Interactive module guide

Critical Value Calculator — z, t, χ², F

This critical value calculator replaces printed z, t, chi-square, and F tables with exact quantiles at any alpha and tail configuration.

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This tool replaces the four printed tables at the back of every statistics textbook: choose a reference distribution (z, t, χ², or F), a significance level, and a tail configuration, and read the exact critical value and rejection region.

It exists because interpolating between table rows is error-prone and unnecessary—the quantile functions are exact—and because the site deliberately offers one interactive lookup instead of four static table pages.

Degrees-of-freedom fields appear only for the distributions that need them.

Use the critical value calculator wherever a printed statistical table would have been used; the exact quantile replaces interpolation between table rows.

When to use this critical value calculator

Use it when

Use it during hand calculations and exams-style practice, where the test statistic is computed manually and only the boundary is needed.
Use it in teaching to show how the boundary moves: sliding α, switching tails, or growing degrees of freedom animates the geometry that tables freeze into rows.
Use it to audit reported results: a paper’s “t exceeded the critical value of 2.06 at df = 24” is checkable here in seconds.

Choose another method when

Avoid using critical values as a substitute for p-values in reporting; the boundary answers reject-or-not at one α, while the p-value carries strictly more information.
Avoid the two-tailed χ² and F regions out of habit; most named tests using these distributions are right-tailed by construction, and the tool defaults accordingly in its guidance.
Avoid quantile lookups for distributions whose assumptions the analysis does not meet — the boundary is only as meaningful as the reference distribution behind it.

Interactive tool

The calculator loads as you approach this section so the guide remains fast on mobile connections.

Example preview

Replacing the z table

Inputs: Normal (z), α=0.05, two-tailed.
Representative result: The rejection boundary is ±1.9600; the central 95% of the distribution lies inside.

Illustrative only. Load the interactive tool to enter your own values and review assumptions.

Watch the explanation

2:25 min

The player loads only after you press play. You can also watch on YouTube.

Read the complete transcript

Critical Values for z, t, Chi-Square, and F. A critical value is a quantile of a reference distribution. It converts a pre-chosen alpha and tail rule into the boundary of a rejection region. Choose the distribution that your test statistic actually follows: standard normal z, Student t, chi-square, or F. Then choose alpha and the alternative direction. Two-tailed tests split alpha in half; right- and left-tailed tests place all of it in the named tail. For symmetric z and t references, two-tailed boundaries are minus and plus the one-minus-alpha-over-two quantile. One-tailed rules use one-minus-alpha or alpha. Student t also needs degrees of freedom. Smaller df makes heavier tails and pushes the boundary farther from zero; as df grows, t approaches z. Chi-square needs one df and is skewed, so a two-tailed region has distinct lower and upper cutoffs. Plus-or-minus notation would be wrong. F needs ordered numerator and denominator degrees of freedom. Swapping them changes the reference distribution, and two-tailed F regions also need two different cutoffs.

At alpha point zero five, the two-tailed z boundary is plus or minus one point nine six zero zero, leaving ninety-five percent in the center. With ten degrees of freedom, the matching two-tailed t boundary widens to plus or minus two point two two eight one. That point two six eight widening is the small-sample premium. By one thousand degrees of freedom, the t boundary is one point nine six two three, nearly z. For chi-square with five degrees of freedom, the usual right-tail five-percent boundary is eleven point zero seven zero five. For F with numerator df five and denominator df ten, the right-tail five-percent boundary is three point three two five eight. If the problem truly needs two tails, chi-square df five uses point eight three one two and twelve point eight three two five. F five-ten uses point one five one one and four point two three six one. At a continuous boundary, the tail p-value equals alpha. But report the p-value, effect size, and interval too, and use specialized references when the model requires them. Look up critical values free at Distri Scope dot com.

How to read the result

For z and t, the two-tailed boundary at α = 0.05 is the familiar ±1.96 and its t-inflated cousins; watching t approach z as df grows is the whole small-sample story in one number.

For χ² and F, right-tail boundaries grow with df in the numerator sense; the left-tail values near zero are the ones tables rarely print and hand calculations most often need for two-sided variance regions.

A statistic exactly at the boundary has p exactly α; the boundary is the p-value’s level set, not a separate concept.

How to use the critical value calculator

Degrees-of-freedom inputs appear only when the chosen distribution uses them: one df for t and χ², a numerator and denominator pair for F.

For F, the order of the two df fields matters: the numerator df belongs to the effect being tested and the denominator df to the error term, and swapping them changes the boundary.

The tail configuration changes the meaning of α: two-tailed splits it across both sides, one-tailed places it entirely in the named tail.

Any α strictly between 0 and 0.5 is accepted, covering the conventional 0.10, 0.05, and 0.01 plus anything a problem sheet invents.

Formula, hypotheses, and assumptions

P(|Z| > z_{1-\alpha/2}) = \alpha

Conditions to review

The test statistic follows the selected reference distribution
Degrees of freedom come from the intended analysis
The tail configuration matches the planned alternative
Critical values complement, not replace, exact p-values

Calculator parameters

Distribution: default Normal (z).
Significance Level (α): default 0.05.
Tails: default Two-tailed.
Degrees of Freedom: default 10.
Numerator df (d₁): default 5.
Denominator df (d₂): default 10.

What the method is doing

Values come from the same quantile functions the calculators use — z and t inverses, χ² and F inverses — already pinned against scipy in the numerical test suite.

Two-tailed χ² and F regions report both boundaries, since those distributions are not symmetric and ± notation would be wrong.

The conclusion sentence states the full rejection rule in words, so the number is never quoted without its direction.

Worked example: replacing three table lookups

A problem set requires: the two-tailed z boundary at α = 0.05, the two-tailed t boundary at α = 0.05 with df = 10, and the right-tailed χ² boundary at α = 0.05 with df = 5.

1Normal (z), two-tailed, α = 0.05: the boundary is ±1.9600 — the constant behind “±2 standard errors”.
2t with df = 10, two-tailed: the boundary widens to ±2.2281, the small-sample premium over the normal.
3χ² with df = 5, right-tailed: the boundary is 11.0705; a goodness-of-fit statistic beyond it rejects at the 5% level.
4Cross-check any of the three against the corresponding test module — the same quantile functions produce both, so they must agree exactly.

Interpretation

Three lookups that would have meant three tables, three rows, and two interpolations reduce to three exact numbers with their rejection rules attached. The t-versus-z comparison in the first two steps is the pedagogical payoff: the 0.27 widening at df = 10 is precisely the price of estimating the standard deviation.

Common mistakes

Do not mix up one- and two-tailed boundaries; at the same α they differ substantially, and the tool prints the rule in words to prevent silent mismatches.
Do not interpolate from memory of old tables when df is unusual; the exact quantile is always available here.
Do not treat the boundary as the analysis; effect sizes and intervals carry the substantive content that a reject/retain verdict compresses away.

Limits and independent validation

Four reference distributions only; specialized tables (studentized range, Dunnett, exact nonparametric) are not included.

The tool reports boundaries, not p-values; the test calculators compute those from data.

Noncentral distributions used in power analysis are not exposed here.

Before using the result

Check the anchors every statistics course memorizes: z boundaries 1.6449 (one-tailed) and 1.9600 (two-tailed) at α = 0.05.
Confirm t approaches z: at df = 1000 the two-tailed boundary should read 1.9623, nearly the normal value.
Verify one value against the matching test module’s reported critical value; they share code and must agree.

t distribution →Explore the distribution whose quantiles the t boundaries come from.chi-square distribution →See the skewed shape behind the asymmetric χ² regions.power and sample size →Use boundaries prospectively in study planning.

← Confidence Interval Calculator One Sample Z-Test →

See every option in the Statistical Hypothesis Test Calculator or review the DistriScope methodology. Educational information; last reviewed 2026-08-09. Verify consequential calculations independently.