The t-test and z-test answer the same kind of question — "is this mean different from that one?" — and their formulas look nearly identical. The confusion is understandable. But choosing between them comes down to one thing: do you know the population standard deviation?
The core difference
Both tests compute a standardized distance between your sample mean and a hypothesized value. The difference is which distribution you compare it against:
- Z-test — uses the normal distribution. Valid when the population standard deviation (σ) is known, or the sample is large enough that the sample SD is essentially σ.
- T-test — uses the t-distribution, which has fatter tails to account for the extra uncertainty of estimating the standard deviation from your sample.
The deciding question: Is the population standard deviation known? If yes → z-test. If no (you're estimating it from the data) → t-test. In practice, σ is almost never known, so the t-test is the default in real research.
The "n > 30" rule of thumb
You'll often hear "use a z-test when n > 30." The reason: as the sample grows, the t-distribution converges to the normal distribution, so the two tests give almost the same answer. By df ≈ 30 the difference in critical values is tiny (t = 2.04 vs z = 1.96 at 95%). So for large samples it barely matters — but the t-test is never wrong, whereas the z-test assumes something you usually can't justify. When in doubt, use the t-test.
| Z-Test | T-Test | |
|---|---|---|
| Population SD (σ) | Known | Unknown (estimated) |
| Reference distribution | Normal | Student's t (df) |
| Best for | Large n, known σ | Any n, unknown σ |
| Real-world use | Rare | The default |
Worked intuition
Suppose a factory's bolts have a historically known σ = 0.5 mm, and you're checking whether a new batch's mean length drifted. Because σ is known from years of data, a z-test is defensible. But if you're testing a brand-new process with no history, you must estimate the SD from your 25 samples — that uncertainty is exactly what the t-test accounts for.
Which calculator do I use?
Our T-Test Calculator covers one-sample, two-sample, and paired designs — the situations you'll actually meet. For a pure standard-normal lookup (a known-σ scenario, or converting a value to a percentile), use the Z-Score Calculator or the z-table.
Run a T-Test →One-sample, two-sample, or paired — with an interpreted p-value.Related: Choosing a statistical test · Interpreting p-values · T-table.