P-Value from Z-Score: formula
Significance thresholds are conventions, and results near 0.05 are fragile. Testing many hypotheses raises the chance of false positives, which multiple-comparison corrections address. The z-based calculation assumes known variance or large samples, and small samples call for the t distribution.
Publication and choice of tail can bias conclusions.
P-Value from Z-Score: worked example
| Z-score | 2 |
|---|---|
| Type of test | Two-tailed |
| p-value | 0.05 |
| p-value as a percent | 5.0% |
Enter the z-score from your test and choose the type of test: left-tailed if you test whether a value is lower than expected, right-tailed if higher, and two-tailed if you test for any difference. Pick the tail before you see the result, based on your hypothesis. The z-score comes from your test statistic, which assumes a normal distribution or a large sample.
The tool converts it to a probability.
P-Value from Z-Score: changing Z-score
| Z-score | p-value | p-value as a percent |
|---|---|---|
| 1.6 | 0.116 | 11.6% |
| 1.8 | 0.078 | 7.8% |
| 2 (example) | 0.05 | 5.0% |
| 2.2 | 0.031 | 3.1% |
| 2.4 | 0.019 | 1.9% |
P-Value from Z-Score: compare input ranges
| Input | Tested from | To | p-value: low input | p-value: high input | Swing |
|---|---|---|---|---|---|
| Type of test | Left-tailed | Two-tailed | 0.975 | 0.05 | 0.925 (1900%) |
| Z-score | 1.8 | 2.2 | 0.078 | 0.031 | 0.048 (95%) |
P-Value from Z-Score: individual input comparisons
| Input | Value | p-value | p-value as a percent |
|---|---|---|---|
| Z-score | 1.6 | 0.116 | 11.6% |
| 1.8 | 0.078 | 7.8% | |
| 2 | 0.05 | 5.0% | |
| 2.2 | 0.031 | 3.1% | |
| 2.4 | 0.019 | 1.9% | |
| Type of test | Left-tailed | 0.975 | 97.5% |
| Right-tailed | 0.025 | 2.5% | |
| Two-tailed | 0.05 | 5.0% |
P-Value from Z-Score: which assumptions move the result?
P-Value from Z-Score: values and definitions
| Quantity and definition | Value used |
|---|---|
| Z-score — The test statistic from your data. Negative values are allowed. | 2 |
| Type of test — Use two-tailed unless you decided on a direction before you saw the data. | Two-tailed |
| p-value | 0.05 |
| p-value as a percent | 5.0% |
Common error: 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.
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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.
Decide the type of test and the threshold before running it. Picking the tail after seeing the sign of the result inflates false positives.
Find the score on the z-score calculator, the area under the curve on the normal probability page, and plan the study on the sample size calculator. The A/B test significance page applies p-values to conversion rates, and the confidence interval calculator shows the effect size range.
How should I read a p-value of 0.03?
If there were truly no effect, results as extreme as yours would occur about 3% of the time by chance.
Is p less than 0.05 proof of an effect?
No. It is evidence against the null hypothesis, and it does not show the size or importance of the effect.
When should I use a two-tailed test?
When you want to detect a difference in either direction. Use one-tailed tests only when you decided in advance that only one direction matters.