P-Value Calculator

Calculate exact p-values from z-test statistics for one-tailed and two-tailed hypothesis tests with automated statistical significance determinations and alpha comparisons.

Hypothesis Test Parameters

Live calculation
Preset Scenarios:

Two-tailed tests check for deviations in both directions, making them more conservative than one-tailed tests.

Calculated P-Value (p)
0.049996
Hypothesis Test Decision
Reject H₀
Evidence: Moderate evidence against the null hypothesis (p < 0.05)Formula: p = 2 × [1 − Φ(|z|)]
Z-Score (z)1.96
Tail TypeTwo-Tailed
Significance α0.05
Result StatusSignificant

Statistical Inference Summary

With a computed p-value of 0.049996 and a significance threshold of α = 0.05, the test concludes to reject the null hypothesis (statistically significant) because 0.05 < 0.05.

What Is a P-Value in Hypothesis Testing?

In inferential statistics, a p-value quantifies the level of statistical evidence against a null hypothesis (H₀). It represents the probability of observing data at least as extreme as your sample findings assuming the null hypothesis is true.

1. Two-Tailed Hypothesis Test (H₁: μ ≠ μ₀)
p=2 × [ 1 − Φ(|z|) ]
2. One-Tailed Directional Tests (Left / Right)
Left Tail: p = Φ(z)|Right Tail: p = 1 − Φ(z)
Step-by-Step Calculation Breakdown (Example: z = 2.10, Two-Tailed Test, α = 0.05)
Step 1: Evaluate Normal Cumulative Distribution Function Φ(|z|)
• For z = 2.10, cumulative probability Φ(2.10) = 0.982136
Step 2: Calculate Single Tail Probability (1 − Φ)
• Upper Tail Area = 1 − 0.982136 = 0.017864
Step 3: Double for Two-Tailed Test and Compare with Alpha
• Two-Tailed p-value = 2 × 0.017864 = 0.035728 (3.57%)
• Since p (0.0357) < α (0.05), we Reject H₀.

Decision Rule Summary

  • If p ≤ α: Reject the null hypothesis H₀. The result is statistically significant.
  • If p > α: Fail to reject the null hypothesis H₀. There is insufficient evidence of an effect.

Frequently Asked Questions

What is a p-value in statistical testing?
A p-value is the probability of obtaining test results at least as extreme as the results actually observed during the test, assuming that the null hypothesis is completely correct. A low p-value (e.g. p < 0.05) indicates that observed data would be very unlikely under the null hypothesis, justifying rejection of the null.
What is the difference between one-tailed and two-tailed tests?
A one-tailed test evaluates an effect in a single designated direction (greater than OR less than), whereas a two-tailed test evaluates for differences in either direction. Two-tailed tests divide significance across both curve tails and are standard in most scientific research.
What does p < 0.05 actually mean?
It means there is less than a 5% probability that the observed sample effect occurred strictly due to random sampling chance under the null hypothesis. It does not prove the alternative hypothesis, but provides strong evidence against the null.
Can a p-value prove a hypothesis?
No. A p-value measures evidence against the null hypothesis model; it does not measure effect size, physical magnitude, or practical significance. Always interpret p-values alongside confidence intervals and effect sizes.
Why does sample size impact p-values?
As sample size increases, standard error shrinks, making even tiny, practically negligible differences statistically significant with very small p-values. That is why effect size must always be reported alongside p-values.
How are critical alpha levels chosen?
Common alpha thresholds are 0.05 (general research), 0.01 (medical and high-stakes trials), and 0.10 (exploratory studies). Alpha represents your maximum acceptable risk of making a Type I error (false positive).

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