Z-Score to P-Value Calculator
Enter your z score, choose the test direction, and get the p value instantly.
What is a p value from a z score?
A p value from a z score tells you how extreme your observed result is under the standard normal distribution. In hypothesis testing, the z score measures how many standard deviations your statistic is from the null hypothesis mean. The p value converts that distance into a probability.
If the p value is small, your result would be unlikely under the null hypothesis, which is why people often compare p to thresholds like 0.05, 0.01, or 0.001.
How this calculator works
The calculator uses the standard normal cumulative distribution function (CDF), usually written as Φ(z):
From that CDF, p values are computed as:
- Left-tailed:
p = Φ(z) - Right-tailed:
p = 1 - Φ(z) - Two-tailed:
p = 2 × (1 - Φ(|z|))
This page computes Φ(z) numerically using a standard error-function approximation, which is accurate for practical statistics use.
When to choose one-tailed vs two-tailed
Two-tailed test
Use this when your alternative hypothesis is “different” (not specifically greater or smaller). Example: “The new process has a mean that differs from 50.”
Left-tailed test
Use when your alternative is specifically “less than.” Example: “Mean battery life is less than 10 hours.”
Right-tailed test
Use when your alternative is specifically “greater than.” Example: “Conversion rate is greater than 3%.”
Quick example
Suppose your z score is 2.10.
- Left-tail p ≈ Φ(2.10) ≈ 0.9821
- Right-tail p ≈ 1 − Φ(2.10) ≈ 0.0179
- Two-tail p ≈ 2 × 0.0179 ≈ 0.0358
At α = 0.05, a two-tailed p of 0.0358 would typically be considered statistically significant.
Common z scores and two-tailed p values
- z = 1.645 → p ≈ 0.10 (two-tailed)
- z = 1.96 → p ≈ 0.05 (two-tailed)
- z = 2.576 → p ≈ 0.01 (two-tailed)
- z = 3.291 → p ≈ 0.001 (two-tailed)
How to interpret the output
Once you calculate:
- p < 0.05: commonly called statistically significant evidence against H₀
- p < 0.01: stronger evidence
- p ≥ 0.05: not enough evidence to reject H₀ at the 5% level
Remember: p values do not tell you effect size, practical importance, or the probability that H₀ is true. They only describe data extremeness under H₀.
Frequently asked questions
Can p values be exactly zero?
Theoretical p values are greater than zero. For very large |z|, software may display an extremely tiny value in scientific notation.
Can I use this for a t test?
Not directly. This calculator assumes a standard normal z distribution. For small samples or unknown population variance, use a t-distribution calculator.
Is this the same as a normal distribution calculator?
It’s a specialized normal distribution calculator focused on converting a z score to one-tailed or two-tailed p values quickly.