How do degrees of freedom affect the p-value?

Lower degrees of freedom create wider (heavier-tailed) t-distributions, resulting in larger p-values for the same t-statistic. As df → ∞, the t-distribution converges to the standard normal.

When should I use t-test vs z-test?

Use the t-test when the population standard deviation is unknown (most cases). Use the z-test only when σ is known or n is very large (n > 100+). In practice, always use the t-test - it reduces to z for large n.

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P-Value Calculator

Calculate p-values from z-score, t-statistic, chi-square, or F-statistic with distribution visualization.

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Quick Reference:

p < 0.05: Significant (reject H₀)
p < 0.01: Very significant
p < 0.001: Highly significant

Frequently Asked Questions

What does a p-value of 0.05 mean?

A p-value of 0.05 means there is a 5% probability of observing data as extreme as (or more extreme than) your results, assuming the null hypothesis is true. It does NOT mean there is a 5% chance the null hypothesis is true.

Is a smaller p-value always better?

A smaller p-value provides stronger evidence against the null hypothesis, but it doesn't measure effect size or practical significance. A tiny p-value with a negligible effect size may not be practically meaningful.

What is the difference between one-tailed and two-tailed p-values?

A one-tailed test checks for an effect in one direction (e.g., mean > μ₀). A two-tailed test checks both directions (mean ≠ μ₀). The two-tailed p-value is exactly double the one-tailed p-value for symmetric distributions like z and t.

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