One Mean Z Test Calculator

Calculator inputs

Hypothesized mean

Sample mean

Population standard deviation

Sample size

Alpha level

Alternative hypothesis

Confidence level

Population size

True mean for power

Measurement label

Decimal places

Use raw data below

Raw sample data

Raw data overrides sample mean and sample size.

Example data table

Claimed mean	Sample mean	Population standard deviation	Sample size	Alpha	Tail	Expected result
100	104.2	12	36	0.05	Two tailed	z = 2.1000, p near 0.0357
50	47.8	8	64	0.01	Left tailed	z = -2.2000, compare with alpha
25	26.1	3.5	49	0.05	Right tailed	z = 2.2000, compare with alpha

Formula used

Z statistic: z = (x̄ - μ₀) / SE

Standard error: SE = σ / √n

Finite population correction: SE = (σ / √n) × √((N - n) / (N - 1))

Two tailed p value: p = 2 × min(Φ(z), 1 - Φ(z))

Right tailed p value: p = 1 - Φ(z)

Left tailed p value: p = Φ(z)

Confidence interval: x̄ ± z_critical × SE

How to use this calculator

Enter the claimed population mean in the hypothesized mean box.
Enter the sample mean, population standard deviation, and sample size.
Select the correct alternative hypothesis before running the test.
Enter alpha as a decimal, such as 0.05, or as 5.
Add population size only when finite population correction is needed.
Add a true mean only when you want a power estimate.
Press Submit to show the result above the form.
Use CSV or PDF buttons to save the output.

Article: One Mean Z Test Guide

What the test measures

A one mean z test checks one sample mean against a claimed population mean. It is useful when the population standard deviation is known. It also works best when the sample is large or the population is near normal.

The test compares two values. The first value is the sample mean. The second value is the hypothesized mean. The difference is divided by the standard error. This creates the z statistic. A large positive or negative z value shows stronger evidence against the claim.

Test options and output

This calculator supports left tailed, right tailed, and two tailed tests. It also accepts raw sample data. When data is entered, the tool counts values and finds the mean. The population standard deviation is still required. That condition keeps the method a true z test.

Alpha controls the rejection rule. Common alpha values are 0.10, 0.05, and 0.01. A smaller alpha needs stronger evidence. The calculator returns the p value, critical values, decision, margin of error, and confidence interval. It can also estimate power when a true mean is supplied.

The confidence interval gives a practical range for the population mean. It uses the same standard error. It is not the same as the hypothesis decision. A test answers a claim. An interval estimates likely values. Using both views improves reporting.

Input quality matters

Finite population correction is included for surveys. Use it only when sampling without replacement from a limited population. It reduces the standard error when the sample is a large part of the population. Leave it blank for most textbook problems.

Good input matters. Use a known population standard deviation. Do not enter a sample standard deviation unless your assignment allows it. Use a one sample t test when the population standard deviation is unknown. Check units before comparing values.

Export and reporting

The export buttons help document work. CSV is useful for spreadsheets. PDF is useful for reports. The example table shows typical entries. Review the formula section before using results in research, audits, production checks, or class assignments. Always state the test direction before reading the p value. Keep alpha fixed before seeing results. Report the z statistic with rounding. Also show the practical conclusion in plain words for readers and decision makers clearly.

FAQs

What is a one mean z test?

It is a hypothesis test for one population mean. It compares a sample mean with a claimed mean when the population standard deviation is known.

When should I use this test?

Use it when you have one sample, a known population standard deviation, and a mean claim. The sample should be large or come from a near normal population.

What does the p value mean?

The p value shows how unusual the sample result is under the null hypothesis. Smaller values give stronger evidence against the claimed mean.

What alpha should I choose?

Many problems use 0.05. Stricter studies may use 0.01. Choose alpha before viewing results to avoid biased decisions.

What is a two tailed test?

A two tailed test checks whether the population mean is different from the claimed mean. It looks for evidence in both directions.

Can I enter raw data?

Yes. Check the raw data option and enter values separated by commas, spaces, or new lines. The calculator will compute the mean and sample size.

Why is population standard deviation required?

A z test assumes the population standard deviation is known. Use a one sample t test when only the sample standard deviation is available.

What does finite population correction do?

It adjusts the standard error when sampling without replacement from a limited population. Use it only when the sample is a meaningful part of the population.