Z-Test Calculator
One-sample z-test with p-value and confidence interval.
Enter your values
Enter the sample mean you are working with.
Enter the population mean μ you are working with.
Enter the population sd σ you are working with.
Enter the sample size you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- z statistic
- 1.6
- p-value (two-tailed)
- 0.1096
- p-value (one-tailed)
- 0.0548
- Verdict
- Not significant at the 5% level
- Standard error
- 2.5
- 95% CI for the mean
- 99.1 to 108.9
Quick answer
The z-test uses z = (x̄ − μ) ÷ (σ ÷ √n) when the population standard deviation is known, comparing z against the normal distribution.
z = (x̄ − μ) ÷ (σ ÷ √n)
At a glance
| What it does | One-sample z-test with p-value and confidence interval. |
|---|---|
| Category | Statistics |
| Inputs needed | Sample mean, Population mean μ, Population SD σ, Sample size |
| Main output | z statistic |
| Formula | z = (x̄ − μ) ÷ (σ ÷ √n) |
| Cost | Free — no sign-up, no download |
How to use the Z-Test Calculator
- 1Enter your sample mean and sample size.
- 2Enter the population mean you are testing against and the known population SD.
- 3Read the two-tailed p-value unless you have a directional hypothesis stated in advance.
Inputs explained
- Sample mean
- Enter the sample mean you are working with.
- Population mean μ
- Enter the population mean μ you are working with.
- Population SD σ
- Enter the population sd σ you are working with.
- Sample size
- Enter the sample size you are working with.
Worked example
Using the values the calculator loads with:
Inputs
- Sample mean104
- Population mean μ100
- Population SD σ15
- Sample size36
Results
- z statistic1.6
- p-value (two-tailed)0.1096
- p-value (one-tailed)0.0548
- VerdictNot significant at the 5% level
- Standard error2.5
- 95% CI for the mean99.1 to 108.9
Frequently asked questions
z-test or t-test?
Use z when the population SD is genuinely known — rare outside textbooks and quality control. Otherwise use a t-test.
Is n = 30 the cutoff?
It is a rule of thumb for normality of the sampling distribution, not a licence to ignore an unknown sigma.
What do I need to enter into the Z-Test Calculator?
Just 4 values: sample mean, population mean μ, population sd σ, sample size. Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Z-Test Calculator work out the answer?
The z-test uses z = (x̄ − μ) ÷ (σ ÷ √n) when the population standard deviation is known, comparing z against the normal distribution. It applies the formula z = (x̄ − μ) ÷ (σ ÷ √n) and shows the working so you can check each step by hand.
Is the Z-Test Calculator free, and do I need an account?
It is completely free with no sign-up, no download and no usage limit. Everything is calculated in your browser, so the numbers you type never leave your device.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.