Student's t-Distribution Calculator

Critical T-Value Calculator

Calculate the critical t-value for one-tailed or two-tailed hypothesis tests using degrees of freedom and a significance level. Enter your values below to get an instant result using the Student's t-distribution.

Enter Your Values

Degrees of Freedom (df) Usually n − 1

Significance Level (α) Between 0 and 0.5

Test Type

What Is a Critical T-Value?

A critical t-value is a cutoff point from the Student's t-distribution. It helps determine whether a calculated test statistic is far enough from the expected value to be considered statistically significant.

The critical value depends on the significance level, degrees of freedom, and whether the statistical test is one-tailed or two-tailed. Lower significance levels normally produce larger absolute critical values.

When Is the T-Distribution Used?

The Student's t-distribution is commonly used when the population standard deviation is unknown, particularly when working with sample means. It is especially useful for smaller samples.

As the number of degrees of freedom increases, the t-distribution becomes progressively closer to the standard normal distribution.

How the Critical T-Value Calculator Works

This calculator determines the appropriate percentile of the Student's t-distribution based on the information you provide.

Enter Degrees of Freedom

Enter the degrees of freedom for your statistical test. For a basic one-sample t-test, this is commonly calculated as n − 1.

Select Alpha and Test Type

Choose your significance level and select a two-tailed, right-tailed, or left-tailed hypothesis test.

Calculate the T-Value

The calculator evaluates the Student's t-distribution and displays the appropriate critical value instantly.

Example: For df = 10 and α = 0.05 with a two-tailed test, the critical t-values are approximately −2.2281 and +2.2281.

One-Tailed vs Two-Tailed T-Values

The probability used to calculate the critical value depends on where the rejection region appears in your hypothesis test.

Test Type	Probability Used	Rejection Region	Output
Two-Tailed	1 − α/2	Both tails	±t*
Right-Tailed	1 − α	Right tail	+t*
Left-Tailed	α	Left tail	−t*

Critical T-Value Formula

A critical t-value is obtained from the quantile of the Student's t-distribution rather than from one simple arithmetic equation.

Two-Tailed Test

For a two-tailed hypothesis test, alpha is divided equally between the two tails: t* = t_{1−α/2, df}

Right-Tailed Test

For a right-tailed hypothesis test: t* = t_{1−α, df}

Left-Tailed Test

For a left-tailed hypothesis test: t* = t_{α, df}

Frequently Asked Questions

The exact critical t-value depends on the degrees of freedom. For a two-tailed test with α = 0.05, the calculator evaluates the 97.5th percentile because 2.5% of the probability is placed in each tail.

The calculation depends on your statistical procedure. For a basic one-sample t-test, degrees of freedom are generally calculated using df = n − 1, where n is the sample size.

Alpha, written as α, represents the significance level selected for a hypothesis test. Common values include 0.10, 0.05, 0.025, and 0.01.

Yes. A left-tailed hypothesis test normally has a negative critical t-value. A two-tailed test includes both a negative and positive critical value.

A two-tailed test divides the total alpha between two tails. This moves the cutoff points farther from the center of the distribution, normally creating a larger absolute critical value.

Generally, yes. At a fixed significance level, the absolute critical t-value normally decreases as degrees of freedom increase and gradually approaches the corresponding normal-distribution value.