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Critical T-Values Reference Table
Critical values for two-tailed tests at common significance levels. If |t| exceeds the critical value, reject the null hypothesis.
| df | α = 0.10 | α = 0.05 | α = 0.01 | α = 0.001 |
|---|---|---|---|---|
| 5 | 2.015 | 2.571 | 4.032 | 6.869 |
| 10 | 1.812 | 2.228 | 3.169 | 4.587 |
| 20 | 1.725 | 2.086 | 2.845 | 3.850 |
| 30 | 1.697 | 2.042 | 2.750 | 3.646 |
| 60 | 1.671 | 2.000 | 2.660 | 3.460 |
| ∞ | 1.645 | 1.960 | 2.576 | 3.291 |
Frequently Asked Questions
The t-distribution is used for hypothesis testing and confidence intervals when the sample size is small (typically n < 30) or when the population standard deviation is unknown. It has heavier tails than the normal distribution, accounting for extra uncertainty.
Degrees of freedom (df) determine the shape of the t-distribution. With lower df, the distribution has heavier tails. As df increases toward infinity, the t-distribution approaches the standard normal distribution (z-distribution).
A one-tailed test checks if the mean is specifically greater than or less than a value. A two-tailed test checks if the mean is simply different (either direction). Two-tailed tests are more conservative and most commonly used in research.
Use a t-test when your sample size is small (n < 30) or when you don't know the population standard deviation. Use a z-test when n ≥ 30 and the population standard deviation is known. For most practical purposes, t-tests are preferred.
For a two-tailed 95% confidence interval (α = 0.05), the critical t-value depends on degrees of freedom. For df = 10 it is ±2.228; df = 20 it is ±2.086; df = 30 it is ±2.042; and for large samples it approaches ±1.96.
A critical value is the t-value that defines the boundary of the rejection region. If your calculated t-statistic exceeds the critical value in absolute terms, you reject the null hypothesis. It depends on your significance level and degrees of freedom.