Please provide any one value below to compute p-value from z-score or vice versa for a standard normal distribution.
In statistics, the p-value (probability value) is the probability of obtaining test results at least as extreme as the results observed during an experiment, assuming that the null hypothesis (H₀) is true.
A small p-value provides strong evidence against the null hypothesis, suggesting that the observed effect or difference is unlikely to be the result of random chance alone.
The most widely accepted threshold in scientific research. If p ≤ 0.05, there is less than a 5% probability that the result occurred by random chance.
Used in clinical trials and rigorous testing. If p ≤ 0.01, there is less than a 1% chance of a false positive result (Type I error).
Represents overwhelming evidence against the null hypothesis, with less than 1 in 1,000 odds of occurrence under H₀.
| Test Type | Hypothesis Statement | Rejection Region | Example Application |
|---|---|---|---|
| Two-Tailed Test | H₁: μ ≠ μ₀ (Any difference) | Both tails (α/2 in each tail) | Testing whether a new manufacturing method changes product weight. |
| Right-Tailed Test | H₁: μ > μ₀ (Increase) | Upper tail (α in right tail) | Testing whether a new fertilizer increases crop yield. |
| Left-Tailed Test | H₁: μ < μ₀ (Decrease) | Lower tail (α in left tail) | Testing whether a diet drug decreases blood cholesterol. |
By evaluating the Standard Normal Cumulative Distribution Function Φ(Z). For a right-tailed test, p = 1 - Φ(Z). For a two-tailed test, p = 2 × (1 - Φ(|Z|)).
A p-value of 0.03 means there is a 3% probability of observing data at least as extreme as yours if the null hypothesis were true. Since 0.03 is less than the standard 0.05 threshold, the result is considered statistically significant.
In theoretical continuous normal distributions, the probability density function has infinite tails, meaning the p-value is strictly between 0 and 1 (0 < p < 1), though it can be rounded to 0 or 1 for extreme test statistics.