Authoritative Methods and Learning Resources
Data Source and Methodology
- NIST Digital Library of Mathematical Functions (DLMF), Chapters on the Gamma and Beta functions and the Error Function, Release 1.2.2 (2024). https://dlmf.nist.gov/
- NIST/SEMATECH e-Handbook of Statistical Methods (2023). https://www.itl.nist.gov/div898/handbook/
All calculations are strictly based on the formulas and data provided by this source.
Implementation details: p-values are computed via cumulative distribution functions (CDFs) using numerically stable forms of the regularized incomplete beta I_x(a, b), the regularized incomplete gamma P(s, x), and the error function erf(x). Tail probabilities are assembled according to the selected alternative hypothesis.
The Formula Explained
Z (Standard Normal): Two–tailed p-value
$$p = 2\bigl(1 - \Phi(|z|)\bigr), \quad \Phi(z) = \frac{1}{2}\bigl[1 + \operatorname{erf}\!\left(\frac{z}{\sqrt{2}}\right)\bigr]$$
Student’s t with ν degrees of freedom:
$$F_t(t;\,\nu) = \tfrac{1}{2} + \operatorname{sgn}(t)\,\tfrac{1}{2}\, I_{\nu/(\nu+t^2)}\!\left(\tfrac{\nu}{2}, \tfrac{1}{2}\right)$$
Chi-square with k degrees of freedom:
$$F_{\chi^2}(x;\,k) = P\!\left(\tfrac{k}{2}, \tfrac{x}{2}\right)$$
F distribution with (d1, d2) degrees of freedom:
$$F_F(x;\,d_1,d_2) = I_{\frac{d_1 x}{d_1 x + d_2}}\!\left(\tfrac{d_1}{2}, \tfrac{d_2}{2}\right)$$
Binomial X ~ Bin(n, p):
$$\Pr(X \le k) = I_{1-p}(n-k,\,k+1), \quad \Pr(X \ge k) = I_{p}(k,\,n-k+1)$$
Two-tailed (conservative): $$p = 2 \min\{\Pr(X \le k), \Pr(X \ge k)\}$$
Glossary of Variables
- Tail Type: alternative hypothesis (two-tailed, right-tailed, left-tailed).
- z: observed Z statistic.
- t, ν: observed Student’s t statistic and degrees of freedom ν > 0.
- X², k: observed chi-square statistic (X² ≥ 0) and degrees of freedom k ≥ 1.
- F, d1, d2: observed F statistic (F ≥ 0) and numerator/denominator degrees of freedom d1,d2 ≥ 1.
- n, k, p: binomial trials (n ≥ 1), successes (0 ≤ k ≤ n), success probability 0 ≤ p ≤ 1.
- α: significance level for decision support (default 0.05).
- P-value: probability, under H0, of an outcome at least as extreme as observed.
How It Works: A Step-by-Step Example
Goal: Two-tailed p-value for a Z-test with z = 2.10.
- Select Distribution = Z (Standard Normal), Tail = Two-tailed.
- Enter z = 2.10, keep α = 0.05.
- Compute p: Φ(2.10) ≈ 0.98214 ⇒ p = 2 × (1 − 0.98214) ≈ 0.0357.
- Decision: p ≈ 0.0357 ≤ 0.05 ⇒ statistically significant at α = 0.05.
Frequently Asked Questions (FAQ)
Which distributions are supported?
Z, Student’s t, Chi-square, F, and Binomial. Each uses exact CDFs based on special functions.
How is the two-tailed p-value computed?
For symmetric distributions (Z, t), p = 2 × one-sided tail using |statistic|. For Chi-square and F, two-tailed is uncommon; if chosen here, it is defined as 2 × min{left, right} and capped at 1. For Binomial, we use the conservative two-tailed p = 2 × min{P(X ≤ k), P(X ≥ k)}.
Do I need to supply sample sizes?
Only indirectly via degrees of freedom for t, chi-square, and F. The calculator expects the test statistic already computed.
What numeric methods are used?
Lanczos approximation for log Γ, Lentz-type continued fractions for incomplete beta, and series/continued-fraction methods for the incomplete gamma—standard techniques used in scientific computing.
Are extreme inputs safe?
Yes within floating-point limitations. The tool guards against invalid domains (e.g., negative df, p outside [0,1]) and returns stable probabilities close to 0 or 1 where appropriate.
What about continuity corrections?
Continuity corrections are not applied automatically; results are exact for the specified distributions.
Can I use this for power analysis?
No. This calculator focuses on p-values. Use a dedicated power calculator for sample size and power computations.