Critical value calculator

What a critical value is

A critical value is the point on a distribution where the area in the tail beyond it equals the significance level, α. Everything past that point is the rejection region: if your test statistic lands there, the result is significant at α. Everything short of it is not.

P(statistic > critical value | H₀ true) = α

That makes the critical value and the p value two ways of stating the same decision. The critical value fixes the threshold on the scale of the statistic before you look at the data; the p value reports how much tail area lies beyond the statistic you actually got. A statistic exactly equal to the critical value has a p value of exactly α, and a statistic beyond it has a p value below α. The two never disagree.

How to use it

Compute your test statistic, then compare its size with the critical value from this page. For a two-tailed test, take the absolute value: reject the null hypothesis when |statistic| exceeds the critical value. For a one-tailed test, reject when the statistic exceeds it in the direction you specified.

With the defaults above — Student's t, df = 18, α = 0.05, two-tailed — the critical value is ±2.101. A t-test on a sample of 19 that returns t = 2.4 is significant at the 5% level; one that returns t = 1.9 is not. That is the whole procedure. The calculator states the rejection region in exactly those terms so there is nothing to translate.

One tail or two

A two-tailed test splits α between both ends of the distribution: 0.025 in each tail at α = 0.05, so the cut-off is the point with 97.5% of the distribution below it. A one-tailed test puts the whole of α in one tail, so the cut-off sits at the 95th percentile instead and is easier to pass. For z at α = 0.05, that is the difference between 1.960 and 1.645.

Two tails are the default because the honest question is nearly always "is there a difference in either direction?" A one-tailed test is only legitimate when the direction was fixed before the data were seen, and a result in the opposite direction would be treated as no result at all. Choosing one tail after noticing which way the data lean is the oldest trick for manufacturing significance, and reviewers know it.

Chi-square and F do not offer the choice. Both statistics are built from squared quantities, cannot be negative, and grow with every kind of departure from the null — so all the evidence is in the upper tail and the calculator puts all of α there regardless of the tails setting.

Which distribution

The distribution follows from the test that produced the statistic, and the degrees of freedom follow from the design.

• Normal (z). Use it when the population standard deviation is known, or the sample is large enough that estimating it barely matters, and for tests on proportions. No degrees of freedom. The two-tailed value at 95% is the familiar 1.960; see the z-score calculator for standardising a value.
• Student's t. Use it when the standard deviation is estimated from the sample — which, in practice, is nearly always. A one-sample or paired t-test has df = n − 1; a pooled two-sample test has df = n₁ + n₂ − 2. The same value is the multiplier in a confidence interval for a mean. As df grows, t converges on z: at df = 30 the two-tailed 5% value is 2.042, and by df = 1,000 it is 1.962.
• Chi-square (χ²). Use it for tests on counts. A goodness-of-fit test across k categories has df = k − 1; a test of independence on an r × c table has df = (r − 1)(c − 1). Always upper-tailed.
• F. Use it for ANOVA and for comparing two variances. It takes two degrees of freedom: df₁ = k − 1 for the numerator (groups minus one) and df₂ = N − k for the denominator (total observations minus groups). Three groups of ten gives df = 2 and 27, and a critical F of 3.354 at α = 0.05. Always upper-tailed.

Reference table

The values below are the ones that come up most often. Each was computed by the same inverse-CDF code that runs the calculator, not copied from a printed table.

Two things are worth noticing in it. The t column shrinks towards 1.960 as the degrees of freedom grow, which is the cost of estimating the standard deviation fading away as the sample gets larger. And the one-tailed z value at α = 0.05 is 1.645 — the same number as the two-tailed value at α = 0.10, because both put 5% in the upper tail.

Confidence level and α

The two describe the same threshold from opposite sides: confidence level = 1 − α. A 95% confidence interval and a test at α = 0.05 use the same critical value, and a two-sided interval that excludes the null value is exactly equivalent to a two-tailed test rejecting it. The calculator reports both figures so that whichever one your textbook or software uses, the cross-check is immediate.

95% confidence ↔ α = 0.05 ↔ z* = 1.960 (two-tailed)

Why tables are giving way to p values

Critical values exist because of printing. A book can hold a page of t values for a handful of α levels and a few dozen degrees of freedom, but it cannot hold the full cumulative distribution for every possible statistic. So the convention was to fix α, look up the single number that mattered, and report only whether the statistic cleared it.

Software removed that constraint. Once a p value can be computed exactly for any statistic, the table adds nothing: it collapses a continuous measure of evidence into a yes or no, and it hides the difference between a statistic that scraped past the line and one that cleared it by a mile. Modern reporting guidelines ask for the exact p value and a confidence interval, and treat the 0.05 line as one reference point rather than a verdict.

Critical values still earn their place in two settings. Building a confidence interval needs the multiplier directly, and learning what a test does is far clearer when the cut-off is visible on the scale of the statistic rather than buried in a probability. This page is for both.