Z-Score Calculator

Critical Value Calculator

Standard-normal critical values for hypothesis-test rejection regions and confidence-interval half-widths.

Test Type

P(|Z| > z*) = α — symmetric rejection in both tails. Reports ±z*.

Common α levels

Solution

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Common Z Critical Values

Reference table of the most-cited z critical values for one- and two-tailed hypothesis tests. Two-tailed values double as confidence-interval multipliers — the two-tailed value at α equals the multiplier for a (1 − α) · 100% confidence interval.

α (significance)One-tailed z*Two-tailed ±z*Confidence Level
0.101.2816±1.644990%
0.051.6449±1.960095%
0.0251.9600±2.241497.5%
0.012.3263±2.575899%
0.0052.5758±2.807099.5%
0.0013.0902±3.290599.9%

Worked Examples

Two-Tailed · α = 0.05

Critical value for a two-tailed test at α = 0.05

The most-cited critical value in introductory statistics — the borderline cutoff for the standard 5% significance level.

  1. Identify the test type: two-tailed at α = 0.05.
  2. Split α across the two tails: α/2 = 0.025 in each tail.
  3. Find z* such that the upper-tail area is 0.025 — equivalently, the cumulative area below it is 0.975.
  4. Apply the inverse normal CDF: z* = Φ⁻¹(0.975) ≈ 1.96.
  5. By symmetry, the critical values are ±1.96.

±1.96 also bounds the half-width of every standard 95% confidence interval based on the normal distribution.

One-Tailed · α = 0.05

Critical value for a right-tailed test at α = 0.05

The standard one-tailed cutoff — used when the alternative hypothesis specifies an upper-direction effect.

  1. Identify the test type: right-tailed at α = 0.05.
  2. All α goes into the upper tail.
  3. Find z* such that the right-tail area equals 0.05.
  4. Apply the inverse normal CDF: z* = Φ⁻¹(0.95) ≈ 1.6449.
  5. Decision rule: reject H₀ if the test statistic exceeds 1.6449.

The one-tailed value (1.6449) is smaller than the two-tailed value (1.96) at the same α because the rejection region is concentrated in a single tail.

Two-Tailed · α = 0.01

Critical value for a two-tailed test at α = 0.01

Used when stronger evidence is required — the textbook 1% significance level.

  1. Identify the test type: two-tailed at α = 0.01.
  2. Split α: α/2 = 0.005 in each tail.
  3. Find z* such that the upper-tail area is 0.005 — equivalently, cumulative area 0.995.
  4. Apply the inverse normal CDF: z* = Φ⁻¹(0.995) ≈ 2.5758.
  5. Critical values: ±2.5758.

The 99% confidence-interval multiplier is the same value: ±2.5758 × SE bounds a 99% CI half-width.

Confidence Level · 99%

Critical value for a 99% confidence interval

Same math as the two-tailed test above, framed in confidence-interval terms.

  1. Identify the confidence level: C = 99%.
  2. Compute the corresponding α: α = 1 − 0.99 = 0.01.
  3. Use the two-tailed critical value: z* = Φ⁻¹(1 − α/2) = Φ⁻¹(0.995) ≈ 2.5758.
  4. Half-width of the CI: ±2.5758 × Standard Error.

The Confidence Level mode is just the two-tailed mode repackaged. The same z* bounds both — they are two views of the same calculation.

Critical z from significance level

A critical value z* is the cutoff that bounds a chosen rejection region in a hypothesis test or the half-width multiplier in a confidence interval. For a right-tailed test it is the z-score whose upper-tail area equals α. For a two-tailed test, the symmetric pair is ±z* where each tail has area α/2. The same value bounds the half-width of a (1 − α) · 100% confidence interval.

z* = Φ⁻¹(1 − α) (right-tailed), z* = Φ⁻¹(1 − α/2) (two-tailed)

How It Works

A critical value calculator returns the z-score that marks the edge of a rejection region — the line beyond which you reject the null hypothesis at a given significance level α. For a right-tailed test the rejection region is the upper tail, so z* satisfies P(Z > z*) = α; for a left-tailed test, z* sits in the lower tail; and for a two-tailed test, the rejection region is split symmetrically across both tails so z* is found from P(|Z| > z*) = α (half of α in each tail). The calculation uses the inverse normal CDF (the probit or quantile function) — the mathematical inverse of the cumulative probability you would read off a z-table. The Confidence Level mode is just the two-tailed case repackaged: a (1 − α) · 100% confidence interval uses the same critical value as a two-tailed test at α.

Example Problem

A researcher plans a two-tailed hypothesis test at the 5% significance level. What critical z-value bounds the rejection region?

  1. Identify the test type: two-tailed at α = 0.05.
  2. Split α across the two tails: each tail receives α/2 = 0.025.
  3. Find z* such that the upper-tail area equals 0.025 — equivalently, the cumulative area below it equals 0.975.
  4. Apply the inverse normal CDF: z* = Φ⁻¹(0.975) ≈ 1.9600.
  5. By symmetry, the critical values are ±1.96.
  6. Decision rule: reject the null hypothesis if the observed test statistic falls below −1.96 or above +1.96.

±1.96 is the most-cited critical value in introductory statistics. It also bounds the half-width of every textbook 95% confidence interval based on the standard normal.

Key Concepts

Critical values are decision thresholds; test statistics are computed from sample data; p-values are tail probabilities. They all answer the same hypothesis-testing question from three angles. A test rejects the null when the test statistic exceeds the critical value, which is equivalent to the p-value falling below α. Critical values exist for every continuous distribution, not just the standard normal — t, F, χ², and others all have their own critical-value tables. This calculator handles only the z-distribution (the standard normal), which is appropriate when the population variance is known or when the sample size is large enough that the t-distribution is well-approximated by the normal.

Applications

  • Hypothesis tests: rejection region boundaries for one- and two-tailed z-tests
  • Confidence intervals: half-width multiplier (e.g., ±1.96 × SE for a 95% CI)
  • Power analysis: critical regions used to compute Type II error and statistical power
  • Quality control: process control limits at chosen Type I error rates
  • Clinical trial design: pre-specified rejection thresholds for primary endpoints
  • Acceptance sampling: cutoff scores for go/no-go decisions in finance, insurance, and engineering

Common Mistakes

  • Forgetting to split α by 2 for two-tailed tests — using α instead of α/2 inflates the rejection region and the Type I error rate
  • Reporting a one-tailed critical value when the test is actually two-tailed (or vice versa)
  • Using a critical z when the population variance is unknown and the sample is small — that situation calls for a t critical value, not z
  • Treating a confidence level (e.g., 95%) as α — the relationship is α = 1 − C, not α = C
  • Comparing the test statistic to α directly instead of to z* (or the p-value to α)
  • Mixing up the sign convention for left-tailed tests — z* should be negative when α puts the rejection region in the lower tail

Frequently Asked Questions

What is the critical value for α = 0.05?

For a one-tailed test, z* ≈ 1.6449 (right-tailed) or z* ≈ −1.6449 (left-tailed). For a two-tailed test, the critical values are ±1.96. The two-tailed value at α = 0.05 is the most-cited critical value in statistics.

What is the critical value for α = 0.01?

For a one-tailed test at α = 0.01, z* ≈ 2.3263. For a two-tailed test at α = 0.01, the critical values are ±2.5758.

What is the difference between a one-tailed and two-tailed critical value?

A one-tailed critical value puts the entire α in one tail. A two-tailed critical value splits α equally between both tails — α/2 in each. Two-tailed critical values are larger in magnitude than one-tailed values at the same α because each tail gets only half the area.

What critical value corresponds to a 95% confidence level?

A 95% confidence interval uses the same critical value as a two-tailed test at α = 0.05: z* ≈ 1.96. The confidence level C and significance level α are related by α = 1 − C/100.

When should I use a critical z instead of a critical t?

Use a critical z when the population standard deviation σ is known, or when the sample size is large enough (typically n ≥ 30) that the sampling distribution of the mean is well-approximated by the normal. Use a critical t when σ is unknown and the sample size is small.

How is the inverse normal CDF computed?

There is no closed-form formula. This calculator uses Acklam's rational approximation, which is accurate to about 1.15 × 10⁻⁹ across the full unit interval. For the most common α values, the result agrees with published z-tables to four decimal places.

What are common z critical values?

One-tailed: 1.2816 at α = 0.10, 1.6449 at α = 0.05, 2.3263 at α = 0.01, 3.0902 at α = 0.001. Two-tailed: 1.6449 at α = 0.10, 1.96 at α = 0.05, 2.5758 at α = 0.01, 3.2905 at α = 0.001. The reference table further down on this page lists these and a few more.

Can a critical value be negative?

Yes — for a left-tailed test, z* is negative. For a two-tailed test, the critical values come in symmetric pairs ±z*, so the lower bound is negative. The Right-tailed mode in this calculator always returns a positive z*; the Left-tailed mode always returns a negative z*.

Reference: Inverse normal CDF computed via Peter Acklam's rational approximation (|error| < 1.15 × 10⁻⁹). Critical-value relationships for one- and two-tailed tests are presented in every introductory statistics text; the equivalence between the two-tailed critical value at α and the (1 − α) confidence-interval multiplier is foundational.

Related Calculators

  • Inverse Z-Score — Same idea, framed around input probability rather than α.
  • P-Value Calculator — Convert a test statistic to a p-value to compare against α.
  • One-Sample Z-Test — Run a full hypothesis test using these critical values.
  • Confidence Interval — Build CIs whose half-widths use the two-tailed critical value.
  • Sample Size — Plan n for a target margin of error using these multipliers.

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