0.025
1.9604
2.226
20.4846
1.9604
0.025
1.9604
2.226
20.4846
1.9604
The Critical Value Calculator finds the cutoff point(s) on a probability distribution that separate the rejection region from the non-rejection region in hypothesis testing. Critical values are the thresholds against which test statistics are compared to make accept/reject decisions.
This calculator supports three major distributions used in statistical inference: the standard normal (Z) distribution for large-sample tests, the Student's t distribution for small-sample means testing, and the chi-square (χ²) distribution for variance testing and goodness-of-fit tests. Both one-tailed and two-tailed options are available.
Understanding critical values is essential for manual hypothesis testing, constructing rejection regions, and interpreting statistical tables. While modern software computes p-values directly, knowing critical values deepens your understanding of the decision framework underlying all hypothesis tests.
The critical value $c$ satisfies $P(X > c) = \alpha$ for one-tailed tests or $P(|X| > c) = \alpha$ for two-tailed tests.
Z critical values come from the inverse standard normal CDF. This calculator uses the rational approximation:
$$z_\alpha \approx \sqrt{-2\ln(\alpha)} - \frac{2.515517 + 0.802853\sqrt{-2\ln\alpha} + 0.010328(-2\ln\alpha)}{1 + 1.432788\sqrt{-2\ln\alpha} + 0.189269(-2\ln\alpha) + 0.001308(-2\ln\alpha)^{3/2}}$$
T critical values are approximated using the Cornish-Fisher expansion from the z-value:
$$t_{\alpha,\nu} \approx z + \frac{z^3 + z}{4\nu} + \frac{5z^5 + 16z^3 + 3z}{96\nu^2}$$
Chi-square critical values use the Wilson-Hilferty approximation:
$$\chi^2_{\alpha,\nu} \approx \nu\left(1 - \frac{2}{9\nu} + z_\alpha\sqrt{\frac{2}{9\nu}}\right)^3$$
For two-tailed tests, α is divided by 2 before computing the critical value, since the rejection region is split between both tails.
The critical value defines the boundary of the rejection region. If your test statistic exceeds the critical value (in absolute value for two-tailed tests), you reject the null hypothesis at significance level α.
The α per Tail shows the actual significance level used for each tail. For a two-tailed test at α = 0.05, each tail gets α/2 = 0.025.
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The two-tailed z-critical value at α=0.05 is 1.96, meaning you reject H₀ if |z| > 1.96.
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The one-tailed t-critical value at α=0.01 with 20 df is approximately 2.528.
A critical value is the point on a test distribution that marks the boundary of the rejection region. If the test statistic is more extreme than the critical value, you reject the null hypothesis. It corresponds to a specified significance level α.
Use a two-tailed test when you are testing for any difference (H₁: μ ≠ μ₀). Use a one-tailed test when you have a directional alternative (H₁: μ > μ₀ or H₁: μ < μ₀). Two-tailed is more conservative and is the default in most situations.
The t-distribution has heavier tails than the normal distribution, especially for small samples. As degrees of freedom increase, the t-distribution approaches the standard normal. With small df, the critical value is larger, reflecting greater uncertainty.
Chi-square critical values are used for: (1) testing population variance, (2) goodness-of-fit tests, (3) tests of independence in contingency tables, and (4) tests of homogeneity. The chi-square distribution is always non-negative and right-skewed.
The z-approximation is accurate to about 4 decimal places. The t-approximation is very good for df ≥ 5 and excellent for df ≥ 30. The chi-square approximation works well for df ≥ 10. For very small df or extreme α values, consult statistical tables or software.
Critical values and p-values are two sides of the same coin. You can either compare your test statistic to the critical value, or compare your p-value to α. Both methods yield the same decision. P-values provide more information because they tell you the exact significance level.
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The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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