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). You can find Z values directly using our Z Score Calculator or explore the full curve with the Normal Distribution Calculator.
- 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. If you need to run a full t test, try our t Test Calculator.
- χ² (Chi-Square): Used for tests about variance or for goodness-of-fit and independence tests. It only takes positive values. Our Chi Square Calculator can help you perform the complete test.
- 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. For full analysis of variance testing, see our ANOVA Calculator.
- 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. You can compute the correlation coefficient itself with our Correlation Coefficient Calculator.
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. Use our p Value Calculator to find the exact probability associated with your result, or build a Confidence Interval around your estimate. If you need to figure out how many observations to collect before running your test, our Sample Size Calculator can help you plan ahead.
For related statistical work, you may also find these tools useful: the Standard Deviation Calculator for measuring spread, the Mean Median Mode Calculator for central tendency, the IQR Calculator for identifying outliers, the Linear Regression Calculator for modeling relationships, the Effect Size Calculator for quantifying practical significance, and the Binomial Distribution Calculator for discrete probability problems.