How to Use the Critical Value Calculator
This critical value calculator finds the cutoff values for five common statistical distributions (Z, T, F, Chi-square, and r). It calculates the two-tailed, right-tailed, and left-tailed critical values simultaneously, so you do not need to run the calculator separately for each tail.
To use the calculator:
- Select the distribution. Choose Z, t, chi-square, F, or Pearson correlation depending on the statistical test you are performing.
- Enter the significance level. For example, enter 0.05 if you are performing a test at the 5% significance level.
- Enter any additional parameters. The t and chi-square distributions require degrees of freedom. The F distribution requires numerator and denominator degrees of freedom, while Pearson correlation requires the sample size.
- Choose the number of decimal places. This determines how many decimal places are shown in the answer.
- Click Calculate.
The calculator instantly returns the two-tailed, right-tailed, and left-tailed critical values together with graphs showing the corresponding critical regions.
The shaded area on each graph represents the region beyond the critical value. This makes it easier to see where your test statistic must fall to be considered statistically significant.
What Is a Critical Value?
A critical value is a cutoff point on a statistical distribution that separates the rejection region from the rest of the distribution.
Suppose you perform a hypothesis test at α = 0.05. The critical value identifies the point beyond which a test statistic is sufficiently extreme to reject the null hypothesis.
For a right-tailed test, the rejection region lies above the critical value. For a left-tailed test, it lies below the critical value. A two-tailed test has rejection regions at both ends of the distribution.
For example, the two-tailed Z critical values at α = 0.05 are approximately −1.96 and 1.96. Therefore, a Z statistic below −1.96 or above 1.96 falls within the rejection region.
The exact critical value depends on the distribution, significance level, tail direction, and degrees of freedom where applicable. This is why there is no single critical value that applies to every hypothesis test.
Which Critical Value Distribution Should You Use?
The distribution you select should match the statistical test you are performing. The table below shows the distributions available in this calculator, the additional information required, and some common situations where each is used.
| Distribution | Required Information | Common Uses |
|---|---|---|
| Z | Significance level | Z-tests and procedures based on the standard normal distribution |
| t | Significance level and degrees of freedom | One-sample t test, independent samples t test, paired t test, and regression t-tests |
| Chi-square (χ²) | Significance level and degrees of freedom | Chi-square tests of independence, goodness-of-fit tests, and some variance procedures |
| F | Significance level, numerator df, and denominator df | ANOVA, regression F-tests, and comparisons involving variances |
| Correlation (r) | Significance level and sample size | Testing whether a Pearson correlation is statistically significant |
If you already know which distribution you need and want to see the calculation in more detail, use the following corresponding dedicated calculator.
- Z Critical Value Calculator
- T Critical Value Calculator
- Chi-Square Critical Value Calculator
- F Critical Value Calculator
- Correlation Critical Value Calculator
Understanding the Critical Value Results
The calculator reports three sets of critical values because the correct cutoff depends on the direction of the alternative hypothesis. Each of these critical values is discussed below.
1. Two-Tailed Critical Value
A two-tailed test has a rejection region at both ends of the distribution. The significance level is divided equally between the two tails.
For example, when α = 0.05, each tail contains an area of 0.025. For symmetric distributions such as Z and t, the lower and upper critical values have the same magnitude but opposite signs. This is why the two-tailed Z critical value at α = 0.05 can be written as ±1.96.
Chi-square and F distributions are not symmetric. As such, their lower and upper critical values are different.
2. Right-Tailed Critical Value
A right-tailed test places the entire significance level in the upper tail of the distribution. You reject the null hypothesis when the calculated test statistic is greater than the right-tailed critical value.
Right-tailed critical values are commonly used with procedures such as ANOVA F-tests, chi-square tests of independence, or chi-square goodness-of-fit.
3. Left-Tailed Critical Value
A left-tailed test places the entire significance level in the lower tail. You reject the null hypothesis when the test statistic is smaller than the left-tailed critical value.
For Z and t distributions, the left-tailed critical value is usually negative when α is below 0.50. However, Chi-square and F distributions only take nonnegative values.
How to Choose Between Two-Tailed, Right-Tailed, and Left-Tailed Critical Values
The critical value you should use depends on the direction of your alternative hypothesis, H₁. Therefore, first look at what your research question is asking and how the alternative hypothesis is written.
- Two-tailed critical value. Use this when you want to determine whether a population parameter is different from a specified value in either direction. For example, H₁: μ ≠ μ₀.
- Right-tailed critical value. Use this when you want to determine whether a population parameter is greater than a specified value. For example, H₁: μ > μ₀.
- Left-tailed critical value. Use this when you want to determine whether a population parameter is smaller than a specified value. For example, H₁: μ < μ₀.
The table below summarizes how the form of the alternative hypothesis determines the type of test and the critical value you should use.
| Alternative Hypothesis | What You Are Testing | Test Type | Critical Value to Use |
|---|---|---|---|
| H₁: μ ≠ μ₀ | Whether the population mean is different from μ₀ | Two-tailed | Two-tailed critical values |
| H₁: μ > μ₀ | Whether the population mean is greater than μ₀ | Right-tailed | Right-tailed critical value |
| H₁: μ < μ₀ | Whether the population mean is smaller than μ₀ | Left-tailed | Left-tailed critical value |
The direction of the test should be decided from the research question before examining the sample results. You should not choose a right-tailed or left-tailed test simply because the sample result happens to fall in that direction.
Our Critical Value Calculator displays all three possibilities at the same time. Once you identify the alternative hypothesis for your test, use the critical value that corresponds to the appropriate tail.
Critical Values and the Significance Level
The significance level, α, determines how much of the probability distribution is placed in the rejection region. Common significance levels include 0.10, 0.05, and 0.01. A smaller α requires stronger evidence before the null hypothesis can be rejected.
For a one-tailed test (left or right-tailed), the full significance level is placed in one tail. Therefore, if α = 0.05, the critical region contains 5% of the distribution. However, for a two-tailed test, α is divided equally between the two tails. If α = 0.05, each tail contains 0.025.
As the significance level becomes smaller, the critical boundary generally moves farther into the tail. This makes it more difficult for a test statistic to enter the rejection region.
Critical Values and Degrees of Freedom
For some distributions, the critical value depends on both the significance level and the degrees of freedom (df). This is because changing the degrees of freedom changes the shape of the distribution and, therefore, the location of the critical value.
The t distribution, for example, requires one degrees-of-freedom value. The way you calculate df depends on the statistical test you are performing. For a one-sample t-test, df = n − 1, while other t-tests may use a different formula.
The chi-square distribution also requires degrees of freedom. Again, the correct df depends on the procedure. For example, a chi-square test of independence calculates degrees of freedom from the number of rows and columns in the contingency table.
For the F distribution, two degrees-of-freedom values are required: the numerator degrees of freedom, df₁, and the denominator degrees of freedom, df₂. Both values are needed because an F distribution is determined by these two parameters.
Pearson correlation critical values also depend on degrees of freedom. For a correlation based on n paired observations, df = n − 2. In this calculator, you only need to enter the sample size, and the corresponding degrees of freedom are determined automatically.
The Z distribution is different because it does not depend on degrees of freedom. Therefore, when finding a Z critical value, you only need to enter the significance level.
If you need to determine the appropriate degrees of freedom before finding a critical value, use our Degrees of Freedom Calculator.
How to Use a Critical Value in Hypothesis Testing
Once you have found the critical value, compare it with the test statistic calculated from your sample. The decision rule depends on the direction of the test:
- Right-tailed test. Reject H₀ when the test statistic is greater than the right-tailed critical value.
- Left-tailed test. Reject H₀ when the test statistic is less than the left-tailed critical value.
- Two-tailed test. Reject H₀ when the test statistic falls below the lower critical value or above the upper critical value.
In other words, you reject the null hypothesis when the test statistic falls within the rejection region.
Tip: For symmetric two-tailed distributions such as the Z and t distributions, you can simplify the decision by comparing absolute values. Reject H₀ if the absolute value of the test statistic is greater than the absolute value of the critical value. For example, if the critical values are ±1.96, reject H₀ when |z| > 1.96.
For example, suppose a two-tailed Z-test at α = 0.05 gives a test statistic of z = 2.30. The critical values are approximately −1.96 and 1.96. Since |2.30| > 1.96, the test statistic falls in the rejection region. Therefore, we reject H₀.
The graphs produced by this calculator show the same decision visually. The shaded areas represent the rejection regions, while the critical values mark the boundaries between the rejection and non-rejection regions.
Critical Value vs P-Value
The critical-value method and the p-value method are two ways of making the same hypothesis-testing decision.
With the critical-value approach, we compare the calculated test statistic with the appropriate critical value. If the test statistic falls in the rejection region, we reject H₀. However, with the p-value approach, we compare the p-value with the significance level. If p < α, we reject H₀.
For example, a test statistic that falls beyond the critical boundary should also produce a p-value below the corresponding significance level, provided both calculations use the same test and alternative hypothesis.
If you already have a test statistic and want to calculate its probability instead of finding a cutoff, use our P-Value Calculator.
Critical Value Calculator vs Statistical Tables
Critical values can also be obtained from statistical distribution tables. Tables are useful when you want to look up common significance levels and degrees of freedom manually.
However, printed tables usually contain only selected values. A calculator is more convenient when you have an unusual significance level, decimal degrees of freedom, or simply want the result without searching through rows and columns.
StatCalc.net also provides interactive statistical tables if you prefer the traditional lookup method. You can use the Z Table, T-Table, Chi-Square Table, F Distribution Table, or the Correlation critical value table.
The calculator and tables should return the same critical values when the same distribution parameters are used, although table values may differ slightly because of rounding.
Frequently Asked Questions
A critical value calculator is an online tool that finds the cutoff point on a statistical distribution for a chosen significance level. This calculator supports Z, t, chi-square, F, and Pearson correlation critical values and displays the two-tailed, right-tailed, and left-tailed results.
No. A critical value is a cutoff on the distribution used to define the rejection region. On the other hand, a p-value measures how extreme the observed result is under the null hypothesis. Both can be used to make the same hypothesis-testing decision.
Yes. A two-tailed test has a lower and an upper critical value because there is a rejection region in both tails. For symmetric distributions such as Z and t, these values have equal magnitudes and opposite signs.
It depends on the distribution. Z critical values do not require degrees of freedom. The t and chi-square distributions require df, while the F distribution requires both numerator and denominator degrees of freedom. Pearson correlation critical values can be determined from the sample size, where df = n − 2.
The chi-square and F distributions are right-skewed and only take nonnegative values. As a result, their lower and upper critical values are not mirror images of each other.
