Math calculators

Critical Value Calculator

Updated Aug 31, 2026 By Jehan Wadia
Rate Formulas
Distribution Type
Test Parameters
Enter a valid α between 0.001 and 0.5
Positive integer (1–1000).
Enter a valid integer between 1 and 1000
Critical Value(s)
±1.9600
Significance Level (α)
0.05
Confidence Level
95.00%
Distribution
Z (Normal)
Detailed Results
Test TypeTwo-tailed
Effective α (per tail)0.0250
Degrees of FreedomN/A
Rejection RegionZ < −1.9600 or Z > 1.9600
Non-rejection Region−1.9600 ≤ Z ≤ 1.9600
p-value at Critical Value0.0500
Critical Value (upper)1.9600
Critical Value (lower)−1.9600
Distribution Visualization
Common Critical Values Reference

Introduction

A critical value is a point on a distribution that marks the boundary between rejecting and not rejecting a hypothesis. Our Critical Value Calculator finds critical values for five distributions: Z (Normal), t (Student's t), χ² (Chi-Square), F (Fisher), and R (Pearson Correlation). Just pick your distribution, set your significance level, choose a tail type, and enter your degrees of freedom. The calculator gives you the exact critical value, shows the rejection region on a chart, and displays a reference table of common critical values. Use it for homework, research, or any time you need to run a hypothesis test.

How to Use Our Critical Value Calculator

Enter your test details below and this calculator will give you the critical value, rejection region, a distribution chart, and a reference table of common critical values.

Distribution Type: Pick the distribution you need. Choose from Z (Normal), t (Student), χ² (Chi-Square), F (Fisher), or R (Pearson Correlation).

Tail Type: Select whether your test is two-tailed, right-tailed, or left-tailed. This tells the calculator which side of the distribution to place the rejection region.

Significance Level (α): Choose your alpha value from the dropdown. Common choices are 0.01, 0.025, 0.05, and 0.10. Pick "Custom" if you need a different value between 0.001 and 0.5.

Degrees of Freedom (df): Enter the degrees of freedom for your test. This field appears for t, χ², F, and R distributions. For the F distribution, you will also need to enter a second value for the denominator degrees of freedom (d₂). Enter a whole number from 1 to 1000.

Calculate: Press the "Calculate" button to get your results. You can also press Enter on your keyboard while in any input field.

Reset: Press the "Reset" button to clear all inputs and return the calculator to its default settings.

What Is a Critical Value?

A critical value is a number on a statistical distribution that marks the boundary between "reject" and "do not reject" a hypothesis. When you run a hypothesis test, you compare your test statistic to the critical value. If your test statistic falls in the rejection region (past the critical value), you reject the null hypothesis. If it does not, you fail to reject it.

The critical value depends on three things: the significance level (α), the type of distribution, and the degrees of freedom. The significance level is the probability of rejecting the null hypothesis when it is actually true. Common values for α are 0.01, 0.05, and 0.10. A smaller α means you need stronger evidence to reject.

Types of Distributions

  • Z (Normal): Used when the population standard deviation is known and the sample size is large (typically n ≥ 30).
  • t (Student's t): Used when the population standard deviation is unknown and the sample size is small. It requires degrees of freedom, which is usually n − 1.
  • χ² (Chi-Square): Used for tests about variance or for goodness-of-fit and independence tests. It only takes positive values.
  • F (Fisher): Used to compare two variances or in ANOVA tests. It has two degrees of freedom values: one for the numerator and one for the denominator.
  • R (Pearson Correlation): Used to test whether a correlation between two variables is statistically significant. The degrees of freedom equal n − 2, where n is the sample size.

Tail Types

A two-tailed test checks if a value is significantly higher or lower than expected. It splits α into both sides of the distribution. A right-tailed test checks if a value is significantly greater. A left-tailed test checks if a value is significantly less. The tail type you choose changes where the critical value falls on the distribution.

How to Use This Calculator

Pick your distribution type, choose a tail type, set your significance level, and enter the degrees of freedom if needed. Click Calculate to get the critical value, rejection region, and a visual chart of the distribution. The reference table below the chart shows critical values for common α levels and degrees of freedom so you can quickly look up what you need.

Once you have your critical value, you can compare it to your test statistic, find the exact probability associated with your result as a p value, or build a confidence interval around your estimate.

Formulas used

Z Critical Value (Standard Normal Inverse)
z_{\text{crit}} = \Phi^{-1}(1 - \alpha) \quad \text{(one-tailed)}, \quad z_{\text{crit}} = \pm\,\Phi^{-1}\!\left(1 - \frac{\alpha}{2}\right) \quad \text{(two-tailed)}
t Critical Value (Student's t Inverse)
t_{\text{crit}} = t^{-1}_{\,df}\!\left(1 - \alpha\right) \quad \text{(one-tailed)}, \quad t_{\text{crit}} = \pm\,t^{-1}_{\,df}\!\left(1 - \frac{\alpha}{2}\right) \quad \text{(two-tailed)}
Chi-Square Critical Value
\chi^{2}_{\text{crit}} = F^{-1}_{\chi^{2},\,df}\!\left(1 - \alpha\right) \quad \text{(right-tailed)}
F Critical Value (Fisher–Snedecor Inverse)
F_{\text{crit}} = F^{-1}_{d_1,\,d_2}\!\left(1 - \alpha\right)
Pearson r Critical Value from t
r_{\text{crit}} = \frac{t_{\text{crit}}}{\sqrt{t_{\text{crit}}^{2} + df}}, \quad df = n - 2
Regularized Incomplete Beta Function (used for t, F, χ² CDFs)
I_{x}(a,b) = \frac{1}{B(a,b)} \int_{0}^{x} t^{\,a-1}(1-t)^{\,b-1}\,dt
Confidence Level from Significance Level
\text{Confidence Level} = (1 - \alpha) \times 100\%

Frequently asked questions

What is the difference between a one-tailed and two-tailed test?

A one-tailed test checks for an effect in only one direction (either greater or less than expected). A two-tailed test checks for an effect in both directions. A two-tailed test splits your significance level (α) in half, putting half on each side of the distribution. This makes two-tailed critical values larger in absolute value than one-tailed critical values for the same α.

How do I find the degrees of freedom for my test?

It depends on the distribution. For a t-test, degrees of freedom usually equals your sample size minus 1 (n − 1). For Chi-Square, it depends on your test type (for goodness-of-fit, it is the number of categories minus 1). For the F distribution, you need two values: d₁ for the numerator and d₂ for the denominator. For Pearson R, degrees of freedom equals n − 2, where n is your sample size.

Why does the Z distribution not need degrees of freedom?

The Z (standard normal) distribution has a fixed shape. It does not change based on sample size. It assumes you know the population standard deviation and have a large enough sample. Because the shape never changes, there is no degrees of freedom to set.

What does the significance level (α) mean?

The significance level (α) is the chance of rejecting the null hypothesis when it is actually true. This is called a Type I error. An α of 0.05 means there is a 5% risk of a false rejection. A smaller α (like 0.01) is stricter and needs stronger evidence to reject.

How do I compare my test statistic to the critical value?

After you calculate your test statistic from your data, compare it to the critical value from this calculator. For a right-tailed test, reject the null hypothesis if your test statistic is greater than the critical value. For a left-tailed test, reject if it is less. For a two-tailed test, reject if your test statistic is less than the lower critical value or greater than the upper critical value.

Can I use a custom significance level?

Yes. Select "Custom…" from the significance level dropdown. Then type any value between 0.001 and 0.5. This lets you use non-standard α values for specialized tests.

When should I use the Chi-Square distribution instead of the Z or t distribution?

Use the Chi-Square distribution when you are testing a claim about a population variance, doing a goodness-of-fit test, or testing for independence between two categorical variables. Use Z or t when you are testing a claim about a population mean.

When should I use the F distribution?

Use the F distribution when you are comparing two population variances or running an ANOVA test to compare means across three or more groups. The F distribution requires two degrees of freedom values: one for the numerator (d₁) and one for the denominator (d₂).

Why is my Chi-Square critical value always positive?

The Chi-Square distribution only takes positive values. It starts at zero and extends to the right. Unlike the Z or t distribution, it is not symmetric and never goes below zero.

What is the relationship between confidence level and significance level?

The confidence level equals 1 minus the significance level. So if α = 0.05, the confidence level is 95%. If α = 0.01, the confidence level is 99%. A higher confidence level means a stricter test with a larger critical value.

How does sample size affect the critical value?

A larger sample size gives you more degrees of freedom, which generally makes the critical value smaller (closer to the Z value). For example, a t critical value with df = 5 is larger than one with df = 100. As degrees of freedom increase, the t distribution approaches the standard normal (Z) distribution.

What does the Pearson R critical value tell me?

The Pearson R critical value tells you the minimum absolute correlation needed for statistical significance. If the correlation coefficient you calculated from your data is larger in absolute value than the R critical value, the correlation is statistically significant at your chosen α level.

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