P-Value Calculator
Calculate p-values from z-scores and t-scores for hypothesis testing
For one-sample t-test: df = n - 1
Formula:
P-value = 2 × (1 - CDF(||))
P-value = CDF()
P-value = 1 - CDF()
P-Value
At α = 0.05:
Significance Scale
P-Value
As Percentage
Confidence Level
1 in X Chance
Significance at Different α Levels
| Significance Level (α) | Confidence Level | Decision | Result |
|---|---|---|---|
Critical Z-Values Reference
| α Level | Two-Tailed Critical Z | One-Tailed Critical Z |
|---|---|---|
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About P-Value Calculator
What is a P-Value?
A p-value (probability value) is the probability of obtaining test results at least as extreme as the observed results, assuming that the null hypothesis is true. It's a fundamental concept in statistical hypothesis testing.
Key Points:
- Lower p-values indicate stronger evidence against the null hypothesis
- P-values range from 0 to 1
- A p-value does NOT measure the probability that the null hypothesis is true
How to Calculate P-Values
From Z-Score (Standard Normal Distribution)
Two-tailed test:
P-value = 2 × (1 - Φ(|z|))
Left-tailed test:
P-value = Φ(z)
Right-tailed test:
P-value = 1 - Φ(z)
Where Φ(z) is the cumulative distribution function (CDF) of the standard normal distribution.
From T-Score (Student's t-Distribution)
T-tests are used when:
- Sample size is small (n < 30)
- Population standard deviation is unknown
The p-value is calculated using the t-distribution CDF with (n-1) degrees of freedom.
Significance Levels (α)
Common significance levels and their interpretations:
| α Level | P-value Threshold | Confidence Level | Interpretation |
|---|---|---|---|
| 0.10 | p < 0.10 | 90% | Marginally significant |
| 0.05 | p < 0.05 | 95% | Statistically significant |
| 0.01 | p < 0.01 | 99% | Highly significant |
| 0.001 | p < 0.001 | 99.9% | Very highly significant |
How to Interpret P-Values
Decision Rules
- If p-value ≤ α: Reject the null hypothesis. The result is statistically significant.
- If p-value > α: Fail to reject the null hypothesis. The result is not statistically significant.
Common Interpretations
- p < 0.001: Very strong evidence against null hypothesis
- p < 0.01: Strong evidence against null hypothesis
- p < 0.05: Moderate evidence against null hypothesis
- p < 0.10: Weak evidence against null hypothesis
- p ≥ 0.10: Little or no evidence against null hypothesis
One-Tailed vs Two-Tailed Tests
Two-Tailed Test
- Tests for any difference (greater or smaller)
- Use when: "Is there a difference?"
- P-value considers both tails of the distribution
One-Tailed Test (Left)
- Tests if the parameter is less than the hypothesized value
- Use when: "Is it less than?"
One-Tailed Test (Right)
- Tests if the parameter is greater than the hypothesized value
- Use when: "Is it greater than?"
Frequently Asked Questions
What is a good p-value?
There's no universal "good" p-value. Whether a p-value is meaningful depends on your field, study design, and the consequences of errors. Traditionally, p < 0.05 is considered statistically significant.
Can a p-value be negative?
No. P-values are probabilities and always range from 0 to 1.
What does p = 0.05 mean?
A p-value of 0.05 means there's a 5% chance of obtaining results as extreme as observed, if the null hypothesis were true.
Why use 0.05 as a threshold?
The 0.05 threshold is a convention introduced by Ronald Fisher. It represents a balance between avoiding false positives (Type I errors) and false negatives (Type II errors).
What's the difference between p-value and significance level?
The significance level (α) is chosen before the study and represents the threshold for rejecting the null hypothesis. The p-value is calculated from the data and compared to α.