Interactive t-Table Calculator
Find Critical t-Value
Student’s t-Distribution Table
The t-table provides critical values for the t-distribution, used when the population standard deviation is unknown and the sample size is small.
How to Use This Table
- Calculate degrees of freedom: df = n - 1 (for one-sample) or df = n₁ + n₂ - 2 (for two-sample)
- Choose your significance level (α) and whether the test is one-tailed or two-tailed
- Find the intersection of df row and α column
Critical Values (Upper Tail)
This table gives t-values where P(T > t) = α (area in the upper tail).
| df | α = 0.10 | α = 0.05 | α = 0.025 | α = 0.01 | α = 0.005 | α = 0.001 |
|---|---|---|---|---|---|---|
| 1 | 3.078 | 6.314 | 12.706 | 31.821 | 63.657 | 318.309 |
| 2 | 1.886 | 2.920 | 4.303 | 6.965 | 9.925 | 22.327 |
| 3 | 1.638 | 2.353 | 3.182 | 4.541 | 5.841 | 10.215 |
| 4 | 1.533 | 2.132 | 2.776 | 3.747 | 4.604 | 7.173 |
| 5 | 1.476 | 2.015 | 2.571 | 3.365 | 4.032 | 5.893 |
| 6 | 1.440 | 1.943 | 2.447 | 3.143 | 3.707 | 5.208 |
| 7 | 1.415 | 1.895 | 2.365 | 2.998 | 3.499 | 4.785 |
| 8 | 1.397 | 1.860 | 2.306 | 2.896 | 3.355 | 4.501 |
| 9 | 1.383 | 1.833 | 2.262 | 2.821 | 3.250 | 4.297 |
| 10 | 1.372 | 1.812 | 2.228 | 2.764 | 3.169 | 4.144 |
| 11 | 1.363 | 1.796 | 2.201 | 2.718 | 3.106 | 4.025 |
| 12 | 1.356 | 1.782 | 2.179 | 2.681 | 3.055 | 3.930 |
| 13 | 1.350 | 1.771 | 2.160 | 2.650 | 3.012 | 3.852 |
| 14 | 1.345 | 1.761 | 2.145 | 2.624 | 2.977 | 3.787 |
| 15 | 1.341 | 1.753 | 2.131 | 2.602 | 2.947 | 3.733 |
| 16 | 1.337 | 1.746 | 2.120 | 2.583 | 2.921 | 3.686 |
| 17 | 1.333 | 1.740 | 2.110 | 2.567 | 2.898 | 3.646 |
| 18 | 1.330 | 1.734 | 2.101 | 2.552 | 2.878 | 3.610 |
| 19 | 1.328 | 1.729 | 2.093 | 2.539 | 2.861 | 3.579 |
| 20 | 1.325 | 1.725 | 2.086 | 2.528 | 2.845 | 3.552 |
| 21 | 1.323 | 1.721 | 2.080 | 2.518 | 2.831 | 3.527 |
| 22 | 1.321 | 1.717 | 2.074 | 2.508 | 2.819 | 3.505 |
| 23 | 1.319 | 1.714 | 2.069 | 2.500 | 2.807 | 3.485 |
| 24 | 1.318 | 1.711 | 2.064 | 2.492 | 2.797 | 3.467 |
| 25 | 1.316 | 1.708 | 2.060 | 2.485 | 2.787 | 3.450 |
| 26 | 1.315 | 1.706 | 2.056 | 2.479 | 2.779 | 3.435 |
| 27 | 1.314 | 1.703 | 2.052 | 2.473 | 2.771 | 3.421 |
| 28 | 1.313 | 1.701 | 2.048 | 2.467 | 2.763 | 3.408 |
| 29 | 1.311 | 1.699 | 2.045 | 2.462 | 2.756 | 3.396 |
| 30 | 1.310 | 1.697 | 2.042 | 2.457 | 2.750 | 3.385 |
| 35 | 1.306 | 1.690 | 2.030 | 2.438 | 2.724 | 3.340 |
| 40 | 1.303 | 1.684 | 2.021 | 2.423 | 2.704 | 3.307 |
| 45 | 1.301 | 1.679 | 2.014 | 2.412 | 2.690 | 3.281 |
| 50 | 1.299 | 1.676 | 2.009 | 2.403 | 2.678 | 3.261 |
| 60 | 1.296 | 1.671 | 2.000 | 2.390 | 2.660 | 3.232 |
| 70 | 1.294 | 1.667 | 1.994 | 2.381 | 2.648 | 3.211 |
| 80 | 1.292 | 1.664 | 1.990 | 2.374 | 2.639 | 3.195 |
| 90 | 1.291 | 1.662 | 1.987 | 2.369 | 2.632 | 3.183 |
| 100 | 1.290 | 1.660 | 1.984 | 2.364 | 2.626 | 3.174 |
| 120 | 1.289 | 1.658 | 1.980 | 2.358 | 2.617 | 3.160 |
| ∞ | 1.282 | 1.645 | 1.960 | 2.326 | 2.576 | 3.090 |
Common Confidence Interval Values
For two-tailed tests (most common for confidence intervals):
| Confidence Level | α (two-tailed) | Use column |
|---|---|---|
| 80% | 0.20 | α = 0.10 |
| 90% | 0.10 | α = 0.05 |
| 95% | 0.05 | α = 0.025 |
| 98% | 0.02 | α = 0.01 |
| 99% | 0.01 | α = 0.005 |
Two-Tailed Critical Values
For two-tailed tests, use these columns:
| df | 80% CI | 90% CI | 95% CI | 98% CI | 99% CI |
|---|---|---|---|---|---|
| 5 | 1.476 | 2.015 | 2.571 | 3.365 | 4.032 |
| 10 | 1.372 | 1.812 | 2.228 | 2.764 | 3.169 |
| 15 | 1.341 | 1.753 | 2.131 | 2.602 | 2.947 |
| 20 | 1.325 | 1.725 | 2.086 | 2.528 | 2.845 |
| 25 | 1.316 | 1.708 | 2.060 | 2.485 | 2.787 |
| 30 | 1.310 | 1.697 | 2.042 | 2.457 | 2.750 |
| ∞ | 1.282 | 1.645 | 1.960 | 2.326 | 2.576 |
When to Use t vs z
| Use t-distribution when: | Use z-distribution when: |
|---|---|
| σ (population SD) is unknown | σ (population SD) is known |
| Sample size is small (n < 30) | Sample size is large (n ≥ 30) |
| Using sample SD (s) | Or for proportions |
Degrees of Freedom
| Test Type | Degrees of Freedom |
|---|---|
| One-sample t-test | df = n - 1 |
| Paired t-test | df = n - 1 (n = number of pairs) |
| Two-sample t-test (pooled) | df = n₁ + n₂ - 2 |
| Two-sample t-test (Welch’s) | Complex formula (software calculated) |
Example: Using the t-Table
Problem: A sample of 16 students has a mean test score of 75. Test if this differs from the population mean of 70 at α = 0.05 (two-tailed).
Solution:
- df = n - 1 = 16 - 1 = 15
- Two-tailed test at α = 0.05, use column α = 0.025
- Critical value: t = ±2.131
- If |calculated t| > 2.131, reject H₀
Step-by-Step: How to Find a t-Value
Here’s a complete walkthrough for finding critical t-values:
Step 1: Identify Your Test Type
- One-sample t-test: Comparing a sample mean to a known value
- Paired t-test: Comparing two related samples (before/after)
- Two-sample t-test: Comparing two independent group means
Step 2: Calculate Degrees of Freedom
For most t-tests: df = n - 1 where n is your sample size.
Step 3: Determine One-Tailed or Two-Tailed
- Two-tailed (most common): Testing if values differ in either direction
- One-tailed: Testing if values are greater than OR less than (not both)
Step 4: Find Your Critical Value
- Locate your df in the leftmost column
- Find your α level in the top row
- The intersection is your critical t-value
Most Common t-Values (Quick Reference)
For quick lookups, here are the most frequently needed critical values:
| Description | df | Critical t |
|---|---|---|
| 95% CI, n=10 | 9 | 2.262 |
| 95% CI, n=20 | 19 | 2.093 |
| 95% CI, n=30 | 29 | 2.045 |
| 99% CI, n=10 | 9 | 3.250 |
| 99% CI, n=20 | 19 | 2.861 |
| 99% CI, n=30 | 29 | 2.756 |
Frequently Asked Questions
What is the t-table used for?
The t-table (Student’s t distribution table) is used to find critical values for t-tests and confidence intervals when the population standard deviation is unknown. It’s essential for hypothesis testing with small sample sizes.
How do I read the t-table?
To read the t-table: (1) Find your degrees of freedom (df) in the left column, (2) Find your significance level (α) in the top row, (3) The value where the row and column meet is your critical t-value.
What is the t-value for 95% confidence with df=10?
For a 95% confidence interval with df=10, use α=0.025 (two-tailed). The critical t-value is 2.228.
What is the difference between a t-table and z-table?
The t-table is used when the population standard deviation is unknown and sample size is small (less than 30). The z-table is used when the population standard deviation is known or sample size is large (30 or more).
Why is it called “Student’s t”?
The t-distribution was published by William Sealy Gosset in 1908 under the pseudonym “Student” because his employer (Guinness Brewery) didn’t allow employees to publish under their own names.
How do I find the t-value for a one-tailed test?
For a one-tailed test at α=0.05, use the column labeled α=0.05 directly. For a two-tailed test at α=0.05, use the column labeled α=0.025 (since 0.05÷2 = 0.025).
What if my degrees of freedom isn’t in the table?
If your exact df isn’t listed, use the next smaller df value to be conservative, or interpolate between adjacent values. For df > 120, use the infinity (∞) row.
What is the critical t-value for a 99% confidence interval?
For a 99% CI (two-tailed), use α=0.005. Common values: df=10 → t=3.169, df=20 → t=2.845, df=30 → t=2.750.
Related Resources
- T-Test Calculator - Perform t-tests automatically
- t-Distribution Lesson - Understand when and why to use t
- Z-Table - For large samples or known σ
- Chi-Square Table - For categorical data analysis
- F-Table - For comparing variances and ANOVA
- Confidence Intervals - Building intervals with t