Z-test Calculator

One-sample • Two-sample • p-value • Critical z

Settings

🧪 Test type

α is optional (default 0.05) to show critical z threshold(s).

Quick summary

Enter summary stats to compute Z-test.

📌 One-sample inputs

x̄

μ₀

σ must be known (population SD). n must be integer ≥ 1.

📌 Two-sample inputs

x̄₁

x̄₂

σ₁

σ₂

n₁

n₂

Δ₀

Δ₀ is hypothesized (μ₁−μ₂). Use 0 for “no difference”.

Results

z statistic

—

p-value

—

Standard error (SE)

—

Effect

—

Critical z

—

Based on α and tail selection.

Decision

—

“Reject H₀” if p ≤ α (α default 0.05).

Updating…

Action

Hard reset: cancels pending updates and clears everything on the first click.

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

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