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Probability & Statistics Studio

Distributions, descriptive stats, confidence intervals and t, chi-square and z tests with p-values.

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.

Defaults: normal σ = 1, F d2 = 10, binomial p = 0.5, uniform b = 1.

Distribution: compute pdf/pmf and P(X ≤ x).

Optional: also compute P(x ≤ X ≤ x₂).

Distribution: the x with P(X ≤ x) = p (e.g. 0.975).

Numbers separated by spaces, commas or new lines. For the independence test: one row of counts per line.

Two-sample and paired tests: the second sample. Goodness of fit: expected counts or proportions (optional; equal by default).

Optional, for a z interval for the mean.

Also sets α = 1 − confidence for the tests.

Try an example: Normal: P(Z ≤ 1.96) and the 97.5% quantile · Binomial: exactly / at most 5 heads in 10 flips · Descriptive statistics · One-sample t-test · Welch two-sample t-test · Chi-square test of independence · Confidence interval for a proportion · Two-proportion z-test

How it works

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.

  • 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.

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.

For scripts and AI agents

  • JSON API: POST https://probabilitystatisticsstudio.usefulweb.site/api/v1/calculate (same calculation as this page)
  • Interface description: OpenAPI 3.1 · AI guide: llms.txt
  • MCP server (Streamable HTTP): https://probabilitystatisticsstudio.usefulweb.site/mcp · tool name probability_statistics_studio
Example request
curl -s -X POST https://probabilitystatisticsstudio.usefulweb.site/api/v1/calculate \
  -H 'content-type: application/json' \
  -d '{"analysis":"distribution","distribution":"normal","param1":0,"param2":1,"x":1.96,"probability":0.975}'

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