How to Read the t-Distribution Table
We put two headers on the t-table intentionally, not by accident. Degrees of freedom run down the left side. Each column header appears twice: one-tailed α on top, two-tailed α directly below — because the same cell answers a different question depending on whether your hypothesis had a direction before you saw the data.
Start with something concrete. Eleven observations puts you at df = 10. If you need a 95% confidence interval, that is a two-tailed question, and the cell you want is 2.228. The same row answers a one-tailed question too — that answer is 1.812, which is meaningfully different. The table gives you both because the framing of your hypothesis, not the data, decides which number you quote.
When to Use the t-Distribution Instead of z
Real studies almost always use t, and the reason is that z assumes you already know the population standard deviation — a condition that basically never holds. The second σ gets estimated from the sample itself, that estimate adds its own uncertainty to the calculation, and the t-distribution is the version of the math that takes that seriously.
William Gosset worked this out in 1908 while running quality control at the Guinness brewery in Dublin, publishing under the pen name "Student" because his employer treated the method as a trade secret. His problem was exactly the modern one: tiny samples, no known σ, and a need to know how far a small-batch average could stray before it meant something.
What the table shows directly is the penalty for small samples. At df = 4 the 95% critical value is 2.776 — roughly 1.4× the z-value of 1.960. That difference exists because estimating σ from four observations is genuinely unreliable, and the distribution widens its tails to compensate. The penalty shrinks as df grows.
How t Converges to z
Scan the 95% column from top to bottom and watch what happens to the gap above 1.960.
| df | 95% CI (t*) | Gap above z = 1.960 |
|---|---|---|
| 5 | 2.571 | +31% |
| 10 | 2.228 | +14% |
| 30 | 2.042 | +4% |
| 100 | 1.984 | +1% |
| ∞ (z) | 1.960 | — |
One thing the table makes concrete that textbook derivations often skip: the z-distribution is not a separate tool — it is the t-distribution with df set to infinity. That bottom row labeled ∞ is just what happens when you no longer need to estimate σ from your own sample.
A Worked Example
The numbers for the example: df = 14, t = 2.50, a one-sample test of whether a new growth medium shifts cell yield across fifteen plates.
At df = 14, two-tailed 0.05 column, the critical value is 2.145.
Because 2.50 clears 2.145, the result is significant at the 5% level — the null gets rejected. The exact two-tailed p is 0.0255.
That 0.0255 is close enough to feel uncomfortable, and that is the point — a smaller sample would have pushed the critical value higher and the same t = 2.50 might not have cleared it. Sample size does real work here, not just at the planning stage. For your own numbers, the t-test calculator handles the statistic and the p-value together, and the p-value calculator takes any t-statistic you have and converts it to a probability.
Frequently Asked Questions
When should I use the t-distribution instead of the z-distribution?
Here is the rule: if you are estimating σ from your sample, use t. If someone is handing you the true population σ from outside, use z. The first situation describes almost every real study. The second describes almost no real study.
What are degrees of freedom in a t-test?
For a one-sample or paired test, df = n − 1. For a pooled two-sample test, df = n₁ + n₂ − 2. For Welch’s t-test, your software calculates an adjusted non-integer df automatically. The df value picks the row you read from the table.
What is the difference between a one-tailed and two-tailed critical value?
Two-tailed splits α across both ends; at df = 10 with α = 0.05 you need ±2.228. One-tailed puts all of α in one direction and needs only 1.812. The two-tailed critical value at α equals the one-tailed value at α/2 — same number, different question.
Why is the t critical value larger than 1.96 for small samples?
We get this question most often from students who expect the critical value to match the z-table. At df = 4 it is 2.776, not 1.960 — the extra width is the t-distribution paying for the uncertainty in your estimate of σ. Past df = 100 the two values are nearly identical.
How do I find the p-value from a t-statistic?
The lookup field above the table handles this. Type in df and the t-statistic and both p-values come back. Quick sanity check: t = 2.50 at df = 14 gives a two-tailed p of 0.0255.