Critical Value Calculator

Find the threshold value for a given significance level α across Z, t, chi-square, and F distributions. Get instant results with step-by-step working.

Critical value vs p-value

This calculator finds critical valuesbefore your test. You set α (e.g., 0.05), and we give you the test-statistic threshold(s). If your observed statistic exceeds this, you reject H₀. This is the inverse of the p-value calculator, which goes the other way: you give a statistic, we compute the probability. Both tools answer the question "Is this result unlikely under H₀?", just from different starting points.

Critical Value —
α —
Enter α and required parameters to calculate.
Distribution curve with critical value and rejection region shaded
Fig. 1 — The sampling distribution under H₀, showing the critical value(s) and rejection region in red.

How this was calculated

  1. α = 0.05 Set the significance level.

The formula

c = F⁻¹(α | distribution)
c
Critical value (or critical values for two-tailed tests)
F⁻¹
Inverse cumulative distribution function (quantile function)
α
Significance level (e.g., 0.05 for 5%)

Worked example

Find the critical value for a two-tailed t-test at α = 0.05 with df = 20.

Step 1. Identify the distribution and tail(s). Two-tailed t-test means we split α equally: 0.05 / 2 = 0.025 in each tail.

Step 2. Find the upper quantile. We want t0.975 (since 1 − 0.025 = 0.975) with 20 degrees of freedom.

Step 3. Using the inverse t CDF: t0.975,20 ≈ 2.0860

Result: The critical values are ±2.086. Reject H₀ if |t| > 2.086.

Check: Enter α = 0.05, distribution = t, df = 20, tail = two-tailed into the calculator above. You should see ±2.086.

Assumptions & when this applies

One-tailed vs two-tailed

A two-tailed test divides α equally into both tails of the distribution (reject if the statistic is extreme in either direction). A one-tailed test puts all of α in one direction only—use this when your hypothesis predicts a direction (e.g., "the treatment improves outcomes", not just "the treatment has an effect").

F-distribution note

F-tests are conventionally one-tailed (upper tail only). This is because the F statistic is defined as a ratio of variances, which is always positive, and we reject H₀ only when the ratio is unusually large. There is no "lower-tail" F-test; the calculator shows upper-tail critical values only.

Rounding and numerical precision

Critical values are computed numerically to high precision (~1e-14 relative error). Values displayed to 4 decimal places are sufficient for practical purposes. Older printed tables may show fewer decimal places; small differences are normal.

Degrees of freedom

The t, χ², and F distributions all depend on degrees of freedom (df). For t-tests, df = n − 1; for chi-square goodness-of-fit, df = (categories − 1); for ANOVA F-tests, df₁ = groups − 1 and df₂ = n − groups. Non-integer df values are rounded to the nearest integer for chi-square and F (a caveat in practice when df is specified by a formula).

Frequently asked questions

What is a critical value and why do I need one?

A critical value is the threshold for your test statistic. Before you collect data, you choose a significance level α (usually 0.05). The critical value is the boundary: if your observed statistic falls outside this boundary, the result is rare enough that you reject the null hypothesis H₀. It separates "likely under H₀" from "unlikely under H₀".

How do I decide between one-tailed and two-tailed critical values?

Use a two-tailed critical value when your hypothesis does not predict a direction—you just want to know if there is any difference. Use a one-tailed critical value when your hypothesis predicts the direction of the effect (e.g., "treatment increases performance"). If you choose one-tailed, be clear about it before seeing your data; switching after the fact to get significance is p-hacking. For a given α, a one-tailed critical value is further from zero (more extreme) than a two-tailed one, because all of α is concentrated in one tail.

Can I use this for the normal distribution (Z-test)?

Yes. The Z-distribution is the standard normal distribution. It does not depend on sample size or degrees of freedom—only on α and whether you choose one-tailed or two-tailed. Common critical values: Z₀.₀₅ (one-tailed, upper) ≈ 1.645; Z₀.₀₂₅ (two-tailed, each tail) ≈ 1.960. Use this when you have a large sample or when the population standard deviation is known.

What do I do with critical values for chi-square and F?

For chi-square goodness-of-fit or independence tests: you compare your observed χ² statistic to the critical value. If observed χ² exceeds the critical value, reject H₀. These are always one-tailed upper-tail tests. For F-tests (ANOVA, equality of variances): same logic—compare your observed F to the critical value, and reject H₀ if F is larger. F-tests are always upper-tailed because F is a ratio of non-negative quantities.

Why is my critical value different from the textbook table?

Textbook tables are usually rounded to 2–3 decimal places. This calculator displays 4 decimals. Rounding differences account for almost all mismatches. Also check that you used the correct df and tail direction. If you are still unsure, check the worked example above or consult your textbook's table directly.

Can I use this to compute confidence intervals?

Indirectly, yes. Confidence intervals use critical values as part of their construction (e.g., a 95% CI uses the two-tailed critical value for α = 0.05). See our confidence interval calculator for a tool tailored to that task.

Sources