Critical Value Calculator
Find critical values and rejection regions for Z, t-Student, chi-square, and F distributions for one-tailed and two-tailed tests.
What Is a Critical Value?
A critical value is the cutoff that separates the rejection region from the non-rejection region in a hypothesis test. If your test statistic falls beyond the critical value(s), you reject the null hypothesis at the chosen significance level $\alpha$.
How Critical Values Are Found
Critical values come from the quantile function (inverse CDF) of the test statistic's null distribution:
- Left-tailed: $Q(\alpha)$
- Right-tailed: $Q(1 - \alpha)$
- Two-tailed: $Q(\alpha/2)$ and $Q(1 - \alpha/2)$
This calculator supports the standard normal ($Z$), Student $t$, chi-square ($\chi^2$), and $F$ distributions.
Supported Distributions
- Z: large-sample tests and known-variance mean tests.
- t: mean tests with unknown variance; needs degrees of freedom.
- $\chi^2$: variance tests, goodness-of-fit, and independence tests.
- F: ANOVA and variance-ratio tests; needs numerator and denominator degrees of freedom.
How to Use This Calculator
- Select the distribution of your test statistic.
- Choose two-tailed, right-tailed, or left-tailed.
- Enter $\alpha$ (commonly 0.05 or 0.01).
- Enter degrees of freedom when required.
- Read the critical value(s) and rejection region.
Example
For a right-tailed $t$ test with 15 degrees of freedom and $\alpha = 0.05$, the critical value is about $1.753$. Reject $H_0$ if the test statistic is greater than or equal to $1.753$.
Frequently Asked Questions
What is the Z critical value for 95% confidence?
For a two-tailed test with α = 0.05, the Z critical values are about ±1.96. For a one-tailed test at α = 0.05, the critical value is about ±1.645.
Is a t critical value the same as a Z critical value?
Not exactly. The t distribution has heavier tails. For large degrees of freedom (often above 30), t critical values become very close to Z critical values.
How do I find a t critical value from sample size?
Set degrees of freedom to n − 1 for a one-sample t test. Then choose the tail type and α, and read the quantile from the t distribution.
Critical value vs p-value: which should I use?
Both can lead to the same reject/fail-to-reject decision. The critical value approach compares the test statistic to a cutoff; the p-value approach compares the observed tail probability to α.
Why do chi-square and F tests often use right-tailed critical values?
Many classic χ² and F procedures (goodness-of-fit, independence, ANOVA) place unusual results in the upper tail, so the rejection region is on the right.
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