P Value Calculator
The tail area behind any statistic you already hold — z, t, chi-square or F — computed from the special functions to machine precision, with the small-p boundary stated instead of faked.
P Value Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Forward tail areas only — no inverse (critical value) machinery beyond z*
- Small-p display boundaries at 1e-4 (format) and 1e-12 (honesty floor)
In short: A statistic of t = 1.6 on 15 degrees of freedom carries a two-sided p of 0.130445 — half in each tail, 0.065223 — computed as a regularized incomplete beta function, not read off a table. The same machinery gives z = 1.96 a two-sided p of 0.049996 (through the erf series), chi-square 4 on df 5 a right-tail p of 0.549416, and F = 3.885 on 2 and 12 a p of 0.050009. Below 1e-12 the page says “below 1e-12” and stops — a p-value smaller than that is printed as a boundary, never as a pretend zero.
Formula
z: p = 2(1 − Φ(|z|)) · t: p = I(DF/(DF+t²))(DF/2, ½) · χ²: p = Q(DF/2, x/2) · F: p = I(d₂/(d₂+d₁f))(d₂/2, d₁/2)
Every route is a forward special-function evaluation: erf Taylor series, NR continued fractions for the incomplete beta and gamma — no tables, no quantile inversion.
Worked Example
- Pick the distribution your statistic lives on.
- Enter the statistic and its degrees of freedom.
- Choose sides (ignored for χ² and F).
- Read p, its split into tails, and the reading sentence.
t = 1.6, df = 15, two-sided: p = 0.130445 (each tail 0.065223). z = 1.96 two-sided: 0.049996. χ² = 4, df 5: 0.549416. F = 3.885 on 2 and 12: 0.050009. χ² = 200 on df 1: below 1e-12, printed as a boundary.
Strengths & Limits Of This Model
Where this engine is strong
- All four curves on one panel from one special-function core
- Boundary cases stated, never rounded into fakes
Where it stops
- No exact per-tail CIs on the p itself
- No Bayesian or permutation alternatives
Practical Use Cases
Reading papers
recompute the p behind a quoted t or F
Checking software
six decimals to diff against
Coursework
every route shown, not just the answer
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
P Value Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
Is a p-value the probability the null is true?
No — that reversal is the classic misreading. p = P(data at least this strange | the claim), a property of the DATA given the claim. The probability the claim is true GIVEN the data is a different conditional that a p-value cannot reach without priors.
Why do χ² and F ignore the sides field?
Because their alternative is directional by construction: excess dispersion pushes the statistic RIGHT, so the question is always the area to the right of what you observed. A left tail of a χ² answers “is the fit suspiciously GOOD?” — a different question this page leaves alone.
Why does the page stop at 1e-12?
Because honest arithmetic does. The continued fractions carry roughly sixteen digits; beyond 1e-12 the tail is computed from differences of nearly-equal quantities and digits die. Printing “below 1e-12” is the truth; printing 0.000000 would be the first digit of a fabrication.
Which statistic goes with which curve?
z standardizes with a KNOWN sigma; t standardizes with an estimated one (T Score page builds it); χ² sums squared count deviations (Chi Square page builds it); F ratios two variances (ANOVA builds it). If you hold a raw sample instead of a statistic, the hypothesis test page runs the whole trial for you.
What is the difference between one-sided 0.05 and two-sided 0.05?
Evidence bar versus bar width. A one-sided p of 0.05 concentrates the whole 5% in one tail, so the same data that scores p = 0.07 two-sided can score 0.035 one-sided — but only if the direction was declared before the data. Switching sides after seeing the result doubles your real error rate while claiming 5%.
Can this page invert a p back into a statistic or a critical value?
No — and that is a stated boundary, not an oversight. Quantile inversion needs different machinery; the site computes z* by bisection where it is exact and refuses elsewhere. Forward areas, yes; invented quantiles, never.
Why is 0.05 the magic bar?
It is not magic — it is a convention Fisher popularized a century ago. The honest sentence names the bar and the consequence: at α = 5%, roughly one true-nothing examination in twenty reads as a discovery. The p is exact arithmetic; the bar is a decision.
Can the same data give different p-values?
Yes — legitimately. The p depends on the curve (z vs t), the sides (one vs two), and what statistic was chosen before the data. Two analysts computing different p-values from the same data are usually asking different questions; the sin is choosing the question after seeing the answers.