T Distribution Table

The t distribution table (the t table) gives critical t values by degrees of freedom and tail probability — the values that decide confidence intervals and t-tests whenever the population standard deviation is unknown, which in real work is nearly always.

How to read the t table

Rows are degrees of freedom (df = n − 1 for a single sample). Columns are one-tailed probabilities: the cell is the t value with that much area in the right tail. The two-tailed tests most people run read the column at half their α — a two-tailed test at 0.05 uses the 0.025 column, because 0.025 sits in each tail. Worked example: a sample of 11 (df = 10) needs the 95% confidence critical value — row 10, column 0.025 — t = 2.228. Compare the bottom row: at infinite df the same column reads 1.960, the z value, because with unlimited data the t distribution becomes the normal. Every column converges to its z counterpart down the rows; the table is really a picture of uncertainty shrinking as samples grow. The blog post how to read a t table rehearses the mechanics with more examples.

When t replaces z

The rule is about σ. If the population standard deviation is genuinely known — a calibrated process, a published standard — use z. If σ is estimated from the same sample you are testing (the s in the standard error), the estimate itself is uncertain, the statistic's tails grow heavier, and the honest critical values come from this table. Small samples widen the gap dramatically: at df = 5 the 0.025 column reads 2.571 against z's 1.960, a 31% larger margin. The z table vs t table page runs one problem through both tables to show what the choice costs when it is made wrongly, and the z score vs t score page clears up the separate, confusingly named T-score scale used in test reporting.

A test worked against the table

A one-sample t-test, start to finish. Nine measurements (n = 9, df = 8) give x̄ = 26 and s = 6 against a claimed mean of 20. The statistic: t = (26 − 20) / (6/√9) = 6/2 = 3.00. Two-tailed at 0.05, the table's df 8 row, 0.025 column, gives a critical value of 2.306. The statistic clears it comfortably — the p-value is in fact about 0.017 — so the claim is rejected at the 5% level. Note the z-table mirage one more time: judged against 1.96, the same statistic looks overwhelming; against the honest 2.306 it is merely solid. Small samples, exact table — that discipline is the whole page.

Printing and downloads

This page prints cleanly from the browser (navigation and tools hide themselves). For the z family, finished files live on the z table PDF page, and the custom z table generator will build a printable table to any range. Values here are computed from the t cumulative distribution and anchor-checked (df 10, 0.025 → 2.228; df 24, 0.025 → 2.064).

Values computed from the t distribution's cumulative function · Verified November 2026

Sources and further reading

Critical values here are computed from the t distribution's cumulative function; the distribution's definition and properties are documented in the NIST/SEMATECH e-Handbook of Statistical Methods.

Frequently asked questions