Core inference
Critical Value Calculator
A critical value is the boundary of the rejection region — the threshold your test statistic has to beat to count as significant. Pick a distribution (z, t, chi-square or F), a significance level α and the degrees of freedom, and this returns the cut-off: a symmetric ±pair for a two-tailed test, or a single upper- or lower-tail value. It is the table-lookup twin of the p-value, drawn on the curve so you can see exactly where “significant” begins.
Two-tailed splits α between both tails and gives a symmetric cut-off for z and t (a lower/upper pair for χ² and F). Upper puts all of α in the right tail; lower puts it all in the left. Chi-square and F tests are usually upper-tailed.
Result
In plain English
Every significance test reduces your data to a single number — the test statistic — and then asks whether that number is extreme enough to be surprising under the null hypothesis. The critical value is the line in the sand. If the statistic lands past it, you are in the rejection region and call the result significant; if it falls short, you do not reject. Choosing the critical value is the same act as choosing how much risk of a false alarm you are willing to run, which is exactly what α is.
- critical value
- The cut-off that bounds the rejection region. Reject the null when the test statistic is at least this extreme. It depends on the distribution, α and the degrees of freedom — never on your data.
- rejection region
- The set of test-statistic values that would lead you to reject the null. Its total probability under the null equals α.
- significance level α
- The false-alarm rate you accept in advance: the chance of rejecting a true null. Common choices are 0.05, 0.01 and 0.10. It is a decision, not something the data hand you.
- one- vs two-tailed
- A two-tailed test looks for a difference in either direction and splits α between both tails (so each holds α∕2). A one-tailed test commits to a direction and spends all of α on one side, giving a smaller cut-off — but only detects effects that way.
- degrees of freedom
- For the t, χ² and F distributions the shape — and therefore the critical value — depends on the degrees of freedom, which come from your sample sizes and the number of estimated quantities.
- the p-value twin
- Comparing the statistic to the critical value gives the identical verdict to comparing the p-value to α. The statistic clears the critical value at the exact moment the p-value drops below α. They are two views of one decision.
- what it cannot tell you
- Clearing the critical value does not measure how big or important the effect is, does not prove the null is false, and is only valid if the test’s assumptions hold. A bare “significant” is the start of the story, not the end.
Frequently asked
What is a critical value?
A critical value is the threshold a test statistic must reach for a result to be declared statistically significant. It marks the edge of the rejection region — the range of outcomes so unlikely under the null hypothesis that you would reject the null if you saw them. The total probability beyond the critical value(s) equals your significance level α, so a 5% test puts 5% of the null distribution into the rejection region. Critical values are read from (or computed for) a specific distribution: the standard normal for a z-test, Student’s t for a t-test, chi-square for goodness-of-fit and independence tests, and F for ANOVA and variance ratios.
How do I choose between a one-tailed and two-tailed critical value?
Decide before you see the data. Use a two-tailed test when a difference in either direction matters — the usual default — and the calculator splits α so each tail holds α∕2, giving the familiar ±1.96 at α = 0.05 for z. Use a one-tailed test only when you have a directional hypothesis fixed in advance and genuinely do not care about the other direction; all of α then sits in one tail, lowering the cut-off to 1.645 and making it easier to reach significance. Switching to one-tailed after seeing which way the data point is a form of p-hacking and inflates your false-positive rate.
Why is the t critical value larger than the z critical value?
Because the t distribution has heavier tails. When you estimate the population standard deviation from a small sample, you add extra uncertainty, and the t distribution widens to absorb it — so you need to go further out to enclose the same central probability. At α = 0.05 two-tailed, z gives ±1.96, but t with 10 degrees of freedom gives ±2.23 and with 5 df gives ±2.57. As the sample grows the degrees of freedom rise, the tails thin, and the t critical value slides down toward the z value: by df ≈ 1000 they are practically identical.
Is the critical value the same as the p-value?
They are two routes to the same decision. The critical-value approach fixes α, finds the cut-off, and checks whether your statistic is beyond it. The p-value approach computes how much of the null distribution is as extreme as your statistic, then checks whether that probability is below α. The verdicts always agree: statistic beyond the critical value ⇔ p-value below α. The p-value carries a little more information (how far past the line you landed), while the critical value is what you draw on the curve to see the boundary itself. This calculator gives you the boundary; the p-value calculator gives you the tail probability.