Calculate Your Critical Z Value
Enter a confidence level and choose the tail type. The calculator returns the correct critical z score for hypothesis testing and confidence intervals.
What is a critical z value?
A critical z value is the cutoff point on the standard normal distribution used to decide whether a test statistic falls in a rejection region. In plain language, it marks the line between “likely by chance” and “statistically significant” based on your chosen confidence level or significance level.
Why it matters
You use critical z values in confidence intervals and z-tests. If your computed test statistic is beyond the critical value, you reject the null hypothesis. If it is inside the boundary, you fail to reject it.
How this calculator works
The calculator uses your confidence level to compute:
- Significance level: α = 1 − confidence level
- Two-tailed critical value: z* = z(1 − α/2)
- Right-tailed critical value: z = z(1 − α)
- Left-tailed critical value: z = z(α)
It then uses an inverse normal distribution approximation to return the corresponding z quantile.
Common critical z values
| Confidence Level | Two-Tailed Critical z (±) | One-Tailed Critical z |
|---|---|---|
| 90% | 1.645 | 1.282 |
| 95% | 1.960 | 1.645 |
| 99% | 2.576 | 2.326 |
Z vs. T: which should you use?
Use a z critical value when the population standard deviation is known or when sample size is large and normal approximation is appropriate. Use a t critical value when the population standard deviation is unknown and sample size is smaller.
Quick decision guide
- Known population standard deviation (σ): use z
- Unknown population standard deviation, small sample: use t
- Large sample and approximation conditions met: z is often acceptable
Example
Suppose you need a 95% two-tailed confidence interval. Then α = 0.05 and α/2 = 0.025 in each tail. The critical z value is 1.96, so your interval takes the familiar form: estimate ± 1.96 × standard error.
Final thoughts
The biggest source of mistakes is mixing up one-tailed and two-tailed setups. Always define your hypothesis first, then pick the tail type. With that in place, the critical z value is straightforward—and this calculator gives you the exact cutoff instantly.