P-Value Calculator
A z-score of 1.96 gives a two-tailed p-value of 0.05, exactly the classic significance cutoff. This p-value calculator converts any z-score into a p-value under the standard normal distribution, for two-tailed, left-tailed, or right-tailed tests. It also tells you at a glance whether the result clears the 0.05 and 0.01 significance levels.
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 1.96 standard deviations holds 2.5 percent of outcomes.
A strong result
Three standard deviations out clears both the 0.05 and 0.01 bars comfortably.
A directional test
A one-tailed test at z = 1.65 just clears 0.05, while the two-tailed version (0.099) does not.
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 |
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.