t-Distribution Calculator
Calculate probabilities, critical values, and p-values for the Student's t-distribution. The t-distribution is used in hypothesis testing, confidence intervals, and when working with small sample sizes. This calculator supports both one-tailed and two-tailed tests with customizable degrees of freedom.
What Is the t-Distribution?
The Student's t-distribution is a probability distribution used in statistics when the sample size is small and the population standard deviation is unknown. It is similar to the normal distribution but has heavier tails, meaning it is more prone to producing values far from the mean.
The t-distribution is defined by its degrees of freedom (df). As df increases, the t-distribution approaches the standard normal distribution.
Common Applications
- Hypothesis Testing: t-tests for comparing means.
- Confidence Intervals: Estimating population means with small samples.
- Regression Analysis: Testing the significance of coefficients.
- Quality Control: Assessing process variability.
How the Calculator Works
This calculator uses numerical integration and approximation methods to compute the cumulative distribution function (CDF) and inverse CDF of the t-distribution. The results are accurate to the specified decimal precision.
Why Use This t-Distribution Calculator?
- Two Modes: Find probabilities from t-values, or find critical t-values from probabilities.
- One-Tailed & Two-Tailed: Supports left-tailed, right-tailed, and two-tailed tests.
- Visual Curve: See the t-distribution curve with the shaded area representing your result.
- Critical Values Table: Quick reference for common confidence levels.
- Free & Private: No registration, no data storage.
❓ t-Distribution Calculator FAQ
What is the t-distribution?
The t-distribution is a probability distribution used when estimating the mean of a normally distributed population with unknown standard deviation and small sample size.
What are degrees of freedom (df)?
Degrees of freedom represent the number of independent observations in a sample. For a t-test, df = n − 1, where n is the sample size. As df increases, the t-distribution approaches the normal distribution.
What is the difference between one-tailed and two-tailed tests?
A one-tailed test (left or right) checks for an effect in one direction. A two-tailed test checks for an effect in either direction (both positive and negative).
When should I use a t-distribution instead of a normal distribution?
Use the t-distribution when the sample size is small (typically n < 30) or when the population standard deviation is unknown.
What is a critical t-value?
A critical t-value is the value that separates the rejection region from the non-rejection region in a hypothesis test. It depends on the significance level (α) and the degrees of freedom.
How is the t-distribution related to the normal distribution?
As degrees of freedom increase, the t-distribution approaches the standard normal distribution. For df ≥ 30, the difference is negligible for most practical purposes.
What is the difference between a t-test and a z-test?
A t-test is used when the population standard deviation is unknown and the sample size is small. A z-test is used when the population standard deviation is known or the sample size is large.
Is this calculator free?
Yes, this calculator is completely free to use. No registration or personal data storage is required. All calculations are performed in your browser.