StatisticsFree

Z-Test Calculator

Use this free online z-test calculator to run a one-sample z-test when the population standard deviation is known. Enter the sample mean, population standard deviation, sample size, and hypothesized mean to get the z-statistic, p-value, and hypothesis test decision.

Enter Values

Sample Mean (x-bar)

The average of your sample data

Population Standard Deviation (sigma)

The known population standard deviation (not the sample standard deviation)

Sample Size (n)

The number of observations in your sample

Hypothesized Mean (mu0)

The population mean under the null hypothesis

Significance Level (alpha)

The threshold for statistical significance (commonly 0.05)

Result

Enter values above and click Calculate to see your result.

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Formula

Core Formula

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

How it works: Subtract the hypothesized mean from the sample mean, then divide by the standard error (population standard deviation divided by the square root of sample size). The z-statistic measures how many standard errors the sample mean falls from the hypothesized value.

Review and Methodology

Updated Mar 26, 2026

This calculator runs locally in your browser. Inputs are converted into the units required by the formula, and the result is paired with supporting references so you can verify the method before using it for planning or estimates.

Worked Example

Sample mean = 105, sigma = 15, n = 64, mu0 = 100, alpha = 0.05

1Step 1: Standard error = 15 / sqrt(64) = 15 / 8 = 1.875

2Step 2: z = (105 - 100) / 1.875 = 2.667

3Step 3: For a two-tailed test, p-value = approximately 0.0077

4Step 4: Since p = 0.0077 < 0.05, reject the null hypothesis

Result: z = 2.667, p = 0.0077. Strong evidence that the population mean differs from 100.

What Is a Z-Test and How Is It Different from a T-Test?

The z-test is a hypothesis test that uses the standard normal distribution as its reference distribution. It is appropriate when the population standard deviation is known. The z-test produces a z-statistic that measures how far the sample result is from the null hypothesis in standard error units.

The z-test uses the standard normal distribution, which has lighter tails than the t-distribution
It requires the population standard deviation to be known, which is rare in practice but common in textbook problems
For large samples (n > 30), the z-test and t-test give nearly identical results
Critical z-values: 1.645 (one-tailed, alpha = 0.05), 1.96 (two-tailed, alpha = 0.05), 2.576 (two-tailed, alpha = 0.01)

While the z-test is less commonly used than the t-test in real-world research, it is fundamental to understanding hypothesis testing.

You can also calculate changes using our T-Test Calculator, Test Statistic Calculator or Degrees of Freedom Calculator.

Frequently Asked Questions

When do I use a z-test instead of a t-test?

Use a z-test when the population standard deviation (sigma) is known. Use a t-test when you only have the sample standard deviation. In practice, t-tests are far more common.

Can I use a z-test with small samples?

Technically yes, if the population standard deviation is truly known. However, small samples with estimated standard deviations should always use the t-test.

What are the common critical z-values?

Two-tailed: z = 1.96 at alpha = 0.05, z = 2.576 at alpha = 0.01. One-tailed: z = 1.645 at alpha = 0.05, z = 2.326 at alpha = 0.01.

How do I interpret the z-statistic?

The z-statistic tells you how many standard errors the sample mean is from the hypothesized mean. Values beyond the critical z-value lead to rejection of the null hypothesis.

What is the relationship between z-tests and confidence intervals?

A 95% confidence interval uses z = 1.96. If the hypothesized mean falls outside this interval, the z-test at alpha = 0.05 will reject. The confidence interval and hypothesis test always agree.

Can I embed this Z-Test Calculator on my website?

Yes. Click the "Embed" button at the top of this page to customize the size, colors, and theme, then copy the iframe code. Paste it into any HTML page, WordPress site, or CMS. It is completely free, requires no signup, and works on all devices. You can also visit our embed guide at calculory.com/services/embed-calculators for more details.

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