What the z-Test Is
A z-test is a hypothesis test that uses the standard normal distribution (the z distribution) to decide whether a statistic (like a mean or a proportion) is consistent with a null hypothesis. The heart of every z-test is the same idea: take what you observed, subtract what the null claims should happen, then scale that difference by a standard error so the result is measured in “standard deviation units.”
This z-Test Calculator covers the four most common cases: (1) one-sample z test for a mean (with known population standard deviation σ), (2) two-sample z test for independent means (with known σ₁ and σ₂), (3) one-proportion z-test, and (4) two-proportion z-test. Each mode reports the same decision-friendly outputs: z statistic, p-value, critical z, confidence interval, and effect size.
When a z-Test Is Appropriate
In practice, z-tests are most appropriate in two situations. First, when testing a mean and the population SD (σ) is truly known from a stable process, historical calibration, or a trusted specification. Second, when testing proportions, where the normal approximation to the binomial distribution is reasonable due to sufficiently large sample sizes.
If you do not know σ for mean tests, the t distribution is usually the right tool and a t-test is the typical default. For small samples, heavy outliers, or severely skewed data, you may need a different method altogether—especially if assumptions like independence or approximate normality are questionable.
z-Test Modes Supported by This Calculator
| Mode | Tests | Null hypothesis | Common use |
|---|---|---|---|
| One-sample mean z | Mean vs μ₀ | H₀: μ = μ₀ | Compare an average to a target when σ is known |
| Two-sample mean z | Mean difference | H₀: μ₁ − μ₂ = Δ₀ | Compare two independent means when σ₁ and σ₂ are known |
| One-proportion z | Proportion vs p₀ | H₀: p = p₀ | Check if a rate differs from a benchmark |
| Two-proportion z | Difference in proportions | H₀: p₁ − p₂ = Δ₀ | Compare conversion rates, defect rates, or success rates |
The Core z-Test Structure
Every z-test reduces to:
Here, estimate might be a sample mean (x̄), a difference of means (x̄₁−x̄₂), a sample proportion (p̂), or a difference in proportions (p̂₁−p̂₂). The standard error (SE) measures the typical variation of that estimate under repeated sampling. Under the null hypothesis (and assumptions), z approximately follows a standard normal distribution.
Mean z-Tests With Known Population Standard Deviation
One-sample mean z-test
The one-sample mean z-test compares a sample mean x̄ to a hypothesized mean μ₀ when σ is known:
If your process SD is truly known, this is a clean test. If σ is estimated from the same sample, the t-test is generally more accurate.
Two-sample mean z-test
The two-sample mean z-test compares two independent group means when σ₁ and σ₂ are known:
This is common in quality control or controlled measurement systems. In many real datasets, σ₁ and σ₂ are not known, so the two-sample t-test (Welch) is used instead.
Proportion z-Tests
One-proportion z-test
The one-proportion z-test compares an observed sample proportion p̂ = x/n to a hypothesized proportion p₀. The hypothesis test uses p₀ in the standard error because the null assumes p = p₀:
For the confidence interval, many workflows use p̂ in the SE (because you’re estimating p), which is what this calculator reports by default.
Two-proportion z-test
The two-proportion z-test compares p̂₁ = x₁/n₁ and p̂₂ = x₂/n₂. Under H₀: p₁ = p₂ (especially when Δ₀ = 0), the standard error for the hypothesis test uses a pooled estimate:
For confidence intervals, an unpooled SE is typically preferred because you are estimating p₁ and p₂ separately:
How to Interpret Tail Type, p-Value, and α
A two-tailed test asks whether the true parameter differs in either direction (greater or less). A one-tailed test asks whether the parameter differs in a single specified direction. Tail choice should be made before analyzing results.
The p-value is computed from the standard normal distribution: it is the probability of a z-statistic at least as extreme as the observed value, assuming the null is true. Your significance level α is the cutoff you choose (commonly 0.05). If p ≤ α you reject H₀; if p > α you fail to reject H₀.
Confidence Intervals and Critical z
A confidence interval provides a range of plausible values for the parameter based on your sample. The standard form is:
The critical value z* depends on your confidence level (for a two-sided interval, z* corresponds to 1 − α/2). This calculator shows z* inside each test mode and also provides a dedicated Critical z tab for quick reference.
Effect Size for z-Tests
Statistical significance answers “is there evidence of a difference?” but not “how big is the difference?” That’s why effect sizes are useful. This tool reports a simple, interpretable effect size per mode:
- Mean z-tests: a standardized difference similar to Cohen’s d using σ (for one-sample, d = (x̄−μ₀)/σ).
- Two-sample means: a standardized difference using an average σ scale (a practical summary for magnitude).
- Proportions: Cohen’s h, an arcsine-based effect size that behaves well near 0 and 1.
Even when p-values are small, effect size helps you decide whether a difference is meaningful in real terms (e.g., operational impact, business value, or clinical relevance).
Step-by-Step: How to Use the z-Test Calculator
Quick workflow
- Set tail type (two-tailed or one-tailed) and choose α and confidence level.
- Select the correct tab: Mean (one-sample or two-sample) or Proportion (one or two).
- Enter summary statistics (means/σ/n) or counts (x and n), then click Calculate.
- Review z, p-value, critical z, CI, and decision.
- Use the effect size output to understand how large the difference is, not just whether it is statistically detectable.
FAQ
z-Test Calculator FAQs
Practical questions about z-tests for means and proportions, assumptions, and interpretation.
A z-test is a hypothesis test based on the standard normal (z) distribution. It is commonly used when population standard deviation is known (for mean tests) or when testing proportions with sufficiently large sample sizes.
Use a z-test for means when the population SD (σ) is known (or n is very large and σ is well-estimated). Use a t-test when σ is unknown and you rely on the sample SD.
Two-tailed tests detect differences in either direction. One-tailed tests detect differences in a specified direction only (greater-than or less-than).
The p-value is the probability (assuming the null hypothesis is true) of observing a z statistic at least as extreme as the one computed from your data.
For two-tailed tests, if the null value is outside the confidence interval (at the same confidence level), the test typically rejects the null at the matching α.
Proportion z-tests assume independent trials and that sample sizes are large enough for a normal approximation (often checked with np and n(1−p) being sufficiently large).
Under the null hypothesis p₁ = p₂, pooling combines successes to estimate a common proportion used in the standard error for the hypothesis test.
For mean z-tests, a common standardized effect is Cohen’s d using σ. For proportions, Cohen’s h is often used to express the difference between proportions on an arcsine scale.