Z Table vs T Table

The z table and the t table answer the same question — how extreme is my statistic? — under different knowledge. Z assumes the population standard deviation is known; t admits it was estimated from the sample and charges you for the uncertainty. Choosing wrongly is not a rounding issue; it changes verdicts.

The decision rule

σ known (a true population value, not a sample estimate): z table, whatever the sample size.
σ unknown, estimated by s: t table with df = n − 1 — at any sample size, strictly speaking; at large n the difference fades, which is why large-sample courses quietly switch to z.
Proportions: z (the standard error comes from the hypothesised p, not an estimated σ in the t sense).
Two samples, regression slopes, most model coefficients: t, with the df your software or formula specifies.

The practical summary taught in most courses — "n ≥ 30, use z; below, use t" — is a simplification of the second rule: by n = 30 the t criticals sit within about 4% of z's, and by n = 120 within 1%. It is a fine rule of thumb for exams and a poor one for publication, where the t value is correct at every n when σ is estimated.

One number that shows the stakes

The two-tailed 0.05 critical value, by table: z says 1.960 always. The t table says 2.571 at df = 5, 2.228 at df = 10, 2.086 at df = 20, 2.042 at df = 30, 2.000 at df = 60, converging on 1.960 only at infinite df. Run a small-sample test with the z critical and your "5% test" actually rejects a true null far more often — at df = 10, using 1.960 instead of 2.228 produces a true error rate near 7.8% while claiming 5%. Confidence intervals suffer the same distortion in width: the z interval at df = 10 is about 12% too narrow, silently overpromising precision the data never provided.

Same data, both tables — sample of 12, x̄ = 52, s = 10, testing μ = 47

The test statistic is t = (52 − 47) / (10/√12) = 1.73. With the t table (df = 11), the two-tailed p-value is about 0.111 — not significant. Judged as a z, the same 1.73 gives p = 0.083 — still not significant at 0.05, but visibly closer, and at a slightly larger statistic the two tables would have delivered opposite verdicts. The statistic's arithmetic is identical; only the reference distribution — the table — changes, because the t distribution's heavier tails honestly price the estimated s. The z score vs t score page handles the unrelated T-score scale (mean 50, sd 10) that shares the letter and confuses everyone once.

A short history that explains the shape

The t distribution exists because of a brewery. William Gosset, a Guinness chemist publishing as "Student" in 1908, needed valid inference from tiny samples — a handful of barley measurements — where treating s as if it were σ produced overconfident conclusions. His distribution priced the estimation honestly, at the cost of the df-indexed table you now have to enter correctly. The z table, by contrast, descends from the era of astronomical and survey data, where samples were huge and σ effectively known. The two tables on this site are thus two centuries of data size made visible: small data buys the t table's caution; abundant data earns the z table's simplicity.

Frequently asked questions

Can I always just use the t table to be safe?

Almost — when σ is estimated, t is the correct choice at any n, and at large n it costs you almost nothing. The exception is when σ is genuinely known (rare outside standardised testing and calibrated processes): there z is exact and t is slightly conservative. "Always t" is a far safer habit than "always z".

Why does the t distribution have heavier tails?

Because the denominator s varies from sample to sample. Occasionally s comes out small, the ratio blows up, and extreme statistics occur more often than the normal predicts. The t distribution builds that extra variability in; as n grows, s stabilises and the tails slim toward the normal.

Which table do I use for a confidence interval for a mean?

t, with df = n − 1, whenever σ is estimated from the sample — the standard case. The interval is x̄ ± t* × s/√n. The z table for confidence intervals page covers the known-σ and large-sample z version, including the 1.645 / 1.96 / 2.576 criticals.

What are degrees of freedom doing in the t table?

They index how much information estimated σ. More data → more df → s tracks σ more tightly → t criticals shrink toward z. df is the dial between the two tables: at df = ∞ they are the same table.