T Distribution Calculator – Student's T-Test
Calculate T-distribution probabilities and critical values for hypothesis testing
How to Use
- Enter your t-value (test statistic)
- Enter the degrees of freedom (n-1 for single sample)
- Click calculate to see probabilities and critical values
- Review one-tailed and two-tailed probabilities
What is the T-Distribution?
The Student's t-distribution (or simply t-distribution) is a probability distribution used in hypothesis testing when the sample size is small and the population standard deviation is unknown. It was developed by William Sealy Gosset under the pseudonym 'Student' in 1908.
The t-distribution is similar to the normal distribution but has heavier tails, meaning it predicts more extreme values. As the sample size increases (degrees of freedom increase), the t-distribution approaches the standard normal distribution.
When to Use the T-Distribution
Use the t-distribution in these situations:
- Small sample sizes (typically n < 30)
- Population standard deviation is unknown
- Testing hypotheses about population means
- Constructing confidence intervals for means
- Comparing means between two groups (t-tests)
- Regression analysis with small samples
Degrees of Freedom
Degrees of freedom (df) determine the shape of the t-distribution. The formula depends on your test:
- One-sample t-test: df = n - 1
- Two-sample t-test (equal variances): df = n₁ + n₂ - 2
- Two-sample t-test (unequal variances): Use Welch's formula
- Paired t-test: df = n - 1 (number of pairs)
Higher degrees of freedom result in a distribution closer to the normal distribution.
Interpreting Results
Understanding your t-distribution results:
- One-tailed probability: Used for directional hypotheses (greater than or less than)
- Two-tailed probability: Used for non-directional hypotheses (different from)
- Critical values: Thresholds for rejecting the null hypothesis
- If |t-value| > critical value, reject the null hypothesis
- Lower p-value (probability) indicates stronger evidence against the null hypothesis
Common Confidence Levels
| Confidence Level | Significance Level (α) | Use Case |
|---|---|---|
| 90% | 0.10 | Preliminary or exploratory studies |
| 95% | 0.05 | Standard for most scientific research |
| 99% | 0.01 | High-stakes decisions requiring strong evidence |
Frequently Asked Questions
- What is the difference between t-distribution and normal distribution?
- The t-distribution has heavier tails than the normal distribution, accounting for the additional uncertainty when estimating population parameters from small samples. As sample size increases, the t-distribution approaches the normal distribution.
- How do I calculate degrees of freedom?
- For a one-sample t-test, degrees of freedom = n - 1, where n is your sample size. For a two-sample t-test with equal variances, df = n₁ + n₂ - 2. For paired samples, df = number of pairs - 1.
- When should I use one-tailed vs two-tailed tests?
- Use a one-tailed test when you have a directional hypothesis (e.g., mean is greater than a value). Use a two-tailed test when testing if a mean is simply different from a value, without specifying direction. Two-tailed tests are more conservative and commonly used.
- What sample size is considered 'small' for using the t-distribution?
- Generally, samples with n < 30 are considered small and benefit from using the t-distribution. However, the t-distribution is appropriate for any sample size when the population standard deviation is unknown. For very large samples (n > 100), t and z distributions are nearly identical.