Critical Value Calculator

Critical Value Calculator 2026 online. Z, t, χ² and F for the selected significance level

This calculator allows you to calculate critical values ​​for the most frequently used distributions in statistics in just a few seconds. Supports standardized normal (Z), Student's t, chi-square (χ²) and F-Snedecor distributions. You choose the significance level α, the type of test (one-tailed or two-sided) and degrees of freedom, and the tool immediately calculates decision limits. Thanks to this, you do not have to look through paper statistical tables, and you have the result at hand in a clear and practical form.

What the calculator calculates and what it is used for

The tool answers a simple but crucial question: at what value of the test statistic the null hypothesis should be rejected. The critical value is the limit that you compare to the calculated test value. If the test exceeds this limit, you conclude that the result is not a coincidence. The calculator is used in:

• mean significance tests when you use the Z or t distribution,
• tests of variance and compliance with the theoretical distribution (χ²),
• comparisons of two variances in F analysis.

It will be used by students, researchers, analysts, academics, as well as people preparing reports. It is especially helpful in learning statistics when you want to quickly check the result of a task, and in practical work when time is of the essence.

Input data and how to enter it

The calculator form requires some information:

1. Distribution type- choose whether you are interested in Z, t, χ², or F. Each corresponds to a different type of test.
2. Test type- bilateral, right-sided or left-sided. The choice depends on your alternative hypothesis.
3. Level of significance α- most often 0.05 (5%), but you can enter any value between 0 and 1.
4. Degrees of freedom- in the case of t and χ² distributions, you enter one integer. For F, you enter two numbers: df1 and df2.

When you click theCalculatebutton, the calculator immediately returns the critical value (one or two, depending on the variant). The result is clear and ready to use in your analysis.

How it works in practice

Instead of using statistical tables, the calculator calculates the quantile of the distribution based on the given α level. For a two-sided test, the Z-distribution at α = 0.05 will return ±1.96. This means that if the calculated test statistic is less than -1.96 or greater than +1.96, you reject the null hypothesis. For one-sided tests, there is only one boundary, on the left or right side of the axis.

Step-by-step examples

Large sample mean, Z distribution, α = 0.05

Select Z, set to two-sided, enter α = 0.05. Result: ±1.96. If your Z_obl = 2.1, it exceeds 1.96, so you reject H₀.

Small sample, t distribution, df = 10, α = 0.05

Select t, two-tailed, α = 0.05, df = 10. Result: ±2.228. If the calculated t = 1.9, you do not reject H₀ because it does not exceed the critical value.

Variance test, χ², df = 20, α = 0.05

Set χ², two-sided, α = 0.05, df = 20. You will get an interval around [8.91, 34.17]. If your χ²_obl = 40, it exceeds the upper limit, so you reject H₀.

Comparison of two variances, F-Snedecor, df1 = 8, df2 = 10, α = 0.05

Select F, two-sided, α = 0.05, df1 = 8, df2 = 10. The calculator will give the lower and upper limits. You compare your result with this range. If F_obl falls outside, you reject H₀.

Indicative table of critical values

This table contains just a few of the most commonly used examples. The calculator calculates exact values ​​for any given parameters.

Why use a calculator

• You save time - you don't have to look through tables.
• You have greater precision - the calculator calculates exactly for the given α and df values.
• Results are readable and report-ready.
• You can quickly check several scenarios by changing α or df with one click.

Frequently asked questions

What is the difference between a two-sided test and a one-sided test

A two-sided test checks whether the result differs in either direction (smaller or larger). One-sided examines only one direction - e.g. whether the result is greater than the expected value.

Which α to choose

The most frequently used is 0.05. This means a 5% risk of committing a type I error. In more rigorous studies, α = 0.01 is chosen.

Are degrees of freedom always necessary

For Z, no. For t and χ² you enter the number df. For F you enter two numbers (df1 and df2). This is a parameter related to the sample size and data structure.

Are the calculator results sufficient to report

Yes. You receive accurate critical values ​​that you can directly transfer to a report or scientific work.

Most common errors

• Confusing α with α/2 in a two-sided test. Remember: the calculator does it for you.
• Incorrect distribution entered. Consider which test you are using and choose the appropriate one.
• Forgetting df at t and χ². Without this, the result will not be correct.

Safety and responsible use

Calculate your critical value now