Calculator
Example Data Table
| Case | Sample Mean | Hypothesized Mean | Sigma | n | Alpha | Alternative |
|---|---|---|---|---|---|---|
| Factory fill weight | 52 | 50 | 10 | 36 | 0.05 | Not equal |
| Battery life claim | 9.8 | 10 | 0.6 | 64 | 0.01 | Less |
| Exam score target | 76 | 74 | 8 | 49 | 0.05 | Greater |
Formula Used
Standard error: SE = σ / √n
Z score: z = (x̄ − μ₀) / SE
Two-tailed p value: p = 2 × P(Z ≥ |z|)
Right-tailed p value: p = P(Z ≥ z)
Left-tailed p value: p = P(Z ≤ z)
Confidence interval: x̄ ± zcritical × SE
Effect size: d = (x̄ − μ₀) / σ
How to Use This Calculator
- Enter the sample mean.
- Enter the hypothesized population mean.
- Enter the known population standard deviation.
- Enter the sample size.
- Select alpha and the alternative hypothesis.
- Add raw data only when you want automatic mean and size.
- Enter a true mean when power is needed.
- Press the calculate button.
- Download the result as CSV or PDF.
Understanding the One Sample Z-Test
A one sample z-test compares a sample mean with a fixed population mean. It is useful when the population standard deviation is known. The method is common in quality control, education, health studies, and business reports. It helps decide whether a measured average is different from a claimed value.
When to Use This Test
Use this test when observations are independent. The variable should be numeric. The population standard deviation must be known or trusted from reliable past data. The sample should be random. A large sample makes the normal model more reasonable. For small samples, the population should be close to normal.
What the Result Means
The calculator reports the z score, standard error, p value, critical value, and confidence interval. The z score shows how far the sample mean is from the hypothesized mean. It uses standard error units. A large absolute z score gives stronger evidence against the null hypothesis. The p value gives the probability of results this extreme, assuming the null hypothesis is true.
Decision Rules
Choose a significance level before testing. Common values are 0.10, 0.05, and 0.01. If the p value is less than or equal to alpha, reject the null hypothesis. If it is greater than alpha, do not reject it. This does not prove the null is true. It only means the sample did not give enough evidence.
One-Tailed and Two-Tailed Choices
A two-tailed test checks for any difference. A left-tailed test checks whether the true mean is lower. A right-tailed test checks whether the true mean is higher. Pick the alternative before seeing the result. Changing it later can mislead the conclusion.
Practical Use
Statistical significance is not the whole story. Review the mean difference and confidence interval. They show the size and likely range of the effect. Also check data quality, sampling method, and context. A small p value can still describe a tiny change. A larger p value may appear with a small sample. Good reporting combines the test result with practical judgment.
Limits to Remember
The test assumes a known sigma. It also assumes independent data. Outliers can distort the mean. Always inspect values before trusting the final decision.
FAQs
What is a one sample 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 calculator?
Use it when your data is numeric, the sample is independent, and the population standard deviation is known or reliably estimated from historical information.
What does the z score mean?
The z score measures how many standard errors the sample mean is away from the hypothesized mean. Larger absolute values show stronger disagreement.
What is the p value?
The p value is the probability of getting a result this extreme, assuming the null hypothesis is true. Smaller values give stronger evidence.
What does alpha mean?
Alpha is the selected significance level. It is the cutoff used for deciding whether the p value is small enough to reject the null hypothesis.
Can I enter raw data?
Yes. Enter values separated by commas, spaces, or semicolons. The calculator will compute the sample mean and sample size automatically.
Does this calculator replace a t-test?
No. Use a t-test when the population standard deviation is unknown and must be estimated from the sample standard deviation.
What does power mean here?
Power is the chance of rejecting the null hypothesis when a selected true mean is correct. Enter a true mean to estimate it.