P-value Calculator
Calculate p-values from Z-scores or T-scores with one-tailed or two-tailed tests. Provides an English-language conclusion of statistical significance.
P-value
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Test Direction
One-tailed
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p-value
0.016586
Z-statistic = 2.13 | α = 0.01
○ Not Statistically Significant
Since p = 0.0166 ≥ α = 0.01, the results are not statistically significant. You fail to reject the null hypothesis.
Decision Rule
If p < α
Reject H₀
Statistically significant
If p ≥ α
Fail to Reject H₀
Not statistically significant
Test Direction
One-tailed
Test Type
Z-test
The Formula
The p-value is the probability of observing a test statistic at least as extreme as the one computed, assuming the null hypothesis is true. A small p-value (p < α) suggests the observed effect is statistically significant.
Variable Definitions
p-value
Probability of observing the data (or something more extreme) under the null hypothesis. Ranges from 0 to 1.
Alpha
The significance threshold (commonly 0.05). If p < α, the result is statistically significant. Set before collecting data.
Test Statistic
The calculated test statistic from the data. Z is for known population variance; T is for unknown variance (requires degrees of freedom).
Degrees of Freedom
A parameter of the t-distribution. Typically n − 1 for a single sample. Required for t-tests but not for z-tests.
How to Use This Calculator
- 1
Select Z-test or T-test based on your data type. Use Z when population standard deviation is known; use T when it is estimated from the sample.
- 2
Enter the test statistic (Z-score or T-score). For t-tests, also enter the degrees of freedom.
- 3
Choose your significance level (α) — 0.05 is standard for most research, 0.01 for stricter tests, 0.10 for exploratory work.
- 4
Select test direction: one-tailed (effect in one direction) or two-tailed (effect in either direction). Two-tailed is more conservative and more common.
- 5
View the p-value and a plain-English conclusion of statistical significance.
The p-value is the probability of observing data at least as extreme as the test statistic
Understanding the Concept
The p-value approach to hypothesis testing is used across scientific research to determine statistical significance. A p-value less than the chosen significance level (α) indicates that the observed result would be unlikely under the null hypothesis — suggesting the effect is real, not due to random chance. The lower the p-value, the stronger the evidence against the null hypothesis. However, the p-value is widely misunderstood. A common misconception is that the p-value is the probability that the null hypothesis is true — it is not. It is the probability of observing data at least as extreme as yours IF the null hypothesis were true. Also, statistical significance does not imply practical significance: a very large sample can detect a trivially small effect as statistically significant. Always consider the effect size alongside the p-value. The calculator computes p-values using numerical approximations of the normal and t-distributions, with the Abramowitz and Stegun formula for the normal CDF and a regularized incomplete beta function for the t-distribution.
Frequently Asked Questions
Sources & References
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