Critical Z Score Calculator
Pick a confidence level — 90%, 95%, 99%, or any custom value — and get the critical z-score your test statistic has to beat, for one- or two-tailed tests, with α shown. Free — no signup.
Find the critical z-score
Or type any level between 1% and 99.9% — for example 92 or 99.5.
Two-tailed splits α across both tails; one-tailed puts it all in one.
The shaded regions are the rejection regions: a test statistic landing there is statistically significant at the matching α.
This critical z score calculator runs entirely in your browser. Nothing is uploaded or stored.
What is a critical z-score?
Every significance test is, underneath, a comparison against a line drawn on the bell curve. The critical value is that line. You choose it before running the test by fixing the significance level α — the risk of a false alarm you are willing to accept. At the conventional α = 0.05 the line sits at z = ±1.96 for a two-tailed test, because exactly 5% of the curve lies beyond ±1.96. Tighten α to 0.01 and the line moves out to ±2.576: fewer false alarms, but real effects must be stronger to cross it. Critical values are also the multipliers inside confidence intervals — a 95% interval is estimate ± 1.96 × standard error — which is why the same three numbers (1.645, 1.96, 2.576) appear everywhere in statistics.
Critical z values for common confidence levels
These are the cutoffs worth knowing. The calculator above handles any level in between, but this table covers the standards that appear in textbooks, exams, and published research:
| Confidence level | α (significance level) | Two-tailed critical z (±) | One-tailed critical z |
|---|---|---|---|
| 80% | 0.20 | 1.282 | 0.842 |
| 90% | 0.10 | 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 |
Notice the overlap pattern: the one-tailed value at one level becomes the two-tailed value at the next level down. That is not a coincidence — a one-tailed 95% test puts 5% in a single tail, and a two-tailed 90% test puts 5% in each tail, so both cutoffs sit at the same point on the curve, z = 1.645. Once you see that, the table stops being six things to memorise and becomes one curve read six ways. For the inverse problem — starting from a percentile rather than a level — use the percentile to z score calculator.
One-tailed vs two-tailed critical values
The tail choice changes the cutoff because it changes how α is spent. A two-tailed test divides α in half: at 95% confidence, 2.5% of the curve sits in each tail, and the boundary of the upper 2.5% is z = 1.96. A one-tailed test spends all of α in a single tail: the boundary of the upper 5% is z = 1.645 — an easier line to cross, which is exactly why one-tailed tests must be justified by a directional prediction made before the data arrived. If a question or paper just says "at the 5% level" without specifying, assume two-tailed and reach for 1.96. When in doubt, the two-tailed value is the honest default: it never assumes you knew the direction of the effect in advance.
How to use this calculator
Critical values are the inverse problem in disguise — the same skill as finding z-scores from percentiles, just with confidence levels instead of percentiles.
- Settle α first. Your confidence level is 1 − α: α = 0.05 means 95% confidence. Most coursework and journals default to 95%.
- Tap a preset or type a custom level — anything from 1% to 99.9%, including values like 92% or 99.5% that no printed table lists.
- Choose one- or two-tailed to match the hypothesis you stated in advance.
- Read the critical value and compare. If your test statistic's absolute value exceeds it, the result is statistically significant at that α. To see how far past the line it fell, feed the same z-score to the z score to p value calculator.
Worked examples
Example 1 — the standard 95% test. A researcher tests at α = 0.05, two-tailed. The critical values are ±1.96. Her test statistic comes out at z = 2.10. Since 2.10 > 1.96, the statistic lands in the rejection region: the result is statistically significant at the 5% level and the null hypothesis is rejected.
Example 2 — building a 99% confidence interval. A poll estimates support at 54% with a standard error of 3 percentage points. The 99% critical value is 2.576, so the margin of error is 2.576 × 3 = 7.7 points, and the interval is 54% ± 7.7%, i.e. roughly 46.3% to 61.7%. The same estimate at 95% confidence would use 1.96 and give a narrower — but less certain — interval.
Example 3 — a one-tailed quality check. A factory only cares whether a machine over-fills: a one-tailed test at α = 0.05. The critical value is 1.645. A sample gives z = 1.40 — below the line, so there is not yet evidence of over-filling at the 5% level. Note that this same z = 1.40 would also fail a two-tailed test (1.40 < 1.96); the one-tailed design gave the effect its best legitimate chance to be detected.
Critical values vs p-values
The critical-value method and the p-value method are two descriptions of the same test, and they can never disagree: |z| beats the critical value exactly when the p-value is at or below α. They differ in what they tell you. The critical value is a fixed fence — quick to apply by hand, and the form confidence intervals are built from. The p-value measures the distance past the fence: z = 1.97 and z = 4.50 both "beat 1.96", but their evidence against the null is worlds apart (p = 0.049 vs p = 0.000007). Exam questions usually want the comparison against the critical value; research reports usually want the exact p-value. This site does both — the fence here, the measurement on the p-value page.
Frequently asked questions
1. What is a critical z-score?
A critical z-score (z critical value) is the cutoff a test statistic must exceed before a result counts as statistically significant. It is set by the significance level α before the test: at α = 0.05 two-tailed, the critical values are ±1.96, so any |z| above 1.96 rejects the null hypothesis.
2. What is the critical z value for a 95% confidence level?
For a two-tailed 95% confidence level (α = 0.05), the critical z-score is 1.96, used as ±1.96. For a one-tailed 95% cutoff it is 1.645. The other standards are 1.645 for 90% two-tailed and 2.576 for 99% two-tailed.
3. What is the difference between one-tailed and two-tailed critical values?
A two-tailed test splits α across both tails, so at 95% confidence each tail holds 2.5% and the critical value is 1.96. A one-tailed test puts all of α in one tail, giving 1.645 at the same confidence level. Two-tailed is the default; one-tailed is only for directional hypotheses fixed in advance.
4. How do critical values relate to confidence intervals?
The critical z-score is the multiplier in the confidence interval formula: estimate ± z* × standard error. A 95% interval uses z* = 1.96, so the interval stretches 1.96 standard errors each side of the estimate and captures the middle 95% of the sampling distribution.
5. Should I use the critical-value approach or the p-value approach?
They always agree — a result is significant exactly when |z| exceeds the critical value and exactly when its p-value is at or below α. The p-value approach reports how far past the line the result fell; the critical-value approach is faster for hand calculations and is how confidence intervals are built. Use whichever your course or report format asks for.