Z-test Calculator

Use this Z-Test Calculator to perform one-sample or two-sample mean z-tests when the relevant population standard deviation is known. It calculates the z statistic, standard error, p-value, and critical value.

Calculation Method

One-sample z-test:

z = (x̄ − μ₀) ÷ (σ ÷ √n)

Two-sample z-test:

z = [(x̄₁ − x̄₂) − Δ₀] ÷ √(σ₁²/n₁ + σ₂²/n₂)

The calculator can use two-tailed, left-tailed, or right-tailed alternatives.

Example

Suppose:

Sample mean = 110
Hypothesized mean = 100
Known population SD = 15
Sample size = 25

Standard error:

SE = 15 ÷ √25 = 3

Z statistic:

z = (110 − 100) ÷ 3 ≈ 3.33

Important Note

This mean-based z-test requires a known population standard deviation. If the population standard deviation is unknown and must be estimated from the sample, a t-test is usually more appropriate.

Two-sample mode assumes independent groups. Statistical significance does not automatically imply practical importance or causation.

Sources

• Penn State STAT 200 — One-Sample Mean Z Test
• NIST/SEMATECH — Testing a Population Mean