P-Value Calculator
A z-score of 1.96 gives an approximate two-tailed p-value of 0.049996 with this calculator. This p-value calculator converts a finite z-score into a left-tailed, right-tailed, or two-tailed probability under the standard normal distribution. It also reports cumulative probability and whether the unrounded result is below 0.05 or 0.01. A p-value describes compatibility between observed data and a specified null model. It does not measure the probability that the null hypothesis is true, the chance that a finding will replicate, or the practical size of an effect. Choose the tail from the hypothesis and analysis plan, then interpret the result with study design, assumptions, effect size, uncertainty, and the full body of evidence.
Quick answer
A p-value is the probability of seeing a result at least as extreme as yours if the null hypothesis were true.
P-value
0.049996
Cumulative probability
0.975002
Significant at 0.05
Yes
Significant at 0.01
No
What this tells you
- •A p-value is the probability of seeing a result at least as extreme as yours if the null hypothesis were true.
- •The z-score measures how many standard deviations your result sits from the null expectation.
- •A two-tailed test counts extreme results in both directions, so its p-value is double the one-tailed value.
- •Smaller p-values mean stronger evidence against the null. Below 0.05 is the common significance bar.
How to Use
- 1Enter your z-score, positive or negative.
- 2Choose the tail type. Use two-tailed unless your hypothesis specified a direction in advance.
- 3Read the p-value, with significance verdicts at the 0.05 and 0.01 levels.
- 4The cumulative probability below shows the percentile of your z-score.
How It Works
Formula
p (two-tailed) = 2 x (1 - CDF(|z|))The standard normal CDF gives the probability of a value at or below z. For a right-tailed test the p-value is 1 minus the CDF, for a left-tailed test it is the CDF itself, and a two-tailed test doubles the right-tail area of the absolute z. At z = 1.96 the right tail holds 2.5 percent, so both tails together give p = 0.05.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
The classic 1.96 threshold
Each tail beyond an absolute z of 1.96 contains about 0.024998 under this approximation. Doubling that area gives the displayed two-tailed result 0.049996.
A strong result
The two-tailed area is about 0.002699935 before rounding. The displayed result is 0.002700, and the unrounded value is below both thresholds.
A directional test
The right-tail area above z = 1.65 is about 0.049471. A two-tailed calculation would be about 0.098943.
Z-Scores and Two-Tailed P-Values
Common z-score thresholds and their two-tailed p-values.
| Z-score | Two-tailed p | Verdict at 0.05 |
|---|---|---|
| 1.00 | 0.3173 | Not significant |
| 1.65 | 0.0989 | Not significant |
| 1.96 | 0.0500 | Borderline |
| 2.58 | 0.0099 | Significant |
| 3.00 | 0.0027 | Significant |
| 3.29 | 0.0010 | Significant |
Interpreting a normal-theory p-value
A p-value is conditional on the null hypothesis and statistical model. It asks how unusual the chosen statistic, or something more extreme, would be if those assumptions held. It does not reverse that condition to give the probability that the null is true. That reverse probability requires a different framework and additional assumptions.
Tail choice belongs in the analysis plan. A right-tailed test looks for unusually large statistics, while a left-tailed test looks for unusually small ones. A two-tailed test counts departures in both directions. Selecting a one-tailed option only after seeing the sign of the z-score can understate uncertainty.
Thresholds are reporting conventions, not physical boundaries. Results just below and just above 0.05 provide nearly the same evidence. The labels apply strict comparisons to the unrounded p-value, but interpretation should consider the exact value, effect estimate, confidence interval, study power, measurement quality, missing data, and analyses performed.
Multiple testing changes the chance of false positives. If many hypotheses, outcomes, subgroups, or stopping points were examined, a small p-value can appear by chance. This calculator converts one z-score and does not adjust for multiplicity, selective reporting, repeated looks at data, clustering, or dependence between tests.
A z-based result is appropriate only when the statistic follows or is well approximated by a standard normal distribution under the null. Small samples, estimated standard errors, sparse counts, paired observations, and complex surveys may require a t distribution, exact test, permutation method, regression model, or design-specific procedure.
Statistical significance does not establish practical importance. A small effect can yield a low p-value in a large sample, while an important estimate can remain uncertain in a small sample. Report the effect magnitude and its uncertainty alongside the p-value whenever possible.
Common mistakes
- Choosing a one-tailed test after seeing the data. The direction must be specified before the experiment, or the p-value is misleading.
- Reading the p-value as the probability the null hypothesis is true. It is the probability of the data given the null, which is a different thing.
- Treating p = 0.049 and p = 0.051 as fundamentally different results. The 0.05 line is a convention, not a law of nature.
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