further. As the sample grows, that uncertainty shrinks and the t-distribution converges towards the normal distribution, which has thinner tails.
To turn a sample mean and standard deviation directly into a test statistic and compare it against a threshold, see the t-test calculator. For the value at a given percentile of a normal distribution rather than a hypothesis-test cutoff, see the inverse normal distribution calculator.