t-Distribution Table (Student's t-Table)

What Is a Critical t-Value?

The critical t-value is the threshold for your test. If your calculated |t-statistic| exceeds the critical value from the table, you reject the null hypothesis at that significance level.

t-Table vs. Z-Table

Use the t-table when the population σ is unknown and n < 30. Use the z-table when σ is known or n ≥ 30. As df → ∞, t-critical values converge to z-critical values.

Degrees of Freedom (df)

For a one-sample t-test: df = n − 1. For a two-sample t-test: df = n₁ + n₂ − 2. For a paired t-test: df = n − 1 pairs. If your df isn't listed, round down.

Compute t = (x̄ − μ₀) / (s / √n) from your sample mean, hypothesized mean, sample SD, and sample size.

For a one-sample t-test: df = n − 1. For two independent samples: df = n₁ + n₂ − 2.

Select your significance level (typically α = 0.05). Use two-tailed for non-directional hypotheses, one-tailed for directional ones.

Find the row for your df and the column for your α. The intersection is the critical t-value. Example: df = 20, α = 0.05 two-tailed → t = 2.086.

If |t-calculated| > t-critical, reject H₀ — the result is statistically significant. Otherwise, fail to reject H₀.

What is the t critical value for df = 20 at α = 0.05 two-tailed?

The critical t-value for df = 20 at α = 0.05 (two-tailed) is 2.086. Use both +2.086 and −2.086 as rejection thresholds in a two-sided test.

When should I use the t-table instead of the z-table?

Use the t-table when the population standard deviation (σ) is unknown. As a rule of thumb, the t-table is preferred for small samples (n < 30). For large samples with unknown σ, the t-distribution is still technically correct — it just approaches the z-distribution.

What if my degrees of freedom aren't listed in the table?

Round down to the nearest df listed in the table. This is the conservative approach — a smaller df gives a larger critical value, making it slightly harder to reject H₀ and reducing Type I error risk.

Is the t-distribution symmetric?

Yes. The t-distribution is bell-shaped and symmetric around 0, similar to the standard normal distribution. It has heavier tails than the normal distribution, especially at low degrees of freedom, which is why its critical values are larger.

Who invented the t-distribution?

The t-distribution was developed by English statistician William Sealy Gosset in 1908 while working at the Guinness Brewery. Because his employer restricted publication, he published under the pseudonym "Student" — giving rise to the name Student's t-distribution.

What does the ∞ row in the t-table represent?

The ∞ (infinity) row shows the z-distribution critical values. As degrees of freedom increase, the t-distribution converges to the standard normal distribution. For example, at df = ∞, α = 0.05 two-tailed, the critical value is 1.960 — exactly the z-score for 95% confidence.