# Probability & Statistics Studio

> Distributions, descriptive stats, confidence intervals and t, chi-square and z tests with p-values. Probabilities, densities and quantiles for the normal, t, chi-square, F, binomial, Poisson, exponential and uniform distributions with plots; descriptive statistics of a data set; confidence intervals for a mean or a proportion; and one-sample, two-sample and paired t-tests, chi-square goodness-of-fit and independence tests and z-tests for proportions, each with a p-value and a plain-language interpretation.

Part of usefulweb.site (https://usefulweb.site/). Results are deterministic calculations from the inputs and the listed assumptions; the API returns exactly what the web page shows.

## Use it
- Human interface: https://probabilitystatisticsstudio.usefulweb.site/ (interactive tool at https://probabilitystatisticsstudio.usefulweb.site/tool)
- JSON API: POST https://probabilitystatisticsstudio.usefulweb.site/api/v1/calculate with a JSON object of inputs (schema: https://probabilitystatisticsstudio.usefulweb.site/api/v1/tool)
- OpenAPI: https://probabilitystatisticsstudio.usefulweb.site/openapi.json
- MCP (Streamable HTTP): https://probabilitystatisticsstudio.usefulweb.site/mcp - tool `probability_statistics_studio`

## Inputs
- analysis (select): Analysis
- distribution (select): Distribution
- param1 (number): Parameter 1 - Defaults: normal μ = 0, t df = 10, chi-square df = 1, F d1 = 5, binomial n = 10, Poisson λ = 1, exponential λ = 1, uniform a = 0.
- param2 (number): Parameter 2 - Defaults: normal σ = 1, F d2 = 10, binomial p = 0.5, uniform b = 1.
- x (number): Value x - Distribution: compute pdf/pmf and P(X ≤ x).
- x2 (number): Second value x₂ - Optional: also compute P(x ≤ X ≤ x₂).
- probability (number): Probability p (for a quantile) - Distribution: the x with P(X ≤ x) = p (e.g. 0.975).
- data (textarea): Data / sample 1 / observed counts / table - Numbers separated by spaces, commas or new lines. For the independence test: one row of counts per line.
- data2 (textarea): Sample 2 / expected counts - Two-sample and paired tests: the second sample. Goodness of fit: expected counts or proportions (optional; equal by default).
- mu0 (number): Hypothesised mean μ0
- sigma (number): Known population SD σ - Optional, for a z interval for the mean.
- successes (integer): Successes x (group 1)
- trials (integer): Trials n (group 1)
- successes2 (integer): Successes x (group 2)
- trials2 (integer): Trials n (group 2)
- p0 (number): Hypothesised proportion p0
- confidence (number): Confidence level - Also sets α = 1 − confidence for the tests.
- alternative (select): Alternative hypothesis
- equalVariances (boolean): Assume equal variances (pooled two-sample t-test)

## Method
- Special functions are implemented directly: log-gamma (Lanczos), the regularised incomplete gamma function (series and continued fraction) and the regularised incomplete beta function (continued fraction). The normal CDF uses erfc via the incomplete gamma function; its quantile uses Acklam’s approximation refined by Halley steps.
- CDFs: t and F via the incomplete beta function, chi-square and Poisson via the incomplete gamma function, binomial via the incomplete beta function. Quantiles of continuous distributions are found by bisection with a Newton polish; discrete quantiles are the smallest k with P(X ≤ k) ≥ p.
- Values are checked in the test suite against published tables (e.g. Φ(1.96) = 0.975002, t₀.₉₇₅ with 10 df = 2.228139, χ²₀.₉₅ with 5 df = 11.0705, F₀.₉₅(5, 10) = 3.3258).
- Descriptive statistics use the sample (n − 1) variance, quartiles by linear interpolation (the R/Excel default), the adjusted skewness and excess kurtosis, and the 1.5 × IQR outlier rule.
- Tests: one-sample, Welch or pooled two-sample and paired t-tests; chi-square goodness-of-fit and independence (without continuity correction); one- and two-proportion z-tests (pooled for the test, unpooled for the interval). The interpretation compares p with α = 1 − confidence.

## Limits
- A p-value is not the probability that H0 is true, and “not significant” does not prove H0. Check the assumptions listed with each test.