Critical Value Calculator

Find the critical value for a z, t, chi-square, or F test from your significance level (alpha), tail choice, and degrees of freedom — the exact cutoff used to accept or reject the null hypothesis.

Quick Facts

Method
Inverse cumulative distribution function (quantile) of the chosen distribution at 1 − α (or 1 − α/2 for two-tailed tests)
Values match standard z, t, χ², and F tables to 4+ decimal places.

Your Results

Calculated
Critical value
-
Reject the null beyond this cutoff
Lower critical value
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Two-tailed lower bound
Tail probability
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Area in the rejection region
Test
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Distribution and tail

Ready

Choose a distribution, tail, and α, then calculate.

What a critical value is

A critical value is the cutoff point on a test statistic's distribution that separates the "fail to reject" region from the rejection region in a hypothesis test. You pick a significance level α (the probability of a false positive you are willing to accept — commonly 0.05), and the critical value is the value of the test statistic that leaves exactly α of the probability in the tail(s). If your computed test statistic is more extreme than the critical value, the result is statistically significant and you reject the null hypothesis.

Formally, the critical value is a quantile of the reference distribution: it is the inverse cumulative distribution function (the inverse CDF, or quantile function) evaluated at 1 − α for a right-tailed test, at α for a left-tailed test, and at 1 − α/2 for a symmetric two-tailed test.

The formulas by distribution

  • Z (standard normal): the critical value is z1−α = Φ⁻¹(1 − α) for one tail, or ±z1−α/2 = ±Φ⁻¹(1 − α/2) for two tails, where Φ⁻¹ is the inverse standard-normal CDF. Z is used when the population standard deviation is known or the sample is large.
  • t (Student's t): tα, df = the inverse of the Student-t CDF with df = n − 1 degrees of freedom. Use t instead of z when the population standard deviation is estimated from a small sample.
  • Chi-square (χ²): the upper critical value χ²1−α, df is the inverse of the chi-square CDF. Used for goodness-of-fit, tests of independence, and variance tests.
  • F: F1−α, d1, d2 is the inverse of the F CDF with numerator df d1 and denominator df d2. Used in ANOVA and for comparing two variances.

The z and t distributions are symmetric about zero, so a two-tailed test has matching negative and positive cutoffs (±value). The chi-square and F distributions are non-negative and right-skewed, so their tests are almost always one-tailed on the upper end.

Common reference values

These are the critical values you will see most often, worth memorizing as sanity checks:

  • z, two-tailed, α = 0.05: ±1.960 (the basis of the 95% confidence interval).
  • z, two-tailed, α = 0.01: ±2.576.   z, one-tailed, α = 0.05: 1.645.   z, one-tailed, α = 0.10: 1.282.
  • t, two-tailed, α = 0.05, df = 10: ±2.228; df = 1: ±12.706. As df grows, t approaches z (for df = 1000 it is about ±1.962).
  • χ², upper α = 0.05, df = 1: 3.841; df = 10: 18.307.
  • F, upper α = 0.05, d1 = 5, d2 = 10: 3.326.

Critical value vs. p-value

The critical-value approach and the p-value approach always agree. With critical values you compare your test statistic to a fixed cutoff; with p-values you compute the tail probability of your statistic and compare it to α directly. You reject the null exactly when the statistic exceeds the critical value, which is the same condition as p ≤ α. Critical values are handy because they can be looked up before collecting data, which is why textbooks tabulate them.

Frequently Asked Questions

How accurate are these critical values?
This calculator computes each critical value from the distribution's inverse CDF using standard numerical methods (Acklam's inverse-normal algorithm, and continued-fraction incomplete beta/gamma functions for t, χ², and F). Results match published statistical tables to at least four decimal places — for example z(0.05, two-tailed) = 1.9600, t(0.05, two-tailed, df=10) = 2.2281, and χ²(0.05, df=1) = 3.8415.
Should I use z or t?
Use z when the population standard deviation σ is known, or when your sample is large (roughly n ≥ 30) so the sample standard deviation is a reliable estimate of σ. Use t when σ is unknown and estimated from a small sample; the t distribution has heavier tails, which produces a slightly larger critical value that correctly accounts for the extra uncertainty. As the degrees of freedom increase, the t critical value converges to the z critical value.
What is the difference between one-tailed and two-tailed?
A two-tailed test splits α between both tails (α/2 in each), so it uses the 1 − α/2 quantile and detects a difference in either direction. A one-tailed test puts all of α in a single tail and is used when you only care about deviations in one direction (e.g. "is the new mean greater than the old?"). A one-tailed test has a smaller critical value and more power for the direction you specified, but it cannot detect an effect in the opposite direction.
How do I find degrees of freedom?
For a one-sample t test, df = n − 1. For a two-sample t test with pooled variance, df = n₁ + n₂ − 2. For a chi-square goodness-of-fit test, df = (number of categories) − 1; for a test of independence, df = (rows − 1)(columns − 1). For an F test in one-way ANOVA, the numerator df = (number of groups) − 1 and the denominator df = (total observations) − (number of groups).

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