Enter a Z-statistic (standard test statistic) and select the tail type and significance level α to compute the P-value and determine whether to reject the null hypothesis H₀.
How to use this calculator
Enter the Z-statistic from your hypothesis test.
Select whether the test is two-tailed, right-tailed, or left-tailed.
Set the significance level α.
Read the P-value and conclusion: reject H₀ if P-value < α.
Formula & method
P-value from normal CDF
Two-tailed: P = 2 × (1 − Φ(|z|)) · Right-tailed: P = 1 − Φ(z) · Left-tailed: P = Φ(z)
Φ(z) = cumulative standard normal distribution. Computed via polynomial approximation.
Example
z = 2.0, two-tailed: P = 2 × (1 − 0.9772) = 0.0456 < α=0.05 → Reject H₀.
P-value ≥ α → fail to reject H₀ (insufficient evidence).
A P-value of 0.03 at α = 0.05 means the result is significant. At α = 0.01, it is not.
A two-tailed test (P = 2 × P_one-tail) is conservative and standard for most tests with no directional hypothesis.
How to interpret your result
P-value meaning
The probability of observing a test statistic as extreme as (or more extreme than) the one computed, assuming H₀ is true. A small P-value means your result is unlikely under H₀.
Common misconception
The P-value is not the probability that H₀ is true. It is the probability of the data given H₀. 'P = 0.05' does not mean 95% chance the result is real.
Frequently asked questions
What is a P-value?
The probability of observing a result as extreme as (or more extreme than) the test statistic, assuming the null hypothesis H₀ is true.
When should I use a one-tailed vs two-tailed test?
Two-tailed when testing if a parameter differs from a value in either direction (most common). One-tailed when you have a directional hypothesis (specifically 'greater than' or 'less than').
What does 'reject H₀' mean?
There is sufficient statistical evidence to conclude that the null hypothesis is unlikely to be true. It does not prove the alternative is definitely true.
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