Calculator Inputs
Formula Used
1) Standardization from a raw value:
z = (x - μ) / σ
2) Z-test for a sample mean:
z = (x̄ - μ₀) / (σ / √n)
3) Tail probabilities from the standard normal distribution:
Left-tail p = Φ(z)
Right-tail p = 1 - Φ(z)
Two-tail p = 2 × min[Φ(z), 1 - Φ(z)]
4) Decision rule:
Reject the null hypothesis when p ≤ α.
This calculator assumes a normal model or a normal sampling distribution. The sample mean mode uses a z-based procedure with a known population standard deviation.
How to Use This Calculator
- Choose a calculation mode based on your available inputs.
- Select the correct tail type for your hypothesis.
- Enter the significance level, usually 0.05 or 0.01.
- Provide the needed values for the chosen mode.
- Press Calculate p-value to display the result above the form.
- Review the p-value, z-score, critical rule, and final decision.
- Use the graph to see the shaded probability region.
- Download the result as CSV or PDF when needed.
Example Data Table
| Scenario | Mode | Inputs | Tail | Z-score | P-value | Decision at α = 0.05 |
|---|---|---|---|---|---|---|
| Benchmark z case | Direct z-score | z = 1.96 | Two-tailed | 1.9600 | 0.0500 | Borderline significance |
| High observation | Raw value | x = 72, μ = 65, σ = 4 | Right-tailed | 1.7500 | 0.0401 | Reject null |
| Low observation | Raw value | x = 58, μ = 60, σ = 5 | Left-tailed | -0.4000 | 0.3446 | Fail to reject |
| Sample mean test | Sample mean | x̄ = 104, μ₀ = 100, σ = 12, n = 36 | Two-tailed | 2.0000 | 0.0455 | Reject null |
| Strong upper-tail signal | Sample mean | x̄ = 51.5, μ₀ = 50, σ = 3, n = 25 | Right-tailed | 2.5000 | 0.0062 | Reject null |
FAQs
1) What does the p-value show?
It measures how extreme your observed result is under the null hypothesis. Smaller values indicate stronger evidence against the null assumption.
2) When should I choose a left-tailed test?
Choose left-tailed when your alternative hypothesis claims the value is smaller than the null benchmark, such as μ < μ₀.
3) When should I use a two-tailed test?
Use two-tailed when deviations in either direction matter. It tests whether the observed value is significantly different, not just higher or lower.
4) What is the difference between raw mode and sample mean mode?
Raw mode converts one observation into a z-score. Sample mean mode tests an average using the sampling distribution and sample size.
5) Does this calculator use a z-test or t-test?
It uses z-based normal calculations. The sample mean mode assumes the population standard deviation is known or the normal model is appropriate.
6) What does α represent?
Alpha is the significance level. It is the threshold used to compare the p-value and make the final reject or fail-to-reject decision.
7) Why does the graph shade only part of the curve?
The shaded area represents the probability region tied to your selected hypothesis tail and observed statistic. That shaded area equals the p-value.
8) Can I export the results?
Yes. After calculation, use the CSV button for tabular output or the PDF button for a shareable report snapshot.