Critical Value Calculator

The same decision, read in the other direction

Critical values and p-values are taught as two procedures. They are one procedure described from two ends, and seeing that makes both easier.

The critical value approach fixes α first and asks: how extreme would the statistic have to be? You compute the boundary, compare, and reject if you are past it.

The p-value approach takes the statistic first and asks: how extreme is it? You compute the tail probability and reject if it falls below α.

These always agree. |statistic| > critical value is true exactly when p < α— not approximately, identically, because the critical value is the inverse of the same function the p-value is a forward evaluation of. This page’s verification suite asserts that equivalence across three hundred generated cases rather than asserting it in prose.

The historical reason both exist is that inverse lookups were cheap from a printed table and forward ones were not. You could find the 5% boundary in a book; you could not find the exact tail area for t = 2.37 with 19 degrees of freedom. Software reversed that, which is why modern reporting quotes p-values.

Critical values remain genuinely useful in two places: they define the rejection region before you see any data, which is the discipline p-values make it easy to abandon; and they are the multiplier in every confidence interval, which is the same number doing a different job.

The tail mistake, and why tables invite it

Nearly every error with critical values is about tails, and printed tables make it easy to commit.

A two-tailed test splits α between both ends. At α = 0.05 that is 2.5% in each tail, so you want the 97.5th percentile: 1.960 for z.

A one-tailed test puts all of α in one end. You want the 95th percentile: 1.645.

Those are different numbers for the same α, and the one-tailed value is smaller — easier to exceed. A statistic of 1.8 is significant one-tailed and not significant two-tailed.

The trap is that many tables are headed only “α = 0.05” without saying which. Others print a two-tailed α row above a one-tailed α row, so the same column serves 0.05 two-tailed and 0.025 one-tailed. Reading the wrong row is easy and silent, and it halves your effective threshold.

The tool keeps both values on screen so the difference is never inferred. And the rule from the p-value page applies with equal force here: choosing one-tailed after seeing which way the data went is not a stricter analysis, it is a weaker one wearing a stricter label.

Why chi-square and F have no plus-or-minus

z and t are symmetric about zero, so one number and a sign describes both boundaries. Chi-square and F are not, and two things follow.

Both are non-negative. Chi-square sums squared quantities and F is a ratio of variances, so neither can be negative and neither is symmetric about anything. A “± critical value” for them is not a thing that exists, and asking for one usually means a step has gone wrong earlier.

The usual tests are one-tailed by construction. A chi-square goodness-of-fit test rejects only for large values, because large means observed and expected counts are far apart. Small values mean they agree, which is not evidence against the null. The same holds for the F test in ANOVA: large F means the between-group variation dominates.

So for these, the whole of α goes in the upper tail even though the test is not directional in the sense a one-tailed t-test is. That is a genuine source of confusion, and it is why the tool stops offering the tail choice when you select them.

Two-sided bounds do exist for a different purpose. Building a confidence interval for a variance needs both a lower and an upper chi-square value, at α/2 in each tail, and those are the pair the tool shows separately. They are for an interval, not for a test.

One curiosity worth knowing: a suspiciously small chi-square — a fit far better than chance should allow — is a recognised signal of fabricated or over-cleaned data. Fisher used it to question Mendel’s pea results.

Choosing the right distribution

Four distributions, four situations, and picking the wrong one is a larger error than picking the wrong tail.

z when the population standard deviation is genuinely known, or for a large-sample test of a proportion. Rarer than its popularity suggests — most of the time σ is estimated and t is correct.

t for a mean when σ came from the sample, which is nearly always. Degrees of freedom are n − 1 for one sample, n₁ + n₂ − 2 for a pooled two-sample test, and a fractional Welch value when the variances are not assumed equal.

chi-square for counts: goodness of fit, independence in a contingency table, and inference about a variance. Degrees of freedom are categories minus one for goodness of fit, and (rows − 1)(columns − 1) for independence.

F for comparing variances, and for ANOVA. It needs two degrees of freedom — numerator and denominator — and they are not interchangeable: F(5, 10) and F(10, 5) are different distributions with different critical values.

A useful relationship that ties three of them together: F with 1 numerator degree of freedom equals the square of t with the same denominator df. The suite asserts it, and it is why a two-group ANOVA and a two-sample t-test always give the same p-value.

How far above z the t value sits

For a mean, the difference between the t and z critical values is exactly the cost of not knowing σ, and it is worth having a feel for its size.

At two-tailed α = 0.05, z is 1.960. The t value is 12.706 at 1 degree of freedom, 2.571 at 5, 2.228 at 10, 2.086 at 20, 2.042 at 30, 2.009 at 50, and 1.984 at 100.

The shape of that sequence is the point. It falls very steeply at first — the difference between 1 and 5 degrees of freedom is enormous — and then flattens into a long slow approach that never quite arrives.

At df = 30 the t value is 4.2% above z. At df = 100 it is 1.2%. At df = 1000 it is 0.1%. It never reaches zero, which is the reason this site uses t at every sample size rather than switching at some threshold; the argument is set out on the Confidence Interval Calculator.

The very small degrees of freedom deserve their own note. At df = 1 the two-tailed 95% critical value is 12.7, and at df = 2 it is 4.30. Those are not typos — with two or three observations you know almost nothing about the variability, and the t distribution is honest about it in a way z is not.

The same number, doing a different job

The multiplier in every confidence interval is a critical value. Recognising that collapses two topics into one.

estimate ± critical value × standard error. The 1.96 in a 95% interval is the two-tailed z critical value at α = 0.05. The t multiplier in a small-sample interval is the two-tailed t critical value at the same α.

This is why an interval and a test always agree: a hypothesised value falls outside the interval exactly when the test rejects it, because both comparisons use the same boundary.

It also explains why a higher confidence level gives a wider interval. 99% needs a critical value further into the tail — 2.576 rather than 1.960 for z — and the interval widens in proportion.

For a variance interval the pattern breaks in an instructive way. Because chi-square is asymmetric, the interval is not symmetric about the sample variance and needs both bounds computed separately — the two-sided pair the tool shows. This is one of the few places in introductory statistics where an interval is genuinely lopsided, and it is a good reminder that the ± form is a convenience of symmetry rather than a general rule.

What a printed table cannot give you

Three limitations, all of which are artefacts of the page rather than the mathematics.

Only the alpha levels the author chose. Usually 0.10, 0.05, 0.025, 0.01 and 0.005. If a pre-registration specifies 0.02, or a Bonferroni correction across seven tests gives α = 0.00714, no table has that column and interpolating is unreliable.

Only some degrees of freedom. Tables typically list every value to 30, then jump to 40, 60, 120 and infinity. A Welch test routinely produces something like df = 23.7, which is not a row anywhere, and rounding down to 23 is conservative but arbitrary.

Nothing at all for most F pairs. F needs a two-dimensional table for every α, so printed versions cover a grid of common combinations and nothing between them.

None of this makes tables wrong — the values in them are correct, and this tool reproduces them, which the suite checks against the published figures for the normal, t, chi-square and F. It just means the constraints of a printed page have shaped what people think is available, and there is no longer any reason to accept them.

Sources and methodology

The published tables this tool is checked against.

• NIST/SEMATECH e-Handbook — critical values of the t distribution
• NIST/SEMATECH e-Handbook — critical values of the chi-square distribution
• NIST/SEMATECH e-Handbook — critical values of the F distribution
• p-value Calculator — the same decision read in the other direction
• Confidence Interval Calculator — where this number becomes a multiplier
• How we calculate — methodology

Method. Every critical value is obtained by inverting the corresponding CDF from this site’s higher-precision distribution engine, by bracketing and bisecting that same function rather than by a second closed-form approximation. That matters: an inverse computed from its own forward function cannot drift away from it, and the suite asserts the round trip — CDF of the inverse returns the probability asked for — across every distribution and a range of probabilities. The published table values for the normal, Student-t, chi-square and F are asserted directly, including the F pairs, which are the ones most often transcribed wrongly. The suite also asserts the equivalence this page rests on: that comparing a statistic against its critical value gives the same verdict as comparing p against α, across three hundred generated cases. That engine is verified on every change against 155 assertions. The count and the per-case breakdown are published on the formula verification page.

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