Reference Table
BY: Statistics Fundamentals Team
Reviewed By: Minsa A (Senior Statistics Editor)
Last Updated: August 2026

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

The Student's t-distribution table gives critical t-values for hypothesis testing and confidence intervals. Use it to find the critical value at your chosen significance level (α) and degrees of freedom (df) when the population standard deviation is unknown. Available for both one-tailed and two-tailed t-tests.

↓ Jump to Table ↓ How to Read ↓ Worked Examples ↓ FAQ
📌 Quick Reference: The most common critical values are: df=10, α=0.05 two-tailed → t = 2.228  |  df=20 → t = 2.086  |  df=29, 99% CI → t = 2.756  |  df=∞ → z = 1.960. Use the lookup tool below or click any cell in the table.

Quick t Critical Value Lookup Tool

Enter your degrees of freedom and significance level to instantly find your critical t-value.

t critical =
—

Click any cell to highlight the critical value and auto-fill the lookup tool above. The last row (∞) shows z-distribution values for comparison. Scroll right on mobile to see all columns.

How to Read the t-Distribution Table

Reading a t-table requires three pieces of information: your degrees of freedom (df), your significance level (α), and whether your hypothesis is one-tailed or two-tailed. Here's the step-by-step process:

1

Calculate the t-statistic

Compute your sample t-statistic using the formula below. You'll compare this value against the critical value from the table.

t-Statistic Formula t = (x̄ − μ₀) / (s / √n)
2

Find degrees of freedom (df)

Degrees of freedom depend on your test type:

  • One-sample t-test: df = n − 1
  • Two-sample t-test: df = n₁ + n₂ − 2
  • Paired t-test: df = n − 1 (pairs)

If your df is not listed, round down to the nearest value.

3

Choose α and tail type

Select your significance level (typically α = 0.05). Use the two-tailed column for non-directional hypotheses (≠) and the one-tailed column for directional hypotheses (> or <).

4

Find the critical value

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

5

Make your decision

If |t-calculated| > t-critical → Reject H₀ (statistically significant result).
If |t-calculated| ≤ t-critical → Fail to reject H₀.

Worked Examples

The following examples walk through common scenarios students and researchers encounter when using the t-table.

📘 Example 1 — One-Sample t-Test (Two-Tailed)

A study has 25 participants. We want to test whether the sample mean differs from the hypothesized mean at α = 0.05, two-tailed. What is the critical t-value?

  1. Sample size n = 25, so df = n − 1 = 24
  2. Test type: two-tailed, α = 0.05
  3. Look up df = 24, α = 0.05 column in the two-tailed table
  4. Critical value: t = 2.064

Reject H₀ if |t-statistic| > 2.064.

📗 Example 2 — One-Tailed Test with df = 11

A research study uses a one-tailed test with α = 0.10 and sample size n = 12. What critical t-value should be compared to the t-statistic?

  1. df = n − 1 = 12 − 1 = 11
  2. Test type: one-tailed, α = 0.10
  3. Look up df = 11, α = 0.10 in the one-tailed table
  4. Critical value: t = 1.363

📙 Example 3 — 99% Confidence Interval with df = 29

You want to construct a 99% confidence interval for a mean, and your sample has 30 observations. What is the critical t-value?

  1. df = n − 1 = 30 − 1 = 29
  2. 99% CI corresponds to two-tailed α = 0.01 (α/2 = 0.005 per tail)
  3. Look up df = 29, α = 0.01 in the two-tailed table
  4. Critical value: t = 2.756

Your confidence interval is: x̄ ± 2.756 × (s / √30).

📕 Example 4 — Two-Sample t-Test

Group A has n₁ = 15 observations, Group B has n₂ = 18 observations. Test at α = 0.05, two-tailed.

  1. df = n₁ + n₂ − 2 = 15 + 18 − 2 = 31
  2. df = 31 is not listed; round down to df = 30
  3. Look up df = 30, α = 0.05 two-tailed
  4. Critical value: t = 2.042

Understanding the t-Distribution Table

What Is a Critical t-Value?

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

One-Tailed vs. Two-Tailed

Use a one-tailed test when your hypothesis has a direction (e.g., "Group A is taller than Group B"). Use a two-tailed test when any difference matters (e.g., "Group A and B differ"). If unsure, default to two-tailed.

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 (e.g., t = 1.960 at df = ∞ equals z = 1.96 for 95% CI).

Why Is It Heavier-Tailed?

The t-distribution is bell-shaped and symmetric, but has heavier tails than the normal distribution — especially at low df. This reflects greater uncertainty with small samples, which is why t-critical values are larger than z-critical values for the same α.

t-Table vs. z-Table vs. Chi-Square Table

Knowing which table to use is essential. Here's a quick comparison:

Feature t-Table (Student's t) z-Table (Normal) Chi-Square Table
When to use σ unknown, any n (preferred for n < 30) σ known, or n ≥ 30 Categorical data, goodness-of-fit, independence tests
Shape Bell-shaped, heavier tails Standard normal, thinner tails Right-skewed, one-tailed
Parameter Degrees of freedom (df) None (fixed distribution) Degrees of freedom (df)
Tails One-tailed or two-tailed One-tailed or two-tailed One-tailed (upper tail only)
Critical value at α=0.05 (two-tail, df=30) 2.042 1.960 43.77 (upper tail only)

The t-Statistic Formula

The t-statistic measures how many standard errors your sample mean is from the hypothesized population mean:

One-Sample t-Statistic t = (x̄ − μ₀) / (s / √n)

Where: x̄ = sample mean, μ₀ = hypothesized population mean, s = sample standard deviation, n = sample size. A larger |t| means the sample mean is further from μ₀, making it more likely you'll reject H₀.

Two-Sample t-Statistic (equal variances) t = (x̄₁ − x̄₂) / (sp × √(1/n₁ + 1/n₂))

Where sp is the pooled standard deviation and df = n₁ + n₂ − 2.

Frequently Asked Questions

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. This is one of the most commonly looked-up values in statistics courses.

What is the t critical value for df = 29 at 99% confidence?

For df = 29 at 99% confidence (α = 0.01 two-tailed), the critical t-value is 2.756. This value (t critical value df=29 99% confidence 2.756) is frequently searched and corresponds to α/2 = 0.005 in each tail.

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 as df increases. Only use the z-table when σ is known.

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

Round down to the nearest df listed in the t-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. For example, if df = 31, use df = 30 from the table.

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. As df → ∞, it converges exactly to the standard normal (z-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. At df = ∞ and α = 0.05 two-tailed, the critical value is 1.960 — exactly the z-score for 95% confidence. This row is useful for comparison.

How do I choose between one-tailed and two-tailed tests?

Choose based on your hypothesis: two-tailed tests check if two groups differ in either direction (H₁: μ ≠ μ₀); one-tailed tests check for a difference in only one direction (H₁: μ > μ₀ or μ < μ₀). If you're unsure, use a two-tailed test — it's more conservative and widely accepted.

Who invented the t-distribution?

The t-distribution was developed by English statistician William Sealy Gosset in 1908 while working as Head Brewer at the Guinness Brewery. Because his employer restricted publication, he published under the pseudonym "Student" in the journal Biometrika — giving rise to the name Student's t-distribution. Earlier foundational work was done by Helmert and Lüroth (1876).

What is the t critical value for df = 7, α = 0.10 two-tailed?

For df = 7 at α = 0.10 two-tailed, the critical t-value is 1.895. You can verify this using the lookup tool above or by locating df = 7 and α = 0.10 in the two-tailed table.

What is the t critical value for df = 17, α = 0.05 two-tailed?

For df = 17 at α = 0.05 two-tailed, the critical t-value is 2.110. This is also written as t.inv.2t(0.05, 17) in Excel/spreadsheet notation, or equivalently as the 97.5th percentile of the t-distribution with 17 df.

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