Critical Z Values
A critical z value is the cutoff a test statistic must cross before a result counts as statistically significant. The complete table is below — by confidence level and by significance level (alpha), one-tailed and two-tailed. The three values that do most of the work are 1.645 (90%), 1.96 (95%) and 2.576 (99%).
| Confidence level | Alpha (α) | Two-tailed z | One-tailed z |
|---|---|---|---|
| 80% | 0.2 | 1.282 | 0.842 |
| 90% | 0.1 | 1.645 | 1.282 |
| 95% | 0.05 | 1.960 | 1.645 |
| 98% | 0.02 | 2.326 | 2.054 |
| 99% | 0.01 | 2.576 | 2.326 |
| 99.9% | 0.001 | 3.291 | 3.090 |
Reading the table: a two-tailed test splits alpha across both tails, so its cutoff is always larger than the one-tailed cutoff at the same level. At 95% confidence (α = 0.05), a two-tailed test needs |z| > 1.96; a one-tailed test needs z > 1.645 in the predicted direction.
Why 1.96 is the z value for 95%
It is not a rounded convenience — it falls straight out of the table. A 95% confidence level leaves 5% outside the interval, split as 2.5% in each tail. The upper cutoff is therefore the z score with 0.9750 to its left, and reading the z score table backwards, 0.9750 sits exactly at z = 1.96. The same method gives every other row: for 90%, find 0.9500 in the table body → z = 1.645; for 99%, find 0.9950 → z = 2.576. Once you see that critical values are just reverse table lookups, you never need to memorise the table — you need to be able to rebuild any row of it.
Critical values by significance level (alpha)
- α = 0.10: two-tailed ±1.645 · one-tailed 1.282
- α = 0.05: two-tailed ±1.960 · one-tailed 1.645
- α = 0.01: two-tailed ±2.576 · one-tailed 2.326
- α = 0.001: two-tailed ±3.291 · one-tailed 3.090
Notice the pattern hiding in this list: the one-tailed value at one level is the two-tailed value at the next level down (one-tailed α = 0.05 → 1.645 → two-tailed α = 0.10). That is because a one-tailed test puts all of alpha in one tail while a two-tailed test splits it — a one-tailed 5% test and a two-tailed 10% test are guarding the same single tail. It is also why choosing one tail after seeing your data is cheating: it quietly doubles your real alpha, and honest researchers set the tails before they compute anything.
How critical values are used — a complete decision
Worked decision at α = 0.05, two-tailed
A wellbeing programme is tested on 25 participants against a reference mean of 50 (standard deviation 10). The group averages 54.5. The standard error is 10 / √25 = 2.0, so the test statistic is z = (54.5 − 50) / 2.0 = 2.25. The critical values at α = 0.05 two-tailed are ±1.96. Because 2.25 lies beyond 1.96, the result falls in the rejection region: reject the null hypothesis. (The p-value route agrees: the tail beyond 2.25 is 0.0122, doubled to 0.0244, which is below 0.05.) The same statistic at α = 0.01 would fail — 2.25 does not reach 2.576 — which is exactly the trade-off a stricter alpha buys: fewer false alarms, more missed effects.
Critical value or p-value — two doors to the same room
Comparing a test statistic with a critical value and comparing a p-value with alpha are the same test in two costumes. If |z| exceeds the two-tailed critical value at your alpha, then the two-tailed p-value is below alpha — always, by construction, because the critical value is defined as the point where the tail area equals alpha. Courses teach the critical-value method because it works with a printed table and no computation beyond a lookup. Software reports p-values because they carry more information: p = 0.0244 tells you the result would survive α = 0.05 but not α = 0.01, while "beyond 1.96" tells you only the first half of that sentence. Learn the critical-value method for exams; read p-values for research. Our z table calculator returns both views: the p-value tool gives the p, and this page gives the fence it has to jump.
Critical values beyond z: a word about t
Everything on this page assumes the standard normal judge — appropriate when σ is known or samples are large. With a small sample and an estimated standard deviation, the fences move outward and belong to the t distribution instead: at 15 degrees of freedom the 95% two-tailed fence is 2.131, not 1.96. Using z fences in t territory manufactures significance; the comparison, with the same dataset judged both ways, is worked on the z score vs t score page.
Frequently asked questions
What is the critical z value for 95% confidence?
1.96 for a two-tailed test or interval (±1.96), and 1.645 for a one-tailed test. It comes from the table: 95% confidence leaves 2.5% in each tail, and the z with 0.9750 to its left is 1.96.
What does z alpha/2 mean?
It is the notation for the two-tailed critical value: zα/2 is the z score with area α/2 in the upper tail. For α = 0.05, zα/2 = z0.025 = 1.96. Searches like "0.05/2" are students looking for exactly this number.
Why is the two-tailed critical value always bigger?
A two-tailed test splits alpha between both ends of the distribution (α/2 each), so each tail must be smaller, and a smaller tail starts further from the mean. A one-tailed test spends all of alpha in one tail, so its fence stands closer in.
What if my z statistic falls exactly on the critical value?
By convention, a statistic that reaches or exceeds the critical value rejects the null hypothesis (p ≤ α counts as significant). In practice, an answer that close to the fence deserves a sentence in your report saying so, whichever way it falls.
Are critical z values ever negative?
Yes — the lower fence of a two-tailed test is the negative of the upper one (−1.96 at 95%), and a left-tailed test's single fence is negative (for example −1.645 at α = 0.05). Tables usually print the positive values and expect you to mirror them, which the symmetry of the curve makes exact.
Building a confidence interval rather than running a test? The confidence intervals page uses these same values inside the interval formula. Need a critical value for a level not listed? Read the main z score table backwards — the method above rebuilds any row.