{ "openapi": "3.1.1", "info": { "title": "Probability & Statistics Studio API", "version": "1.0.0", "summary": "Distributions, descriptive stats, confidence intervals and t, chi-square and z tests with p-values.", "description": "Every endpoint performs the same calculation as the human page and returns inputs, result, units, assumptions, sources, warnings and computedAt. MCP endpoint: /mcp." }, "servers": [ { "url": "https://probabilitystatisticsstudio.usefulweb.site" } ], "paths": { "/api/v1/calculate": { "post": { "operationId": "calculate", "summary": "Distributions, descriptive stats, confidence intervals and t, chi-square and z tests with p-values.", "description": "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.\n\nMethodology:\n- 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.\n- 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.\n- 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).\n- 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.\n- 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.\n\nA p-value is not the probability that H0 is true, and “not significant” does not prove H0. Check the assumptions listed with each test.", "tags": [ "math" ], "x-mcp-tool": "probability_statistics_studio", "requestBody": { "required": true, "content": { "application/json": { "schema": { "$ref": "#/components/schemas/ProbabilityStatisticsStudioInput" }, "examples": { "example1": { "summary": "Normal: P(Z ≤ 1.96) and the 97.5% quantile", "value": { "analysis": "distribution", "distribution": "normal", "param1": 0, "param2": 1, "x": 1.96, "probability": 0.975 } }, "example2": { "summary": "Binomial: exactly / at most 5 heads in 10 flips", "value": { "analysis": "distribution", "distribution": "binomial", "param1": 10, "param2": 0.5, "x": 5, "x2": 7 } }, "example3": { "summary": "Descriptive statistics", "value": { "analysis": "describe", "data": "12, 15, 11, 18, 14, 15, 22, 13, 16, 15, 40" } }, "example4": { "summary": "One-sample t-test", "value": { "analysis": "t-one", "data": "5.1 4.9 5.6 5.8 6.0 5.3 5.5 5.7", "mu0": 5 } }, "example5": { "summary": "Welch two-sample t-test", "value": { "analysis": "t-two", "data": "20.1 22.3 19.8 21.5 23.0 20.7", "data2": "18.2 19.5 17.9 18.8 20.1 19.0 18.4" } }, "example6": { "summary": "Chi-square test of independence", "value": { "analysis": "chisq-indep", "data": "30 10\n20 40" } }, "example7": { "summary": "Confidence interval for a proportion", "value": { "analysis": "ci-proportion", "successes": 42, "trials": 120, "confidence": 0.95 } }, "example8": { "summary": "Two-proportion z-test", "value": { "analysis": "z-two-prop", "successes": 45, "trials": 100, "successes2": 30, "trials2": 100 } } } } } }, "responses": { "200": { "description": "Calculation result envelope.", "content": { "application/json": { "schema": { "$ref": "#/components/schemas/ProbabilityStatisticsStudioResult" } } } }, "400": { "description": "Malformed JSON body.", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } }, "422": { "description": "Invalid input (see errors[]).", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } }, "429": { "description": "Rate limit exceeded (see Retry-After).", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } } } } }, "/api/v1/tool": { "get": { "operationId": "getToolDefinition", "summary": "Inputs, output schema and examples for this tool", "responses": { "200": { "description": "Tool definition.", "content": { "application/json": { "schema": { "type": "object" } } } } } } } }, "components": { "schemas": { "Problem": { "type": "object", "description": "RFC 9457 problem details. 422 responses include per-field `errors`.", "required": [ "type", "title", "status", "detail" ], "properties": { "type": { "type": "string" }, "title": { "type": "string" }, "status": { "type": "integer" }, "detail": { "type": "string" }, "code": { "type": "string" }, "errors": { "type": "array", "items": { "type": "object", "required": [ "field", "message" ], "properties": { "field": { "type": "string" }, "message": { "type": "string" } } } } } }, "ProbabilityStatisticsStudioInput": { "type": "object", "additionalProperties": false, "properties": { "analysis": { "title": "Analysis", "type": "string", "enum": [ "distribution", "describe", "ci-mean", "ci-proportion", "t-one", "t-two", "t-paired", "chisq-gof", "chisq-indep", "z-one-prop", "z-two-prop" ], "default": "distribution" }, "distribution": { "title": "Distribution", "type": "string", "enum": [ "normal", "t", "chisq", "f", "binomial", "poisson", "exponential", "uniform" ], "default": "normal" }, "param1": { "title": "Parameter 1", "description": "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.", "type": "number" }, "param2": { "title": "Parameter 2", "description": "Defaults: normal σ = 1, F d2 = 10, binomial p = 0.5, uniform b = 1.", "type": "number" }, "x": { "title": "Value x", "description": "Distribution: compute pdf/pmf and P(X ≤ x).", "type": "number" }, "x2": { "title": "Second value x₂", "description": "Optional: also compute P(x ≤ X ≤ x₂).", "type": "number" }, "probability": { "title": "Probability p (for a quantile)", "description": "Distribution: the x with P(X ≤ x) = p (e.g. 0.975).", "type": "number", "minimum": 0, "maximum": 1 }, "data": { "title": "Data / sample 1 / observed counts / table", "description": "Numbers separated by spaces, commas or new lines. For the independence test: one row of counts per line.", "type": "string", "maxLength": 200000 }, "data2": { "title": "Sample 2 / expected counts", "description": "Two-sample and paired tests: the second sample. Goodness of fit: expected counts or proportions (optional; equal by default).", "type": "string", "maxLength": 200000 }, "mu0": { "title": "Hypothesised mean μ0", "type": "number", "default": 0 }, "sigma": { "title": "Known population SD σ", "description": "Optional, for a z interval for the mean.", "type": "number", "minimum": 0 }, "successes": { "title": "Successes x (group 1)", "type": "integer", "minimum": 0 }, "trials": { "title": "Trials n (group 1)", "type": "integer", "minimum": 1 }, "successes2": { "title": "Successes x (group 2)", "type": "integer", "minimum": 0 }, "trials2": { "title": "Trials n (group 2)", "type": "integer", "minimum": 1 }, "p0": { "title": "Hypothesised proportion p0", "type": "number", "minimum": 0, "maximum": 1, "default": 0.5 }, "confidence": { "title": "Confidence level", "description": "Also sets α = 1 − confidence for the tests.", "type": "number", "minimum": 0.5, "maximum": 0.9999, "default": 0.95 }, "alternative": { "title": "Alternative hypothesis", "type": "string", "enum": [ "two-sided", "less", "greater" ], "default": "two-sided" }, "equalVariances": { "title": "Assume equal variances (pooled two-sample t-test)", "type": "boolean", "default": false } }, "required": [] }, "ProbabilityStatisticsStudioResult": { "type": "object", "additionalProperties": false, "required": [ "schemaVersion", "tool", "computedAt", "inputs", "summary", "result", "units", "assumptions", "sources", "warnings", "presentation", "links" ], "properties": { "schemaVersion": { "type": "string", "const": "1.0" }, "tool": { "type": "object", "additionalProperties": false, "required": [ "id", "name", "version" ], "properties": { "id": { "type": "string" }, "name": { "type": "string" }, "version": { "type": "string" } } }, "computedAt": { "type": "string", "format": "date-time" }, "inputs": { "type": "object", "description": "Normalised inputs actually used, including applied defaults." }, "summary": { "type": "string" }, "result": { "type": "object", "additionalProperties": false, "required": [ "analysis", "distribution", "descriptive", "interval", "test", "warnings" ], "properties": { "analysis": { "type": "string", "enum": [ "distribution", "describe", "ci-mean", "ci-proportion", "t-one", "t-two", "t-paired", "chisq-gof", "chisq-indep", "z-one-prop", "z-two-prop" ] }, "distribution": { "type": [ "object", "null" ], "additionalProperties": false, "properties": { "name": { "type": "string" }, "label": { "type": "string" }, "discrete": { "type": "boolean" }, "mean": { "type": [ "number", "null" ] }, "variance": { "type": [ "number", "null" ] }, "sd": { "type": [ "number", "null" ] }, "pdfAtX": { "type": [ "number", "null" ] }, "cdfAtX": { "type": [ "number", "null" ] }, "survivalAtX": { "type": [ "number", "null" ], "description": "P(X > x)" }, "between": { "type": [ "number", "null" ], "description": "P(x ≤ X ≤ x₂)" }, "quantile": { "type": [ "number", "null" ] }, "x": { "type": [ "number", "null" ] }, "x2": { "type": [ "number", "null" ] }, "probability": { "type": [ "number", "null" ] } } }, "descriptive": { "type": [ "object", "null" ], "additionalProperties": false, "properties": { "n": { "type": "integer" }, "mean": { "type": "number" }, "median": { "type": "number" }, "modes": { "type": "array", "items": { "type": "number" } }, "sd": { "type": [ "number", "null" ] }, "variance": { "type": [ "number", "null" ] }, "min": { "type": "number" }, "max": { "type": "number" }, "q1": { "type": "number" }, "q3": { "type": "number" }, "iqr": { "type": "number" }, "se": { "type": [ "number", "null" ] }, "skewness": { "type": [ "number", "null" ] }, "kurtosis": { "type": [ "number", "null" ] }, "sum": { "type": "number" }, "outliers": { "type": "array", "items": { "type": "number" } } } }, "interval": { "type": [ "object", "null" ], "additionalProperties": false, "properties": { "method": { "type": "string" }, "confidence": { "type": "number" }, "estimate": { "type": "number" }, "lower": { "type": "number" }, "upper": { "type": "number" }, "margin": { "type": [ "number", "null" ] }, "alternatives": { "type": "array", "items": { "type": "object", "additionalProperties": false, "properties": { "method": { "type": "string" }, "lower": { "type": "number" }, "upper": { "type": "number" } } } } } }, "test": { "type": [ "object", "null" ], "additionalProperties": false, "properties": { "name": { "type": "string" }, "h0": { "type": "string" }, "h1": { "type": "string" }, "statistic": { "type": "number" }, "statisticName": { "type": "string" }, "df": { "type": [ "number", "null" ] }, "pValue": { "type": "number" }, "alpha": { "type": "number" }, "reject": { "type": "boolean" }, "estimate": { "type": "number" }, "ciLower": { "type": [ "number", "null" ] }, "ciUpper": { "type": [ "number", "null" ] }, "effectName": { "type": [ "string", "null" ] }, "effectSize": { "type": [ "number", "null" ] }, "interpretation": { "type": "string" }, "details": { "type": "array", "items": { "type": "object", "additionalProperties": false, "properties": { "name": { "type": "string" }, "value": { "type": "string" } } } } } }, "warnings": { "type": "array", "items": { "type": "string" } } } }, "units": { "type": "object", "additionalProperties": { "type": "string" } }, "assumptions": { "type": "array", "items": { "type": "string" } }, "sources": { "type": "array", "items": { "type": "object", "additionalProperties": false, "required": [ "title" ], "properties": { "title": { "type": "string" }, "url": { "type": "string" }, "publisher": { "type": "string" }, "retrievedAt": { "type": "string" }, "note": { "type": "string" } } } }, "warnings": { "type": "array", "items": { "type": "string" } }, "presentation": { "type": "array", "items": { "type": "object" }, "description": "Display blocks used by the human page (same data)." }, "links": { "type": "object", "additionalProperties": false, "required": [ "human", "api", "openapi", "methodology" ], "properties": { "human": { "type": "string", "format": "uri" }, "api": { "type": "string", "format": "uri" }, "openapi": { "type": "string", "format": "uri" }, "methodology": { "type": "string", "format": "uri" } } } } } } } }