Critical Z-Value Calculator
Use this critical z value calculator to find the z-score that marks the edge of a rejection region. Choose your confidence level or significance level, pick one- or two-tailed, and get z*, the percentile, a shaded normal curve and the matching TI-84 and Excel formulas — instantly, free, with every step shown.
Find your critical value
Confidence level
α = 1 − confidence level. Edit either box — the other updates automatically.
Tail type
Two-tailed: α is split between both tails (α/2 each side).
Critical z-value (z*)
95% confidence · two-tailed
Exact z* = 1.95996
Calculation
- α = 1 − 0.95 = 0.05
- α/2 = 0.025 in each tail
- Cumulative area Φ(z*) = 1 − α/2 = 0.975
- z* = Φ⁻¹(0.975) = 1.95996 ≈ 1.96
Standard normal curve — rejection region shaded
Pink = rejection region (total area α). Dashed lines mark the critical value(s).
A critical z-value is the z-score that marks the boundary of a rejection region on the standard normal curve. For a 95% confidence level, the two-tailed critical value is z* = 1.96, and the one-tailed value is 1.645; at 99% the values are 2.576 and 2.326. Any test statistic beyond z* falls in the rejection region.
What Is a Critical Z-Value?
A critical z-value, written z*, is the cut-off on the standard normal distribution (mean 0, standard deviation 1) that separates ordinary results from results extreme enough to reject a null hypothesis. Everything beyond z* — in one tail, or split across both tails — is the rejection region, and its total area is the significance level, α.
That is why the same confidence level gives two different numbers. A two-tailed test splits α across both ends of the curve, so each tail holds α/2 and the cut-off sits further out (1.96 at 95%). A one-tailed test puts all of α in a single tail, so the cut-off is closer to the centre (1.645 at 95%). Choosing the wrong tail type is the most common mistake with critical values — decide the direction of your test before you look the value up. If you need the value for a single observation rather than a test, our z-score calculator converts a raw score to a z-score and its percentile.
How to Find a Critical Z-Value
Start with your confidence level or significance level. For a two-tailed test, divide α by 2, subtract from 1, and read the matching z-score from the inverse normal function: z* = Φ⁻¹(1 − α/2). For a one-tailed test there is no division — z* = Φ⁻¹(1 − α), or Φ⁻¹(α) for a left-tailed test.
Worked example — 95% confidence, two-tailed: α = 1 − 0.95 = 0.05, so α/2 = 0.025 and the cumulative area is 1 − 0.025 = 0.975. The z-score with 97.5% of the area to its left is 1.96, so z* = ±1.96. On a TI-84 that is invNorm(0.975, 0, 1); in Excel it is =NORM.S.INV(0.975). The normal distribution calculator shows the same inverse-normal step with a picture of the shaded area.
Critical Z-Value Table: Common Values
These are the values most textbooks and exam papers use. Keep the table handy — or let the calculator above work out any level that is not listed, including custom confidence levels such as 92% or 98%.
| Confidence level | α | Two-tailed z* | One-tailed z* |
|---|---|---|---|
| 90% | 0.10 | ±1.645 | 1.282 |
| 95% | 0.05 | ±1.960 | 1.645 |
| 99% | 0.01 | ±2.576 | 2.326 |
Values are rounded to three decimal places. The calculator gives the full five-decimal value — for example, 1.95996 for a 95% two-tailed test and 2.57583 for 99%.
One-Tailed vs Two-Tailed Critical Values
Use a two-tailed test when you are checking for a difference in either direction — a new teaching method could raise or lower scores, for example. Use a one-tailed test only when you have a clear, pre-registered reason to look in one direction, such as testing whether a drug lowers blood pressure. Switching to a one-tailed test after seeing the data is not legitimate: it artificially lowers the bar.
At 95% confidence the difference is stark: ±1.96 two-tailed against 1.645 one-tailed. The one-tailed cut-off is lower because the whole 5% sits in one tail instead of being split into 2.5% at each end. The same z* values also build confidence intervals — the confidence interval calculator for a mean and standard deviation uses ±1.96 for a 95% interval. And once you have a test statistic, you can compare it with z* or turn it into a probability with our p-value calculator from a z-score.
TI-84 and Excel: Critical Z-Value Equivalents
TI-84: press 2ND → DISTR → invNorm( and enter the cumulative area, the mean and the standard deviation: invNorm(area, 0, 1). For a 95% two-tailed test the area is 0.975, so invNorm(0.975, 0, 1) = 1.96. One-tailed, use invNorm(0.95, 0, 1) = 1.645 for the right tail, or invNorm(0.05, 0, 1) = −1.645 for the left tail. The area is the only input that changes — the mean stays 0 and the standard deviation stays 1 for a standard normal curve.
Excel and Google Sheets: type =NORM.S.INV(probability). A 95% two-tailed critical value is =NORM.S.INV(0.975); a 90% one-tailed value is =NORM.S.INV(0.9), and a 99% two-tailed value is =NORM.S.INV(0.995). In very old versions of Excel the function is called =NORMSINV(0.975). The same critical value also appears in the margin of error formula, z* × standard error — read what margin of error means to see how 1.96 turns a standard error into a 95% interval width.
Critical Z-Value vs Z-Score and P-Value
A z-score and a critical z-value are related but not the same thing. A z-score tells you how many standard deviations one observation sits from the mean; a critical value is the boundary that observation — or a test statistic — is judged against. In a one-proportion z-test, for example, you compute the test statistic z and reject the null hypothesis when |z| is greater than z* — for a 95% two-tailed test, when |z| > 1.96.
A p-value answers the same question from the other side: it is the probability of a result at least as extreme as the one observed, assuming the null hypothesis is true. If your test statistic falls beyond z*, the p-value is smaller than α, and the two methods always agree. Learn more in what a p-value is. The definitions and the standard normal probabilities used on this page follow the NIST/SEMATECH e-Handbook of Statistical Methods.
Frequently Asked Questions
What is the critical z-value for 95% confidence?
For a two-tailed test at 95% confidence, the critical value is z* = ±1.96 (1.95996 to five decimals). For a one-tailed test at the same level, it is 1.645.
What is the critical z-value for 90% confidence?
Two-tailed: ±1.645 (1.64485). One-tailed: 1.282 (1.28155).
What is the critical z-value for 99% confidence?
Two-tailed: ±2.576 (2.57583). One-tailed: 2.326 (2.32635).
How do I find the critical z-value on a TI-84?
Press 2ND, DISTR, then invNorm( and type the cumulative area, 0 and 1. For a 95% two-tailed test, invNorm(0.975, 0, 1) gives 1.96; for a one-tailed test use invNorm(0.95, 0, 1), which gives 1.645.
What is the critical value formula in Excel?
Use =NORM.S.INV(probability). For a 95% two-tailed value the probability is 0.975, so =NORM.S.INV(0.975) = 1.96. For a one-tailed 95% value use =NORM.S.INV(0.95) = 1.645.
What is the difference between a critical value and a p-value?
A critical value is a boundary on the z-scale: reject the null hypothesis when the test statistic falls beyond it. A p-value is a probability: reject the null hypothesis when the p-value is smaller than α. The two rules always agree for the same test.
When should I use a one-tailed test instead of a two-tailed test?
Use a one-tailed test only when your research question specifies a direction before you collect data — for example, testing whether a new drug lowers blood pressure. If you are testing for any difference in either direction, use a two-tailed test. Never switch to a one-tailed test after seeing the results.
What is α in a critical value calculation?
α, the significance level, is the total area of the rejection region: α = 1 − confidence level. For a 95% confidence level, α = 0.05. A two-tailed test splits α into α/2 = 0.025 in each tail; a one-tailed test puts the whole 0.05 in a single tail.
More Tools to Explore
Keep working with the normal distribution — these tools pair naturally with critical values:
- Z-Score Calculator — convert any raw score to a z-score and its percentile.
- Normal Distribution Calculator — probabilities, percentiles and inverse normal values.
- P-Value Calculator from Z-Score — turn a test statistic into a p-value.
- One-Proportion Z-Test Calculator — test a sample proportion against a claimed value.
- Confidence Interval Calculator — build an interval from a mean and standard deviation.
- Margin of Error Calculator — find the margin of error for a survey or a sample.