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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

The average of your sample data

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

The number of observations in your sample

The population mean under the null hypothesis

The threshold for statistical significance (commonly 0.05)

Result

Enter values above and click Calculate to see your result.

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Formula

z = (x-bar - mu0) / (sigma / sqrt(n))

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.

Worked Example

Sample mean = 105, sigma = 15, n = 64, mu0 = 100, alpha = 0.05 Step 1: Standard error = 15 / sqrt(64) = 15 / 8 = 1.875 Step 2: z = (105 - 100) / 1.875 = 2.667 Step 3: For a two-tailed test, p-value = approximately 0.0077 Step 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.

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