What Is Critical Value of z?
The critical value of z is the point on the standard normal curve that cuts off a tail area of α. A test statistic beyond it falls in the rejection region.
Enter a confidence level such as 95, or α such as 0.05, and pick the tail. A two-tailed test splits α in half, so 95% confidence gives ±1.960, while a one-tailed test gives 1.645.
The table below lists the usual levels so you can check the result, and the shaded curve shows where the rejection region sits.
How to Use the Critical Value of z Calculator
- Choose whether you know the confidence level or α.
- Enter the value.
- Choose the test type.
- Read the critical value and the shaded curve.
Critical Value of z Formula
Two-tailed critical value
One-tailed critical value
Critical Value of z Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| α | Significance level | |
| z | Critical value on the standard normal scale | |
| Φ⁻¹ | Inverse standard normal CDF | |
| 1 − α | Confidence level |
Critical Value of z Worked Example
For 95% confidence, α = 0.05. A two-tailed test puts 0.025 in each tail, so the critical values are ±Φ⁻¹(0.975) = ±1.960. A right-tailed test puts all 0.05 in one tail, so the critical value is Φ⁻¹(0.95) = 1.645.
With the example values loaded in the calculator above, the steps are:
- Significance level α = 0.05.
- α ÷ 2 = 0.025 in each tail, so z = Φ⁻¹(0.975) = 1.95996.
- Reject H₀ if z < −1.96 or z > 1.96.
Result: Critical values of z = ± 1.95996.
How to Read a Critical z Value
The critical value marks the edge of the rejection region. For 95% confidence, α = 0.05. A two-tailed test puts 0.025 in each tail, so the critical values are ±1.960. A right-tailed test puts all 0.05 in one tail, so the critical value is 1.645.
If your z statistic is beyond the critical value, reject the null hypothesis. For a confidence interval, the critical value is the multiplier on the standard error. A higher confidence level gives a larger critical value and a wider interval.
Where Critical Value of z Is Used
Hypothesis tests
Set the rejection region for a z-test.
Confidence intervals
Find the multiplier z* for 90%, 95% or 99%.
Exams
Check values from a z table.
Reporting
State the cut-off you used.
Common Critical Value of z Mistakes to Avoid
- Two-tailed tests use α ÷ 2 in each tail.
- The left-tailed value is the negative of the right-tailed value.
- A larger confidence level gives a larger critical value.
- Use t instead of z when the SD is estimated from a small sample.
Critical Value of z Terms and Definitions
- Critical value
- The cutoff on the test statistic scale that separates the rejection region from the rest.
Critical Value of z FAQs
What is the critical value of z for 95% confidence?
1.960 for a two-tailed test, and 1.645 for a one-tailed test.
What is the difference between a critical value and a p-value?
The critical value is a fixed cutoff set by α. The p-value is computed from your data. You reject when the statistic passes the cutoff, which is the same as the p-value falling below α.
When should I use t instead of z?
Use t when the population standard deviation is unknown and estimated from a small sample.