Answer first
What this calculator tells you
Calculate the p-value for a z-score in a left-tailed, right-tailed or two-tailed test. Turn a test statistic into the probability of a result at least as extreme under the null hypothesis. Formula: P(Z ≤ z) = Φ(z); right tail = 1 − Φ(z); two-tailed p = 2 × (1 − Φ(|z|)). At the worked-example inputs, the p-value is 0.05. Holding every other input steady, moving z-score from 1.6 to 2.4 moves the result from 0.019 to 0.116.
Transparent method
The formula
Turn a test statistic into the probability of a result at least as extreme under the null hypothesis.
Worked example
Example inputs
How to interpret the result
A p-value is the chance of seeing a result at least as extreme as yours if nothing were going on. A z-score of 1.96 gives a two-tailed p of about 0.05, which is why 1.96 is the familiar cutoff. The tail choice matters: a right-tailed test halves that figure, but only if you chose the direction before you saw the data.
At the worked-example inputs the p-value is 0.05. It falls as type of test and z-score increase.
These are standard formulas from introductory statistics. Real-world inference depends on assumptions, normality, independence, a genuinely random sample, that this calculator does not check for you.
Before you rely on it
What to check
Decide the type of test and the threshold before running it. Picking the tail after seeing the sign of the result inflates false positives.
The common error
Where people go wrong with p-value from z-score calculator
Reading the p-value as the probability that the hypothesis is true. It is the probability of the data under the null hypothesis, which is a different thing.
Sensitivity evidence
How z-score changes the p-value
Holding every other input at the worked-example value, moving z-score from 1.6 to 2.4 moves the p-value from 0.019 to 0.116: a spread of 0.098, or 195% of the worked-example result.
| Z-score | p-value | p-value as a percent |
|---|---|---|
| 1.6 | 0.116 | 11.6% |
| 1.8 | 0.078 | 7.8% |
| 2worked example | 0.05 | 5.0% |
| 2.2 | 0.031 | 3.1% |
| 2.4 | 0.019 | 1.9% |
Every input, tested
Which input moves the p-value most
Of the 2 inputs, type of test moves the p-value most (0.925 across the range tested) and z-score moves it least (0.048).
| Input | Tested from | To | p-value at each end | Swing |
|---|---|---|---|---|
| Type of test | Left-tailed | Two-tailed | 0.975 to 0.05 | 0.925 (1900%) |
| Z-score | 1.8 | 2.2 | 0.078 to 0.031 | 0.048 (95%) |
Step by step
The worked example, input by input
| Input | Value used | What it means |
|---|---|---|
| Z-score | 2 | The test statistic from your data. Negative values are allowed. |
| Type of test | Two-tailed | Use two-tailed unless you decided on a direction before you saw the data. |
| p-value | 0.05 | |
| p-value as a percent | 5.0% | |
Inputs, definitions and assumptions
Z-score
The test statistic from your data. Negative values are allowed. The prefilled worked-example value is 2.
Type of test
Use two-tailed unless you decided on a direction before you saw the data. The prefilled worked-example value is Two-tailed.
How to use this calculator
- 1Verify the inputs. Gather z-score and type of test from your own documents; the prefilled values are examples.
- 2Save a baseline. The worked example puts the p-value at 0.05. Store your own version of it as Scenario A.
- 3Test one change. Start with type of test, the input with the biggest effect here: switching type of test changes the p-value: left-tailed gives 0.975; right-tailed gives 0.025; two-tailed gives 0.05.
- 4Check the boundary. Read the interpretation boundary above before acting on the result.
People also ask
Frequently asked questions
How do you calculate p-value from z-score?
P(Z ≤ z) = Φ(z); right tail = 1 − Φ(z); two-tailed p = 2 × (1 − Φ(|z|)). At the worked-example inputs the p-value is 0.05.
What does the p-value from z-score result mean?
Turn a test statistic into the probability of a result at least as extreme under the null hypothesis. At the worked-example inputs the p-value is 0.05. It falls as type of test and z-score increase.
How much does z-score change the p-value?
Holding every other input at the worked-example value, moving z-score from 1.6 to 2.4 moves the p-value from 0.019 to 0.116, a spread of 0.098.
What are the limits of this p-value from z-score calculator?
These are standard formulas from introductory statistics. Real-world inference depends on assumptions, normality, independence, a genuinely random sample, that this calculator does not check for you. The tables on this page test z-score only from 1.6 to 2.4; a value outside that range is not tabulated here.
Which input moves the p-value most in the p-value from z-score calculator?
Ranked by how far each moves the p-value across the range tested: type of test (0.925, 1900%) and z-score (0.048, 95%).
How much does type of test matter in the p-value from z-score calculator?
The worked example uses Two-tailed. Holding every other input at its worked-example value, switching type of test changes the p-value: left-tailed gives 0.975; right-tailed gives 0.025; two-tailed gives 0.05.
Which inputs change the p-value as a percent in the p-value from z-score calculator?
At the worked-example inputs it is 5.0%. Z-score takes it from 7.8% to 3.1% and type of test takes it from 97.5% to 5.0%.
Why can a positive test on a good test still be unlikely to be right?
When the condition is rare, most positives come from the large group without it. Ninety percent sensitivity and a 5 percent false positive rate can still leave a positive result at about a 15 percent chance of the condition if only 1 percent of people have it.
What does a p-value of 0.05 mean?
It is the chance of seeing a result at least this extreme if there were no real effect. It is not the chance that the effect is real, and a result just under 0.05 is weak evidence on its own.
Sources and evidence
Free Calculators Online is independent and is not affiliated with or endorsed by the source organizations. Educational estimates only.