Z Test Calculator

When to Use a Z-Test

A one-sample z-test checks whether a sample mean differs from a hypothesized population mean when the population standard deviation σ is known. It uses the standard normal distribution because the test statistic is standardized with σ rather than the sample standard deviation. Related tools: Hypothesis Testing Calculator, Z-Score Calculator.

Z-Test Formula

$$z = (\bar{x} - \mu_0)\frac{\sqrt{n}}{\sigma}$$

Compare the z statistic with critical values or convert it to a p-value. Reject $H_0$ when the p-value is less than α or when z falls in the rejection region.

Example

Nine bottles have mean volume 980 mL. If μ₀ = 1000 mL and σ = 30 mL, then $z = (980 - 1000)\sqrt{9}/30 = -2$. A two-tailed test at α = 0.05 gives p ≈ 0.0456, so you reject the claim that the mean is 1000 mL.