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P-Value Calculator

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About This Tool

P-Value Calculator – Convert Test Statistics to P-Values

The p-value is the single most-quoted number in hypothesis testing, yet it comes from a different formula depending on which distribution your test statistic belongs to. This calculator is a general-purpose "distribution lookup" tool: give it a z, t, χ², or F statistic (plus degrees of freedom, where required), and it returns the exact p-value — no printed statistical table or interpolation needed. It also works in reverse, converting a target significance level α into the critical value you'd need to reject the null hypothesis.

Choosing the Right Distribution

The Normal (Z) distribution applies when the population standard deviation is known or the sample is large. The Student's t distribution is the more common real-world case: it's used whenever the standard deviation is estimated from the sample itself, with heavier tails that widen as degrees of freedom shrink. The Chi-Square (χ²) distribution shows up in variance tests and tests of independence between categorical variables, while the F distribution is the ratio of two variances — the basis of ANOVA and variance-comparison tests.

One-Tailed vs. Two-Tailed

A two-tailed test checks for a difference in either direction and splits α across both tails of the distribution; a one-tailed test checks a single direction and puts the full α in one tail, giving more power to detect an effect in that specific direction. Because Chi-Square and F statistics are always non-negative and conventionally interpreted as right-tailed, the tail selector is hidden for those two distributions — there is no ambiguous "other side" to test against.

Statistic → P-Value, or α → Critical Value

In Statistic → P-Value mode, enter your computed statistic and the calculator returns the exact probability of observing a value at least that extreme under the null hypothesis, along with a plain-language "significant" or "not significant" decision and a pass/fail check at the three most common thresholds: α = 0.01, 0.05, and 0.10. In α → Critical Value mode, enter only your significance level and degrees of freedom to get the threshold statistic needed to reject the null hypothesis — the same value you'd otherwise look up in a printed z, t, χ², or F table.

How the Calculation Works

Each distribution's cumulative probability comes from a numerically stable, from-scratch implementation: the normal CDF uses the error function Φ(z) = 0.5(1 + erf(z/√2)); the t and F CDFs use the regularized incomplete beta function; and the χ² CDF uses the regularized incomplete gamma function. Since none of these have a closed-form inverse, critical values in reverse-lookup mode are solved with bisection, converging to full display precision in well under a hundred iterations.

P-values don't measure effect size
A tiny p-value tells you an effect is unlikely to be due to chance — it says nothing about how large or practically meaningful that effect is. Pair the p-value with a confidence interval or an effect-size measure like Cohen's d before drawing conclusions.

Common Use Cases

Students use this tool to check textbook and homework answers without flipping through printed tables; analysts convert a raw statistic reported by R, SPSS, or Excel into an interpretable p-value; and anyone preparing for a statistics exam can use the reverse lookup to drill critical values for any distribution, degrees of freedom, and significance level combination.

Frequently Asked Questions

Is the P-Value Calculator free?

Yes, P-Value Calculator is totally free :)

Can I use the P-Value Calculator offline?

Yes, you can install the webapp as PWA.

Is it safe to use P-Value Calculator?

Yes, any data related to P-Value Calculator only stored in your browser (if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.

How does this P-Value Calculator work?

Choose a distribution — Normal (Z), Student's t, Chi-Square, or F — enter your test statistic (and degrees of freedom, where required), and the calculator converts it into a p-value using the exact cumulative distribution function. You can also work in reverse: enter a significance level (α) to get the critical value needed to reject the null hypothesis.

What is a p-value and how do I interpret it?

A p-value is the probability of observing a test statistic at least as extreme as yours if the null hypothesis (H₀) were true. If the p-value is less than or equal to your chosen significance level α (commonly 0.05), the result is statistically significant and H₀ is rejected; otherwise you fail to reject H₀.

Which distribution should I use — Z, t, Chi-Square, or F?

Use Z when the population standard deviation is known or the sample is large; use t when it's estimated from a small sample (this is the more common case in practice). Chi-Square applies to variance tests and categorical/independence tests, while F applies to comparing variances or to ANOVA-style tests. If you're unsure, check which distribution your test statistic was originally computed from.

What is the difference between one-tailed and two-tailed tests?

A two-tailed test checks for a difference in either direction and splits the significance level across both tails, while a one-tailed test checks for a difference in only one specified direction (greater than or less than) and puts the full significance level in a single tail. Chi-Square and F tests are conventionally right-tailed only, so the tail selector is hidden for those two distributions.

How do I find a critical value instead of a p-value?

Switch the calculation direction to "α → Critical Value," enter your significance level and degrees of freedom, and the calculator returns the statistic threshold needed to reject H₀ for the selected distribution and tail type — the same values found in a printed z/t/χ²/F table.

How accurate are the calculator's results?

All four distributions are computed from numerically stable regularized incomplete gamma and beta functions — the same mathematical objects underlying textbook statistical tables — so results match published tables and statistical software to several decimal places, including for fractional degrees of freedom.