Statistics Reference
How to Read a T-Table
A t-table lookup needs three decisions — degrees of freedom, significance level, and one tail or two — and each is a place where answers silently go wrong. This guide walks through the three steps with real lookups from the t-table on this site, shows how the same table serves tests and confidence intervals, and pairs with the companion guide on reading a z-table.
What the Table Contains
Unlike a z-table, which lists cumulative probabilities for every z-score, a t-table lists only critical values: the handful of cutoffs that tests and intervals actually use. It has to be organized this way because the t-distribution changes shape with the degrees of freedom — one full cumulative table per df would fill a book. So rows carry df, columns carry the significance level α, and each cell answers one question: how many standard errors from zero must a t statistic land to be significant at this level?
The distribution is symmetric, so only positive values are printed. For a left-tailed test, use the negative of the table value.
The Three-Step Lookup
Example: two-tailed one-sample t-test, n = 15, α = 0.05
- Degrees of freedom. A one-sample test uses df = n − 1 = 14, so go to the row labeled 14.
- Pick the correct header. The test is two-tailed, so read the two-tail α = 0.05 header (the same column as one-tail 0.025).
- Read the cell. Row 14 meets that column at 2.145. Reject the null hypothesis when |t| exceeds 2.145.
A one-tailed version of the same test would read the one-tail 0.05 column instead — a smaller cutoff (1.761 at df = 14), because all of the rejection area sits in a single tail. Choosing the tail before seeing the data is what keeps that easier threshold honest.
Degrees of Freedom by Procedure
| Procedure | Degrees of freedom | Example |
|---|---|---|
| One-sample t-test / CI for a mean | n − 1 | n = 15 → df = 14 |
| Paired t-test | n − 1 (pairs) | 10 pairs → df = 9 |
| Two-sample t-test (pooled) | n₁ + n₂ − 2 | 12 and 14 → df = 24 |
| Simple regression slope | n − 2 | 20 points → df = 18 |
Each degree of freedom is a piece of information left over after estimating the means involved — the same accounting explained in the n vs n−1 guide.
The Same Lookup for Confidence Intervals
Confidence intervals always use the two-tail reading: a 95% interval leaves 5% split across both tails. From a sample of 15 with mean 52 and standard deviation 8:
- Critical value: df = 14, two-tail 0.05 → t* = 2.145.
- Standard error: 8 / √15 ≈ 2.0656.
- Margin of error: 2.145 × 2.0656 ≈ 4.43.
- Interval: 52 ± 4.43 → (47.57, 56.43).
Wanting 99% confidence instead moves one column right (two-tail 0.01, t* = 2.977 at df = 14) and widens the margin to 6.15 — the width-for-confidence trade covered in the confidence intervals guide.
The Mistakes That Flip Answers
- Reading the wrong header row. Using one-tail 0.05 (1.761 at df = 14) for a two-tailed test doubles your intended false-alarm rate. The two-tail row is the one that matches “different from” hypotheses and confidence intervals.
- Using n instead of n − 1. With small samples one row matters: at n = 5, df = 4 gives 2.776 while the df = 5 row reads 2.571.
- Grabbing z habits. 1.96 is a z value. The t equivalent is always larger until df grows: 2.145 at df = 14, 2.064 at df = 24, 1.980 at df = 120.
- Sign confusion on one-tailed tests. For a left-tailed test the rejection region is t ≤ −(table value); the table only prints the positive cutoff.
From Lookup to Full Test
The table answers “what is the cutoff?” — the t-test calculator runs the entire procedure, reporting the t statistic, exact p-value, and decision for one-sample, two-sample, and paired designs. For exact critical values at any degrees of freedom (including the fractional df of Welch's test), the critical value calculator evaluates the same inverse function the printed table was built from.
Try the T-Table (Student's t Critical Values)
Critical values for degrees of freedom 1 to 1000 with one-tail and two-tail headers, worked examples, and a printable layout.
Frequently Asked Questions
What do the rows and columns of a t-table mean?
Rows are degrees of freedom (df), which depend on your sample size and test type; columns are significance levels (alpha), usually with separate header rows for one-tailed and two-tailed tests. The cell where your df row meets your alpha column is the critical value: the cutoff your t statistic must exceed for significance, or the multiplier for a confidence interval's margin of error.
How do I find degrees of freedom for the t-table?
It depends on the procedure: n - 1 for a one-sample or paired t-test (and for a confidence interval for one mean), n1 + n2 - 2 for a classic pooled two-sample t-test, and n - 2 for a simple regression slope. A sample of 15 in a one-sample test therefore uses the df = 14 row.
Which column do I use for a 95% confidence interval?
The two-tail alpha = 0.05 column (equivalently one-tail 0.025), because a 95% interval leaves 5% of probability split across the two tails. With df = 14 that cell reads 2.145, so the interval is the sample mean plus or minus 2.145 standard errors. A 99% interval uses the two-tail 0.01 column instead (2.977 at df = 14).
Why does the t-table have both one-tail and two-tail headers?
Both headers describe the same columns from different viewpoints. A one-tailed test puts all of alpha in one tail; a two-tailed test splits it across both tails, so each tail holds alpha/2. That is why one-tail 0.025 and two-tail 0.05 label the same column - the same critical value serves a directional test at 2.5% and a non-directional test at 5%.
What if my degrees of freedom fall between two rows?
Round down to the nearest listed row for a conservative answer - the critical value will be slightly too large rather than too small, so you will never overstate significance. Interpolating between neighboring rows gets closer, and a critical value calculator evaluates the exact value at any df, including non-integer Welch degrees of freedom.
When do I use the t-table instead of the z-table?
Use t whenever the population standard deviation is estimated from the sample - the usual situation. The t critical values are larger to pay for that extra uncertainty, especially at small df: 12.706 versus 1.96 for two-tail 5% at df = 1. By df = 120 the gap has shrunk to 1.980 versus 1.960, and the bottom (infinity) row of a t-table simply is the z-table.