statistics-ai999x — offline, verified statistics solver with step-by-step answers

Desktop app & command line · Compute in your browser · Install (Python) · Guides · Evaluation · AI999X on Hugging Face · Website · Support: Buy Me a Coffee

License: Apache-2.0 · Author: Ibnul Jaif Farabi (creator and engineer) · Organization: AI999X · Version 1.0.3

Field: Mathematics & Statistics · Tasks: Question Answering, Token Classification, Text Classification, Tabular Regression, Sentence Similarity, Feature Extraction

statistics-ai999x is an offline, deterministic Python solver that answers statistics questions with verified results from 110 certified capabilities, gives 609 documented answers identically on Windows and Linux, and refuses rather than guesses when an answer cannot be verified.

statistics-ai999x is a compact, deterministic statistics engine for Python. It reads a question in plain English (or a typed call), picks a certified statistics capability, computes the answer exactly, checks it, and returns the result with its steps, units, LaTeX and assumptions — or an honest refusal that says what is missing. It runs fully offline on any CPU, needs no GPU, no API key and no internet, and is built to give zero wrong answers: when it is not sure, it refuses instead of guessing.

Run it your way. statistics-ai999x comes as the AI999X desktop app and as the command-line program statistics for Windows and Linux, runs in your browser as WebAssembly, and installs as a Python package. The desktop app, the command line and the browser run the same compiled Rust program on your own device: offline, no Python needed.

Get it

Way to run it	For	Get it
AI999X Statistics desktop app 1.2.0	Windows, Linux — this model in its own app: search, ask in words, forms, live figures, explanations, offline	Windows (x86-64) · Linux (x86-64)
AI999X desktop app 1.2.0	Windows, Linux — every AI999X model in one app: search, ask in words, forms, live figures, explanations, offline	Windows (x86-64) · Linux (x86-64)
Command line (statistics)	Windows, Linux — one program per model, offline, no Python	Windows (x86-64) · Linux (x86-64)
In your browser	any device — nothing to install	Compute on the model page
Python	Python 3.10+	pip install statistics-ai999x

Checksums: SHA256SUMS · every version: https://www.ai999x.com/downloads/desktop/latest/ · install guide: desktop & command line

Rust versions

Statistics also comes as native programs compiled from Rust, version 1.2.0 of the AI999X desktop app and command line: they run offline on your computer and need no Python. Each file is in this repository's rust/ folder, on the GitHub release v1.0.3 and on the website.

Program	System	File	Size	Download	Use
Desktop app (Statistics)	Windows x86-64	ai999x-statistics-1.2.0-windows-x86_64.zip	7.3 MB	Hub · GitHub · website	unzip, then double-click ai999x-statistics.exe (no installer)
Desktop app (Statistics)	Linux x86-64	ai999x-statistics-1.2.0-linux-x86_64.tar.gz	8.3 MB	Hub · GitHub · website	extract, then run ./ai999x-statistics (X11 or Wayland desktop)
Command line (statistics)	Windows x86-64	ai999x-statistics-cli-1.2.0-windows-x86_64.zip	3.1 MB	Hub · GitHub · website (all models)	unzip, then statistics.exe ask "<question>" or statistics.exe solve <capability> name=value
Command line (statistics)	Linux x86-64	ai999x-statistics-cli-1.2.0-linux-x86_64.tar.gz	3.0 MB	Hub · GitHub · website (all models)	extract, then ./statistics ask "<question>" or ./statistics solve <capability> name=value

The command-line archive holds statistics with a README of its commands and examples, the licence and the notice. Checksums: rust/SHA256SUMS (sha256sum -c SHA256SUMS).

Quick start

Desktop app

Download AI999X Statistics for your system (above), unpack it and start it; type a question in words or pick a capability, fill in the form and press Compute. Figures and an explanation of the answer appear beside it. The AI999X desktop app has the same, with Statistics among all AI999X models.

Linux, from a terminal:

curl -LO https://www.ai999x.com/downloads/desktop/1.2.0/ai999x-statistics-1.2.0-linux-x86_64.tar.gz
curl -LO https://www.ai999x.com/downloads/desktop/1.2.0/SHA256SUMS
sha256sum -c --ignore-missing SHA256SUMS
tar xzf ai999x-statistics-1.2.0-linux-x86_64.tar.gz
./ai999x-statistics-1.2.0-linux-x86_64/ai999x-statistics

Command line

Linux:

curl -LO https://www.ai999x.com/downloads/desktop/1.2.0/ai999x-cli-1.2.0-linux-x86_64.tar.gz
curl -LO https://www.ai999x.com/downloads/desktop/1.2.0/SHA256SUMS
sha256sum -c --ignore-missing SHA256SUMS
mkdir -p ~/.local/share && tar xzf ai999x-cli-1.2.0-linux-x86_64.tar.gz -C ~/.local/share
echo 'export PATH="$HOME/.local/share/ai999x-cli-1.2.0-linux-x86_64:$PATH"' >> ~/.bashrc && source ~/.bashrc
statistics --version

Windows (PowerShell):

$F = "ai999x-cli-1.2.0-windows-x86_64.zip"
Invoke-WebRequest "https://www.ai999x.com/downloads/desktop/1.2.0/$F" -OutFile $F
Invoke-WebRequest "https://www.ai999x.com/downloads/desktop/1.2.0/SHA256SUMS" -OutFile SHA256SUMS
(Get-FileHash $F -Algorithm SHA256).Hash.ToLower() -eq ((Select-String -SimpleMatch $F SHA256SUMS).Line -split ' ')[0]
Expand-Archive $F -DestinationPath "$env:LOCALAPPDATA\AI999X" -Force
$D = "$env:LOCALAPPDATA\AI999X\ai999x-cli-1.2.0-windows-x86_64"; $env:Path += ";$D"
[Environment]::SetEnvironmentVariable("Path", [Environment]::GetEnvironmentVariable("Path", "User") + ";$D", "User")
statistics --version

Every command below was run as shown; the text under it is what the program printed.

Ask in words — Write the question as you would say it: the program reads the quantities and their units, picks the capability and answers.

$ statistics ask "Perform a one-sample t-test of [5.1, 4.9, 5.3, 5.5, 5.0, 5.2] against mu0 = 5, two-sided."
Read as: One-sample t-test
t = 1.889822365
df = 5
p_value = 0.1173867951
mean = 5.166666667
standard_deviation = 0.2160246899
  by: One-sample t-test  (t statistic with n - 1 degrees of freedom; tail probability from the incomplete beta function)
  mu0 = 5
  alternative = two-sided
  assumes: random sample from a roughly normal population (or large n); the alternative must be stated for a one-sided claim

What a capability needs — The equation or method, every input and output with its SI unit. Read this before solve: it tells you the input names.

$ statistics info t_test_one_sample
One-sample t-test
  t statistic with n - 1 degrees of freedom; tail probability from the incomplete beta function
  in   x        the data values (at least two) (vector) [1]
  in   mu0      hypothesised mean [1]
  in   alternative alternative hypothesis: two-sided (default), greater or less (choice) [1]
  out  t        (mean - mu0)/(s/sqrt(n)) [1]
  out  df       n - 1 [1]
  out  p_value  p-value [1]
  out  mean     sample mean [1]
  out  standard_deviation sample standard deviation [1]

Solve — Inputs as name=value, each with any unit of the right kind. The answer lists every output, the method used, each input converted to SI, and the conditions the answer assumes.

$ statistics solve t_test_one_sample "x=[5.1, 4.9, 5.3, 5.5, 5, 5.2]" mu0=5.0 alternative=two-sided
t = 1.889822365
df = 5
p_value = 0.1173867951
mean = 5.166666667
standard_deviation = 0.2160246899
  by: One-sample t-test  (t statistic with n - 1 degrees of freedom; tail probability from the incomplete beta function)
  mu0 = 5
  alternative = two-sided
  assumes: random sample from a roughly normal population (or large n); the alternative must be stated for a one-sided claim

Draw the figures — The live figures for your own inputs, written as SVG files into the folder given with --out (here the current folder); open them in any browser. Without --out the SVG text comes back as JSON.

$ statistics figures t_test_one_sample "x=[5.1, 4.9, 5.3, 5.5, 5, 5.2]" mu0=5.0 alternative=two-sided --out .
[
  {
    "slug": "t-test-one-sample-live-1",
    "title": "P-value as hypothesised mean varies",
    "caption": "P-value as mu0 runs from 1.25 to 8.75, every other input as you entered it (41 solutions). Marked: your mu0 = 5 gives p_value = 0.1174. Your inputs: x = (list), mu0 = 5.0, alternative = two-sided.",
    "file": "./t-test-one-sample-live-1.svg"
  },
  {
    "slug": "t-test-one-sample-live-2",
    "title": "P-value for each choice of alternative hypothesis: two-sided (default), gr...",
    "caption": "P-value for every choice of alternative, every other input as you entered it (highlighted: your choice). two-sided: 0.1174; greater: 0.05869; less: 0.9413. Your inputs: x = (list), mu0 = 5.0, alternative = two-sided.",
    "file": "./t-test-one-sample-live-2.svg"
  },
  {
    "slug": "t-test-one-sample-live-3",
    "title": "P-value as the data values (at least two), entry 1 varies",
    "caption": "P-value as x[1] runs from 1.275 to 8.925, every other input as you entered it (29 solutions). Marked: your x[1] = 5.1 gives p_value = 0.1174. Your inputs: x = (list), mu0 = 5.0, alternative = two-sided.",
    "file": "./t-test-one-sample-live-3.svg"
  }
]

Many questions at once — One JSON request per line in, one JSON answer per line out, in order: for scripts, notebooks and agents (the program starts once). In PowerShell: Get-Content cases.jsonl | statistics batch

cases.jsonl:

{"key": "t_test_one_sample", "inputs": {"x": "[5.1, 4.9, 5.3, 5.5, 5, 5.2]", "mu0": 5.0, "alternative": "two-sided"}}
{"key": "normal_probability_below", "inputs": {"mu": 100.0, "sigma": 15.0, "x": 120.0}}

$ statistics batch < cases.jsonl
{"outcome":"SOLVED","message":"","value":0.11738679514629244,"unit":"","solved_for":"p_value","si":[["mu0",5.0]],"values":[["t",1.88982236504614,""],["df",5,""],["p_value",0.11738679514629244,""],["mean",5.166666666666667,""],["standard_deviation",0.2160246899469286,""]],"target":"p_value","phrase":"(mean - mu0)/(s/sqrt(n))","assumptions":["random sample from a roughly normal population (or large n)","the alternative must be stated for a one-sided claim"],"name":"One-sample t-test","equation":"t statistic with n - 1 degrees of freedom; tail probability from the incomplete beta function"}
{"outcome":"SOLVED","message":"","value":0.9087887802741321,"unit":"","solved_for":"probability","si":[["mu",100.0],["sigma",15.0],["x",120.0]],"values":[["probability",0.9087887802741321,""],["z",1.3333333333333333,""]],"target":"probability","phrase":"P(X < x)","assumptions":["X normally distributed","sigma > 0"],"name":"Normal probability below a value","equation":"z = (x - mu)/sigma; Phi(z) from the complementary error function"}

In your browser

Open Statistics on ai999x.com, choose a capability and press Compute: the model runs on your own device as WebAssembly; nothing is sent anywhere. Change any input (any unit of the right kind) and the answer, its explanation and the figures follow.

In your own page: statistics_ai999x.wasm and the loader ax-wasm.js (https://www.ai999x.com/compute/rust/ax-wasm.js).

Python

pip install statistics-ai999x

Worked examples in every form

The same worked example from the command line, from Python and in the browser. Each command-line output is what the program printed.

One-way ANOVA

One-way analysis of variance for 2 to 5 independent groups: F statistic, p-value, sums of squares and eta squared.

Command line:

$ statistics solve one_way_anova "g1=[23, 25, 21, 22]" "g2=[30, 28, 31, 29]" "g3=[26, 27, 24, 25]"
f = 22.12
p_value = 0.0003357531392
df_between = 2
df_within = 9
ss_between = 92.16666667
ss_within = 18.75
ms_between = 46.08333333
ms_within = 2.083333333
eta_squared = 0.8309541698
  by: One-way ANOVA  (partition the total sum of squares into between- and within-group parts; F with (k - 1, N - k) df)
  assumes: independent groups; roughly normal populations with equal variances

Python:

import statistics_ai999x as m

a = m.solve("one_way_anova", g1=[23, 25, 21, 22], g2=[30, 28, 31, 29], g3=[26, 27, 24, 25])
print(a.values or (a.value, a.unit))

Browser: open this capability on the model page and press Compute in your browser.

What this means: One-way analysis of variance partitions the total variability into between-group and within-group components. The computed F ratio is 22.12, derived from the mean square between groups of 46.0833 divided by the mean square within groups of 2.08333. This corresponds to degrees of freedom 2 for the numerator and 9 for the denominator. The resulting upper-tail probability is 0.000335753, indicating whether the observed differences among group means are statistically significant relative to the background noise.

Fisher's exact test (2 x 2)

Fisher's exact test of association in a 2 x 2 table, with the sample odds ratio.

Command line:

$ statistics solve fisher_exact_test "table=[[8, 2], [1, 5]]" alternative=two-sided
p_value = 0.03496503497
odds_ratio = 20
  by: Fisher's exact test (2 x 2)  (hypergeometric probabilities of all tables with the observed margins, in exact integer arithmetic)
  alternative = two-sided
  assumes: fixed margins (conditional test); whole-number counts

Python:

import statistics_ai999x as m

a = m.solve("fisher_exact_test", table=[[8, 2], [1, 5]], alternative="two-sided")
print(a.values or (a.value, a.unit))

Browser: open this capability on the model page and press Compute in your browser.

What this means: This calculation determines the probability of observing a table as extreme or more extreme than the one provided, assuming no association between the variables. The result yields an exact p-value of 0.034965 and a sample odds ratio of 20. A low p-value suggests that the observed distribution of counts across the categories is unlikely to have occurred by chance alone under the assumption of independence.

McNemar's test for paired proportions

McNemar's test for a 2 x 2 table of paired outcomes: continuity-corrected chi-square and the exact binomial p-value.

Command line:

$ statistics solve mcnemar_test "table=[[30, 12], [5, 53]]"
chi_square = 2.117647059
p_value = 0.1456100954
exact_p_value = 0.1434631348
discordant = 17
  by: McNemar's test for paired proportions  (only the discordant cells b and c carry information)
  assumes: paired (matched) binary outcomes; b + c > 0

Python:

import statistics_ai999x as m

a = m.solve("mcnemar_test", table=[[30, 12], [5, 53]])
print(a.values or (a.value, a.unit))

Browser: open this capability on the model page and press Compute in your browser.

What this means: McNemar's test evaluates whether the proportion of binary outcomes differs between two related groups, such as before and after treatment. The calculation relies solely on the discordant pairs, where the outcome changed from yes to no or no to yes. Based on these counts, the continuity-corrected chi-square value is computed as 2.11765. The resulting p-value is 0.14561, which measures the probability of observing such an imbalance by chance if there were no true difference. An exact binomial p-value of 0.143463 is also provided for greater accuracy when sample sizes are small.

Confidence interval for a mean (sigma known, z-interval)

The z confidence interval for a population mean when the population standard deviation sigma is known.

Command line:

$ statistics solve ci_mean_known_sigma xbar=52 sigma=8 n=36 confidence=0.95
lower = 49.38671469
upper = 54.61328531
margin_of_error = 2.613285313
critical_value = 1.959963985
  by: Confidence interval for a mean (sigma known, z-interval)  (xbar +/- z_(alpha/2) sigma / sqrt(n))
  xbar = 52
  sigma = 8
  confidence = 0.95
  assumes: sigma known; normal population or large n; random sample

Python:

import statistics_ai999x as m

a = m.solve("ci_mean_known_sigma", xbar="52", sigma="8", n="36", confidence="0.95")
print(a.values or (a.value, a.unit))

Browser: open this capability on the model page and press Compute in your browser.

What this means: This calculation determines a range of plausible values for the population mean using the sample mean and known standard deviation. The interval spans from 49.3867 to 54.6133, centered on the observed sample mean of 52. The width of this interval is defined by the margin of error, which equals 2.61329. This value represents the maximum expected difference between the true population mean and the sample estimate at the specified confidence level.

Documentation

• Rust user guide — the command line, every capability with a command and what it printed: PDF · Markdown
• Python guide — every capability with a Python example: PDF · Markdown
• Desktop app & command line — downloads, install, ask and search, figures, offline use, checksums: https://www.ai999x.com/desktop-cli/
• Model page — compute in your browser, every capability explained: https://www.ai999x.com/models/statistics-ai999x/

Highlights

• Verified, not generated. Every answer is computed by a certified statistics capability and checked (substituted back, dimension-checked or compared with an independent computation) before it is returned.
• Zero wrong answers by design. Seven explicit outcomes — SOLVED, NEEDS_INPUT, AMBIGUOUS, INVALID_INPUT, OUT_OF_SCOPE, UNRECOGNIZED_FORMAT, INTERNAL_ERROR — never a silent guess. Every one of its 609 documented answers is reproduced exactly on every tested platform.
• Plain-English questions and a guaranteed typed interface. ask() reads documented question forms; solve(capability_id, **inputs) always solves a supported task with valid inputs.
• Explains itself. Steps, formulas in LaTeX, units, assumptions and the model used, ready for notebooks, tutoring apps and homework checkers.
• Native and in the browser. Every capability also computes natively in Rust: the desktop app and the statistics command line for Windows and Linux, and WebAssembly on the model page — no Python, no network.
• Small and portable. Pure Python (sympy, mpmath); identical answers on Windows and Linux (Python 3.10, 3.11, 3.12, 3.13, 3.14).
• Coverage: descriptive statistics, probability distributions, confidence intervals, hypothesis tests, regression and correlation, experimental design, quality control and reliability.

Model overview

Property	Value
Package	statistics-ai999x 1.0.3 (import statistics_ai999x)
Type	Deterministic neuro-symbolic statistics engine (rule-based reading + certified symbolic capabilities)
Capabilities	110 certified procedures
Knowledge pack	version 1.0.1
Golden answers	609 documented answers, identical on every tested platform
Platforms tested	Windows (Python 3.13), Linux x86-64 (Python 3.10, 3.11, 3.12, 3.13, 3.14)
Runtime	CPU only, offline; no GPU, no network, no API key
License	Apache-2.0
Creator / engineer	Ibnul Jaif Farabi
Organization	AI999X

Python quick start

pip install statistics-ai999x
# or straight from this repository:
pip install https://huggingface.co/AI999X/statistics-ai999x/resolve/main/dist/statistics_ai999x-1.0.3-py3-none-any.whl

import statistics_ai999x as statistics

statistics.ask("Perform a one-sample t-test of [5.1, 4.9, 5.3, 5.5, 5.0, 5.2] against mu0 = 5, two-sided.").show()
# -> SOLVED — t = 1.88982, df = 5, p_value = 0.117387, mean = 5.16667  (t_test_one_sample)
statistics.ask("For a normal distribution with mean = 100 and standard deviation = 15, find P(X < 120).").show()
# -> SOLVED — probability = 0.908789, z = 1.33333  (normal_probability_below)
statistics.ask("Find the 95% confidence interval for the mean of [12, 15, 11, 18, 14, 16].").show()
# -> SOLVED — lower = 11.6237, upper = 17.043, margin_of_error = 2.70963, critical_value = 2.57058  (ci_mean_t)
statistics.ask("Chi-square test of independence for the table [[20, 30], [25, 25]].").show()
# -> SOLVED — chi_square = 1.0101, df = 1, p_value = 0.314879, expected = [[22.5, 27.5], [22.5, 27.5]]  (chi_square_independence)
statistics.solve("weibull_reliability", beta=2, eta=1000, t=500).show()
# -> SOLVED — reliability = 0.778801, unreliability = 0.221199, hazard_rate = 0.001, mttf = 886.227  (weibull_reliability)
statistics.ask("Who created you?").message
# -> statistics-ai999x was created and engineered by Ibnul Jaif Farabi at AI999X.

The Python package also installs a Python command, statistics-ai999x "<question>"; for the native program use statistics (above). The outputs above were produced by running this exact code on statistics-ai999x 1.0.3.

Find the right capability, and draw it

statistics.search() finds which certified capability a question is about — across every AI999X model, so a question for another model points you to it. It runs ai999x-encoder, AI999X's own capability encoder (4.7M parameters, a 4.9 MB ONNX model, offline; the root model.safetensors of this repository): pip install "statistics-ai999x[search]". ask(question, suggest=True) adds the same pointers to a question it does not answer; an answer itself never changes.

import statistics_ai999x as statistics

statistics.search("Solve x^2 - 5x + 6 = 0", k=2)
# -> Roots of a quadratic equation (math-foundations-ai999x, 0.63); Roots of a cubic equation (math-foundations-ai999x, 0.59)

How every answer is verified

1. Reading — the question is turned into typed quantities (value, unit, kind, role) and one asked quantity; every stated number must be used or the question is refused.
2. Planning — only certified capabilities may answer; a capability applies only under the conditions the wording states (for example a pendulum, a liquid at rest, boiling).
3. Computing — exact symbolic or high-precision numeric evaluation.
4. Checking — dimensions, substitution back into the defining equation, and agreement between independent readings; any disagreement becomes a refusal, never a number.

Capabilities are certified before they ship: each must reproduce hand-worked reference cases, reject a deliberately wrong variant (a control), and leave every benchmark answer unchanged.

Python worked examples, with their steps

Each question was put to statistics-ai999x; the answer is the one it returned, with the steps it took.

One-sample t-test · Hypothesis tests

Perform a one-sample t-test of [5.1, 4.9, 5.3, 5.5, 5.0, 5.2] against mu0 = 5, two-sided.

→ t = 1.889822365, df = 5, p_value = 0.1173867951, mean = 5.166666667, standard_deviation = 0.2160246899

Steps

• One-sample t-test: x = [5.1, 4.9, 5.3, 5.5, 5, 5.2], mu0 = 5, alternative = two-sided → t = 1.889822365, df = 5, p_value = 0.1173867951, mean = 5.166666667, standard_deviation = 0.2160246899

Normal probability below a value · Probability distributions

For a normal distribution with mean = 100 and standard deviation = 15, find P(X < 120).

→ probability = 0.9087887803, z = 1.333333333

Steps

• Normal probability below a value: mu = 100, sigma = 15, x = 120 → probability = 0.9087887803, z = 1.333333333

Confidence interval for a mean from data (t-interval) · Confidence intervals

Find the 95% confidence interval for the mean of [12, 15, 11, 18, 14, 16].

→ lower = 11.62370216, upper = 17.0429645, margin_of_error = 2.709631171, critical_value = 2.570581836, df = 5, mean = 14.33333333

Steps

• Confidence interval for a mean from data (t-interval): x = [12, 15, 11, 18, 14, 16], confidence = 0.95 → lower = 11.62370216, upper = 17.0429645, margin_of_error = 2.709631171, critical_value = 2.570581836, df = 5, mean = 14.33333333

Chi-square test of independence · Hypothesis tests

Chi-square test of independence for the table [[20, 30], [25, 25]].

→ chi_square = 1.01010101, df = 1, p_value = 0.3148786413, expected = [[22.5, 27.5], [22.5, 27.5]]

Steps

• Chi-square test of independence: table = [20 30; 25 25] → chi_square = 1.01010101, df = 1, p_value = 0.3148786413, expected = [[22.5, 27.5], [22.5, 27.5]]

Weibull reliability, hazard rate and MTTF · Reliability and survival

m.solve("weibull_reliability", beta=2, eta=1000, t=500)

→ reliability = 0.7788007831, unreliability = 0.2211992169, hazard_rate = 0.001, mttf = 886.2269255

Steps

• Weibull reliability, hazard rate and MTTF: beta = 2, eta = 1000, t = 500 → reliability = 0.7788007831, unreliability = 0.2211992169, hazard_rate = 0.001, mttf = 886.2269255

One-way ANOVA · Analysis of variance

Perform a one-way ANOVA on groups [23, 25, 21, 22], [30, 28, 31, 29] and [26, 27, 24, 25].

→ f = 22.12, p_value = 0.0003357531392, df_between = 2, df_within = 9, ss_between = 92.16666667, ss_within = 18.75, ms_between = 46.08333333, ms_within = 2.083333333, eta_squared = 0.8309541698

Steps

• One-way ANOVA: g1 = [23, 25, 21, 22], g2 = [30, 28, 31, 29], g3 = [26, 27, 24, 25] → f = 22.12, p_value = 0.0003357531392, df_between = 2, df_within = 9, ss_between = 92.16666667, ss_within = 18.75, ms_between = 46.08333333, ms_within = 2.083333333, eta_squared = 0.8309541698

Figures

Every point was computed by statistics-ai999x.

One-sample t test: statistic against the rejection region

The one-sample t test of x = 5.1, 4.9, 5.3, 5.5, 5.0, 5.2 against μ₀ = 5: the t density with 5 degrees of freedom (from the package's quantiles at 121 cumulative probabilities, differenced), the two-tailed rejection region beyond ±2.57058 for α = 0.05 shaded; marked: t = 1.88982, p-value = 0.117387.

Poisson distribution and an at-least probability

The Poisson distribution with mean λ = 4: P(X = k) for k = 0 … 14, each from the package; the stems with k ≥ 2 carry P(X ≥ 2) = 0.908422; marked: the most probable count k = 4, P = 0.195367.

Three groups as box plots, with their ANOVA

Box plots of 3 groups (g1 = 23, 25, 21, 22; g2 = 30, 28, 31, 29; g3 = 26, 27, 24, 25): each box spans the quartiles and the bar is the median, all from the package's percentiles (g1 median 22.5, quartiles 21.75–23.5; g2 median 29.5, quartiles 28.75–30.25; g3 median 25.5, quartiles 24.75–26.25). One-way ANOVA: F = 22.12 with 2 and 9 degrees of freedom, p-value 0.000335753, η² = 0.830954.

Simple linear regression: data and fitted line

The least-squares line through x = 1, 2, 3, 4, 5, y = 2.2, 4.1, 6.1, 7.9, 10.2: intercept 0.16 (standard error 0.143295), slope 1.98 (standard error 0.0432049), r = 0.999287, R² = 0.998574, p-value of the slope 2.2873e-05, as returned by the package.

Kaplan–Meier survival curve

The Kaplan–Meier survival curve returned by the package for times 6, 6, 6, 7, 10, 13, 16, 22, 23 with event indicators 1, 1, 1, 0, 1, 1, 1, 1, 1 (0 = censored): it steps down at each event time 6, 10, 13, 16, 22, 23; marked: t = 12, S = 0.533333.

Confidence intervals for a mean at five levels

t confidence intervals for the mean of x = 12, 15, 11, 18, 14, 16 at 5 confidence levels, each from the package: 50 % [13.5673, 15.0993]; 80 % [12.7776, 15.8891]; 90 % [12.2093, 16.4574]; 95 % [11.6237, 17.043]; 99 % [10.0831, 18.5836] (sample mean 14.3333, 5 degrees of freedom); higher confidence, wider interval.

Capabilities

110 certified procedures. A few of them:

• Procedures: 2^2 factorial design: main effects and interaction; 2^3 factorial design: main effects and interactions; Arithmetic mean; Bartlett's test for equal variances; Binomial probability: at least k successes; Binomial probability: at most k successes; Binomial probability: exactly k successes; Chi-square goodness-of-fit test; Chi-square test for a variance; Chi-square test of independence; Coefficient of variation; Cohen's d and Hedges' g; Confidence interval for a correlation (Fisher z); Confidence interval for a mean (sigma known, z-interval); Confidence interval for a mean from data (t-interval); Confidence interval for a mean from summary statistics (t-interval); Confidence interval for a proportion (Wald); Confidence interval for a variance and standard deviation; Confidence interval for the difference of two means; Covariance; Critical value of F; Critical value of Student's t; Critical value of chi-square; Critical value of z …

statistics_ai999x.templates() lists every question form ask() reads; the capability guide (PDF, every example executed) is included in this repository.

All 110 formulas

Show every formula, by topic

Analysis of variance

One-way ANOVA

$$F=\frac{S{S}_B\mathrm{/}(k-1)}{S{S}_W\mathrm{/}(N-k)}F\; =\; \backslash frac\{SS\_B/(k-1)\}\{SS\_W/(N-k)\}$$

Categorical data

Fisher's exact test (2 x 2)

$$P(a)=\frac{\left(\genfrac{}{}{0px}{}{a+b}{a}\right)\left(\genfrac{}{}{0px}{}{c+d}{c}\right)}{\left(\genfrac{}{}{0px}{}{n}{a+c}\right)}P(a)\; =\; \backslash frac\{\backslash binom\{a+b\}\{a\}\backslash binom\{c+d\}\{c\}\}\{\backslash binom\{n\}\{a+c\}\}$$

McNemar's test for paired proportions

$${\chi }^2=\frac{(\mathrm{\mid }b-c\mathrm{\mid }-1{)}^2}{b+c}\backslash chi^2\; =\; \backslash frac\{(\backslash vert\; b\; -\; c\backslash vert\; -\; 1)^2\}\{b\; +\; c\}$$

Confidence intervals

Confidence interval for a mean (sigma known, z-interval)

$$\bar{x}\pm z_{\alpha \mathrm{/}2}\frac{\sigma }{\sqrt{n}}\backslash bar\{x\}\; \backslash pm\; z\_\{\backslash alpha/2\}\backslash frac\{\backslash sigma\}\{\backslash sqrt\{n\}\}$$

Confidence interval for a mean from data (t-interval)

$$\bar{x}\pm t_{\alpha \mathrm{/}2,n-1}\frac{s}{\sqrt{n}}\backslash bar\{x\}\; \backslash pm\; t\_\{\backslash alpha/2,\backslash ,n-1\}\backslash frac\{s\}\{\backslash sqrt\{n\}\}$$

Confidence interval for a mean from summary statistics (t-interval)

$$\bar{x}\pm t_{\alpha \mathrm{/}2,n-1}\frac{s}{\sqrt{n}}\backslash bar\{x\}\; \backslash pm\; t\_\{\backslash alpha/2,\backslash ,n-1\}\backslash frac\{s\}\{\backslash sqrt\{n\}\}$$

Confidence interval for a proportion (Wald)

$$\widehat{p}\pm z_{\alpha \mathrm{/}2}\sqrt{\frac{\widehat{p}(1-\widehat{p})}{n}}\backslash hat\{p\}\; \backslash pm\; z\_\{\backslash alpha/2\}\backslash sqrt\{\backslash frac\{\backslash hat\{p\}(1-\backslash hat\{p\})\}\{n\}\}$$

Confidence interval for a variance and standard deviation

$$[\frac{(n-1)s^2}{{\chi }_{1-\alpha \mathrm{/}2,n-1}^2},\frac{(n-1)s^2}{{\chi }_{\alpha \mathrm{/}2,n-1}^2}]\backslash left[\backslash frac\{(n-1)s^2\}\{\backslash chi^2\_\{1-\backslash alpha/2,\backslash ,n-1\}\},\backslash \; \backslash frac\{(n-1)s^2\}\{\backslash chi^2\_\{\backslash alpha/2,\backslash ,n-1\}\}\backslash right]$$

Confidence interval for the difference of two means

$$({\bar{x}}_1-{\bar{x}}_2)\pm t_{\alpha \mathrm{/}2,\nu }\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}(\backslash bar\{x\}\_1\; -\; \backslash bar\{x\}\_2)\; \backslash pm\; t\_\{\backslash alpha/2,\backslash ,\backslash nu\}\backslash sqrt\{\backslash frac\{s\_1^2\}\{n\_1\}\; +\; \backslash frac\{s\_2^2\}\{n\_2\}\}$$

Wilson score interval for a proportion

$$\frac{\widehat{p}+\frac{z^2}{2n}\pm z\sqrt{\frac{\widehat{p}(1-\widehat{p})}{n}+\frac{z^2}{4n^2}}}{1+\frac{z^2}{n}}\backslash frac\{\backslash hat\{p\}\; +\; \backslash frac\{z^2\}\{2n\}\; \backslash pm\; z\backslash sqrt\{\backslash frac\{\backslash hat\{p\}(1-\backslash hat\{p\})\}\{n\}\; +\; \backslash frac\{z^2\}\{4n^2\}\}\}\{1\; +\; \backslash frac\{z^2\}\{n\}\}$$

Confidence intervals and tests

Critical value of F

$$F_{\alpha ;{\nu }_1,{\nu }_2}:P(F_{{\nu }_1,{\nu }_2}>c)=\alpha F\_\{\backslash alpha;\backslash ,\backslash nu\_1,\backslash nu\_2\}:\backslash \; P(F\_\{\backslash nu\_1,\backslash nu\_2\}\; >\; c)\; =\; \backslash alpha$$

Critical value of Student's t

$$t_{\alpha \mathrm{/}2,\nu }:P(T_{\nu }>t)=\alpha \mathrm{/}2t\_\{\backslash alpha/2,\backslash ,\backslash nu\}:\backslash \; P(T\_\backslash nu\; >\; t)\; =\; \backslash alpha/2$$

Critical value of chi-square

$${\chi }_{\alpha ,\nu }^2:P({\chi }_{\nu }^2>c)=\alpha \backslash chi^2\_\{\backslash alpha,\backslash nu\}:\backslash \; P(\backslash chi^2\_\backslash nu\; >\; c)\; =\; \backslash alpha$$

Critical value of z

$$z_{\alpha \mathrm{/}2}={\mathrm{\Phi }}^{-1}(1-\alpha \mathrm{/}2)z\_\{\backslash alpha/2\}\; =\; \backslash Phi^\{-1\}(1\; -\; \backslash alpha/2)$$

Descriptive statistics

Arithmetic mean

$$\bar{x}=\frac{1}{n}\sum _{i=1}^n{x}_i\backslash bar\{x\}\; =\; \backslash frac\{1\}\{n\}\backslash sum\_\{i=1\}^\{n\}\; x\_i$$

Coefficient of variation

$$CV=\frac{s}{\bar{x}}CV\; =\; \backslash frac\{s\}\{\backslash bar\{x\}\}$$

Five-number summary and interquartile range

$$Q_p=x_{(\lfloor h\rfloor )}+(h-\lfloor h\rfloor )(x_{(\lfloor h\rfloor +1)}-x_{(\lfloor h\rfloor )}),h=1+(n-1)pQ\_p\; =\; x\_\{(\backslash lfloor\; h\; \backslash rfloor)\}\; +\; (h\; -\; \backslash lfloor\; h\; \backslash rfloor)\backslash ,(x\_\{(\backslash lfloor\; h\; \backslash rfloor\; +\; 1)\}\; -\; x\_\{(\backslash lfloor\; h\; \backslash rfloor)\}),\backslash \; h\; =\; 1\; +\; (n-1)p$$

Geometric mean

$$G={(\prod x_i)}^{1\mathrm{/}n}G\; =\; \backslash left(\backslash prod\; x\_i\backslash right)^\{1/n\}$$

Harmonic mean

$$H=\frac{n}{\sum 1\mathrm{/}{x}_i}H\; =\; \backslash frac\{n\}\{\backslash sum\; 1/x\_i\}$$

Mean absolute deviation

$$MAD=\frac{1}{n}\sum \mathrm{\mid }{x}_i-\bar{x}\mathrm{\mid }MAD\; =\; \backslash frac\{1\}\{n\}\backslash sum\; \backslash vert\; x\_i\; -\; \backslash bar\{x\}\backslash vert$$

Mean and standard deviation of a frequency table

$$\bar{x}=\frac{\sum f_i{x}_i}{\sum f_i},s=\sqrt{\frac{\sum f_i(x_i-\bar{x}{)}^2}{n-1}}\backslash bar\{x\}\; =\; \backslash frac\{\backslash sum\; f\_i\; x\_i\}\{\backslash sum\; f\_i\},\backslash quad\; s\; =\; \backslash sqrt\{\backslash frac\{\backslash sum\; f\_i\; (x\_i\; -\; \backslash bar\{x\})^2\}\{n\; -\; 1\}\}$$

Median

$$\tilde{x}=x_{((n+1)\mathrm{/}2)}\text{ or }\frac{1}{2}(x_{(n\mathrm{/}2)}+x_{(n\mathrm{/}2+1)})\backslash tilde\{x\}\; =\; x\_\{((n+1)/2)\}\; \backslash \; \backslash text\{or\}\backslash \; \backslash tfrac12\backslash left(x\_\{(n/2)\}\; +\; x\_\{(n/2+1)\}\backslash right)$$

Median absolute deviation

$$MAD=\mathrm{median}(\mathrm{\mid }{x}_i-\tilde{x}\mathrm{\mid })MAD\; =\; \backslash operatorname\{median\}\backslash left(\backslash vert\; x\_i\; -\; \backslash tilde\{x\}\backslash vert\; \backslash right)$$

Mode

$$\mathrm{mode}=\arg \underset{v}{\max }\mathrm{\#}\{i:x_i=v\}\backslash operatorname\{mode\}\; =\; \backslash arg\backslash max\_v\; \backslash \#\backslash \{i\; :\; x\_i\; =\; v\backslash \}$$

Outliers by the 1.5 IQR rule

$$[Q_1-1.5IQR,Q_3+1.5IQR][Q\_1\; -\; 1.5\backslash ,IQR,\backslash \; Q\_3\; +\; 1.5\backslash ,IQR]$$

Percentile of a data set

$$P_p=x_{(\lfloor h\rfloor )}+(h-\lfloor h\rfloor )(x_{(\lfloor h\rfloor +1)}-x_{(\lfloor h\rfloor )})P\_p\; =\; x\_\{(\backslash lfloor\; h\; \backslash rfloor)\}\; +\; (h\; -\; \backslash lfloor\; h\; \backslash rfloor)(x\_\{(\backslash lfloor\; h\; \backslash rfloor+1)\}\; -\; x\_\{(\backslash lfloor\; h\; \backslash rfloor)\})$$

Population standard deviation

$$\sigma =\sqrt{\frac{1}{n}\sum (x_i-\mu {)}^2}\backslash sigma\; =\; \backslash sqrt\{\backslash frac\{1\}\{n\}\backslash sum\; (x\_i\; -\; \backslash mu)^2\}$$

Population variance

$${\sigma }^2=\frac{1}{n}\sum (x_i-\mu {)}^2\backslash sigma^2\; =\; \backslash frac\{1\}\{n\}\backslash sum\; (x\_i\; -\; \backslash mu)^2$$

Range

$$R=x_{\max }-x_{\min }R\; =\; x\_\{\backslash max\}\; -\; x\_\{\backslash min\}$$

Sample standard deviation

$$s=\sqrt{\frac{1}{n-1}\sum (x_i-\bar{x}{)}^2}s\; =\; \backslash sqrt\{\backslash frac\{1\}\{n-1\}\backslash sum\; (x\_i\; -\; \backslash bar\{x\})^2\}$$

Sample variance

$$s^2=\frac{1}{n-1}\sum (x_i-\bar{x}{)}^{2}s^2\; =\; \backslash frac\{1\}\{n-1\}\backslash sum\; (x\_i\; -\; \backslash bar\{x\})^2$$

Skewness and kurtosis

$$G_1=\frac{\sqrt{n(n-1)}}{n-2}\frac{m_3}{m_2^{3\mathrm{/}2}},G_2=\frac{(n-1)((n+1)g_2+6)}{(n-2)(n-3)}G\_1\; =\; \backslash frac\{\backslash sqrt\{n(n-1)\}\}\{n-2\}\backslash ,\backslash frac\{m\_3\}\{m\_2^\{3/2\}\},\backslash quad\; G\_2\; =\; \backslash frac\{(n-1)\backslash left((n+1)g\_2\; +\; 6\backslash right)\}\{(n-2)(n-3)\}$$

Standard error of the mean

$$S{E}_{\bar{x}}=\frac{s}{\sqrt{n}}SE\_\{\backslash bar\{x\}\}\; =\; \backslash frac\{s\}\{\backslash sqrt\{n\}\}$$

Trimmed mean

$${\bar{x}}_{\alpha }=\frac{1}{n-2k}\sum _{i=k+1}^{n-k}{x}_{(i)},k=\lfloor \alpha n\rfloor \backslash bar\{x\}\_\{\backslash alpha\}\; =\; \backslash frac\{1\}\{n\; -\; 2k\}\backslash sum\_\{i=k+1\}^\{n-k\}\; x\_\{(i)\},\backslash \; k\; =\; \backslash lfloor\; \backslash alpha\; n\; \backslash rfloor$$

Weighted mean

$${\bar{x}}_w=\frac{\sum w_i{x}_i}{\sum w_i}\backslash bar\{x\}\_w\; =\; \backslash frac\{\backslash sum\; w\_i\; x\_i\}\{\backslash sum\; w\_i\}$$

z-score of a value

$$z=\frac{x-\mu }{\sigma }z\; =\; \backslash frac\{x\; -\; \backslash mu\}\{\backslash sigma\}$$

Effect sizes

Cohen's d and Hedges' g

$$d=\frac{{\bar{x}}_1-{\bar{x}}_2}{s_p},s_p=\sqrt{\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}}d\; =\; \backslash frac\{\backslash bar\{x\}\_1\; -\; \backslash bar\{x\}\_2\}\{s\_p\},\backslash \; s\_p\; =\; \backslash sqrt\{\backslash frac\{(n\_1-1)s\_1^2\; +\; (n\_2-1)s\_2^2\}\{n\_1+n\_2-2\}\}$$

Experimental design

2^2 factorial design: main effects and interaction

$$\text{Effect}=\frac{\text{contrast}}{2^{k-1}}\backslash text\{Effect\}\; =\; \backslash frac\{\backslash text\{contrast\}\}\{2^\{k-1\}\}$$

2^3 factorial design: main effects and interactions

$$\text{Effect}=\frac{\text{contrast}}{2^{k-1}}\backslash text\{Effect\}\; =\; \backslash frac\{\backslash text\{contrast\}\}\{2^\{k-1\}\}$$

Cronbach's alpha (internal consistency)

$$\alpha =\frac{k}{k-1}(1-\frac{\sum s_i^2}{s_T^2})\backslash alpha\; =\; \backslash frac\{k\}\{k-1\}\backslash left(1\; -\; \backslash frac\{\backslash sum\; s\_i^2\}\{s\_T^2\}\backslash right)$$

Randomized complete block design (two-way ANOVA without replication)

$$F_{\text{trt}}=\frac{S{S}_{\text{trt}}\mathrm{/}(t-1)}{S{S}_E\mathrm{/}((t-1)(b-1))}F\_\{\backslash text\{trt\}\}\; =\; \backslash frac\{SS\_\{\backslash text\{trt\}\}/(t-1)\}\{SS\_E/((t-1)(b-1))\}$$

Hypothesis tests

Bartlett's test for equal variances

$$T=\frac{(N-k)\ln s_p^2-\sum (n_i-1)\ln s_i^2}{1+\frac{1}{3(k-1)}(\sum \frac{1}{n_i-1}-\frac{1}{N-k})}T\; =\; \backslash frac\{(N-k)\backslash ln\; s\_p^2\; -\; \backslash sum\; (n\_i\; -\; 1)\backslash ln\; s\_i^2\}\{1\; +\; \backslash frac\{1\}\{3(k-1)\}\backslash left(\backslash sum\; \backslash frac\{1\}\{n\_i\; -\; 1\}\; -\; \backslash frac\{1\}\{N-k\}\backslash right)\}$$

Chi-square goodness-of-fit test

$${\chi }^2=\sum \frac{(O_i-E_i{)}^2}{E_i},\nu =k-1\backslash chi^2\; =\; \backslash sum\; \backslash frac\{(O\_i\; -\; E\_i)^2\}\{E\_i\},\backslash \; \backslash nu\; =\; k\; -\; 1$$

Chi-square test for a variance

$${\chi }^2=\frac{(n-1)s^2}{{\sigma }_0^2}\backslash chi^2\; =\; \backslash frac\{(n-1)s^2\}\{\backslash sigma\_0^2\}$$

Chi-square test of independence

$${\chi }^2=\sum _{i,j}\frac{(O_{ij}-E_{ij}{)}^2}{E_{ij}},E_{ij}=\frac{R_i{C}_j}{n}\backslash chi^2\; =\; \backslash sum\_\{i,j\}\backslash frac\{(O\_\{ij\}\; -\; E\_\{ij\})^2\}\{E\_\{ij\}\},\backslash \; E\_\{ij\}\; =\; \backslash frac\{R\_i\; C\_j\}\{n\}$$

F-test for equality of two variances

$$F=\frac{s_1^2}{s_2^2}F\; =\; \backslash frac\{s\_1^2\}\{s\_2^2\}$$

Levene's test for equal variances

$$W=\frac{(N-k)\sum n_i({\bar{z}}_{i\cdot }-\bar{z}{)}^2}{(k-1)\sum \sum (z_{ij}-{\bar{z}}_{i\cdot }{)}^2},z_{ij}=\mathrm{\mid }{x}_{ij}-{\tilde{x}}_i\mathrm{\mid }W\; =\; \backslash frac\{(N-k)\backslash sum\; n\_i(\backslash bar\{z\}\_\{i\backslash cdot\}\; -\; \backslash bar\{z\})^2\}\{(k-1)\backslash sum\backslash sum\; (z\_\{ij\}\; -\; \backslash bar\{z\}\_\{i\backslash cdot\})^2\},\backslash \; z\_\{ij\}\; =\; \backslash vert\; x\_\{ij\}\; -\; \backslash tilde\{x\}\_i\backslash vert$$

One-proportion z-test

$$z=\frac{\widehat{p}-p_0}{\sqrt{p_0(1-p_0)\mathrm{/}n}}z\; =\; \backslash frac\{\backslash hat\{p\}\; -\; p\_0\}\{\backslash sqrt\{p\_0(1-p\_0)/n\}\}$$

One-sample t-test

$$t=\frac{\bar{x}-{\mu }_0}{s\mathrm{/}\sqrt{n}},\nu =n-1t\; =\; \backslash frac\{\backslash bar\{x\}\; -\; \backslash mu\_0\}\{s/\backslash sqrt\{n\}\},\backslash \; \backslash nu\; =\; n\; -\; 1$$

One-sample z-test for a mean (sigma known)

$$z=\frac{\bar{x}-{\mu }_0}{\sigma \mathrm{/}\sqrt{n}}z\; =\; \backslash frac\{\backslash bar\{x\}\; -\; \backslash mu\_0\}\{\backslash sigma/\backslash sqrt\{n\}\}$$

Paired t-test

$$t=\frac{\bar{d}}{s_d\mathrm{/}\sqrt{n}},\nu =n-1t\; =\; \backslash frac\{\backslash bar\{d\}\}\{s\_d/\backslash sqrt\{n\}\},\backslash \; \backslash nu\; =\; n\; -\; 1$$

Two-proportion z-test

$$z=\frac{{\widehat{p}}_1-{\widehat{p}}_2}{\sqrt{\widehat{p}(1-\widehat{p})(\frac{1}{n_1}+\frac{1}{n_2})}}z\; =\; \backslash frac\{\backslash hat\{p\}\_1\; -\; \backslash hat\{p\}\_2\}\{\backslash sqrt\{\backslash hat\{p\}(1-\backslash hat\{p\})\backslash left(\backslash frac\{1\}\{n\_1\}\; +\; \backslash frac\{1\}\{n\_2\}\backslash right)\}\}$$

Two-sample t-test (pooled variances)

$$t=\frac{{\bar{x}}_1-{\bar{x}}_2}{s_p\sqrt{1\mathrm{/}{n}_1+1\mathrm{/}{n}_2}},\nu =n_1+n_2-2t\; =\; \backslash frac\{\backslash bar\{x\}\_1\; -\; \backslash bar\{x\}\_2\}\{s\_p\backslash sqrt\{1/n\_1\; +\; 1/n\_2\}\},\backslash \; \backslash nu\; =\; n\_1\; +\; n\_2\; -\; 2$$

Two-sample t-test from summary statistics

$$t=\frac{{\bar{x}}_1-{\bar{x}}_2}{\sqrt{s_1^2\mathrm{/}{n}_1+s_2^2\mathrm{/}{n}_2}}t\; =\; \backslash frac\{\backslash bar\{x\}\_1\; -\; \backslash bar\{x\}\_2\}\{\backslash sqrt\{s\_1^2/n\_1\; +\; s\_2^2/n\_2\}\}$$

Welch's two-sample t-test (unequal variances)

$$t=\frac{{\bar{x}}_1-{\bar{x}}_2}{\sqrt{s_1^2\mathrm{/}{n}_1+s_2^2\mathrm{/}{n}_2}}t\; =\; \backslash frac\{\backslash bar\{x\}\_1\; -\; \backslash bar\{x\}\_2\}\{\backslash sqrt\{s\_1^2/n\_1\; +\; s\_2^2/n\_2\}\}$$

Nonparametric tests

Friedman test

$$Q=\frac{12}{nk(k+1)}\sum _j{R}_j^2-3n(k+1)Q\; =\; \backslash frac\{12\}\{nk(k+1)\}\backslash sum\_j\; R\_j^2\; -\; 3n(k+1)$$

Kruskal-Wallis H test

$$H=\frac{12}{N(N+1)}\sum \frac{R_i^2}{n_i}-3(N+1)H\; =\; \backslash frac\{12\}\{N(N+1)\}\backslash sum\; \backslash frac\{R\_i^2\}\{n\_i\}\; -\; 3(N+1)$$

Mann-Whitney U test (exact)

$$U_1=R_1-\frac{n_1(n_1+1)}{2}U\_1\; =\; R\_1\; -\; \backslash frac\{n\_1(n\_1+1)\}\{2\}$$

Mann-Whitney U test (large-sample)

$$U_1=R_1-\frac{n_1(n_1+1)}{2}U\_1\; =\; R\_1\; -\; \backslash frac\{n\_1(n\_1+1)\}\{2\}$$

Sign test

$$P=2\min (P(K\le k),P(K\ge k)),K\sim \mathrm{B}\mathrm{i}\mathrm{n}(n,\frac{1}{2})P\; =\; 2\backslash min\backslash left(P(K\; \backslash le\; k),\; P(K\; \backslash ge\; k)\backslash right),\backslash \; K\; \backslash sim\; \backslash mathrm\{Bin\}(n,\; \backslash tfrac12)$$

Wilcoxon signed-rank test (exact)

$$R^{+}=\sum _{d_i>0}\mathrm{rank}(\mathrm{\mid }{d}_i\mathrm{\mid })R^+\; =\; \backslash sum\_\{d\_i\; >\; 0\}\; \backslash operatorname\{rank\}(\backslash vert\; d\_i\backslash vert\; )$$

Wilcoxon signed-rank test (large-sample)

$$R^{+}=\sum _{d_i>0}\mathrm{rank}(\mathrm{\mid }{d}_i\mathrm{\mid })R^+\; =\; \backslash sum\_\{d\_i\; >\; 0\}\; \backslash operatorname\{rank\}(\backslash vert\; d\_i\backslash vert\; )$$

Power and sample size

Power of a one-sample z-test

$$1-\beta =\mathrm{\Phi }(d\sqrt{n_{\text{eff}}}-z_{1-\alpha \mathrm{/}2})+\mathrm{\Phi }(-d\sqrt{n_{\text{eff}}}-z_{1-\alpha \mathrm{/}2})1\; -\; \backslash beta\; =\; \backslash Phi\backslash left(d\backslash sqrt\{n\_\{\backslash text\{eff\}\}\}\; -\; z\_\{1-\backslash alpha/2\}\backslash right)\; +\; \backslash Phi\backslash left(-d\backslash sqrt\{n\_\{\backslash text\{eff\}\}\}\; -\; z\_\{1-\backslash alpha/2\}\backslash right)$$

Power of a two-sample z-test

$$1-\beta =\mathrm{\Phi }(d\sqrt{n_{\text{eff}}}-z_{1-\alpha \mathrm{/}2})+\mathrm{\Phi }(-d\sqrt{n_{\text{eff}}}-z_{1-\alpha \mathrm{/}2})1\; -\; \backslash beta\; =\; \backslash Phi\backslash left(d\backslash sqrt\{n\_\{\backslash text\{eff\}\}\}\; -\; z\_\{1-\backslash alpha/2\}\backslash right)\; +\; \backslash Phi\backslash left(-d\backslash sqrt\{n\_\{\backslash text\{eff\}\}\}\; -\; z\_\{1-\backslash alpha/2\}\backslash right)$$

Sample size per group to compare two means (z-test power)

$$n\approx 2{\left(\frac{z_{1-\alpha \mathrm{/}2}+z_{1-\beta }}{d}\right)}^{2}n\; \backslash approx\; 2\backslash left(\backslash frac\{z\_\{1-\backslash alpha/2\}\; +\; z\_\{1-\backslash beta\}\}\{d\}\backslash right)^2$$

Sample size to estimate a mean

$$n=\lceil {\left(\frac{z_{\alpha \mathrm{/}2}\sigma }{E}\right)}^2\rceil n\; =\; \backslash left\backslash lceil\; \backslash left(\backslash frac\{z\_\{\backslash alpha/2\}\backslash ,\backslash sigma\}\{E\}\backslash right)^2\; \backslash right\backslash rceil$$

Sample size to estimate a proportion

$$n=\lceil \frac{z_{\alpha \mathrm{/}2}^{2}p(1-p)}{E^2}\rceil n\; =\; \backslash left\backslash lceil\; \backslash frac\{z\_\{\backslash alpha/2\}^2\backslash ,p(1-p)\}\{E^2\}\; \backslash right\backslash rceil$$

Probability

Probability of at least one occurrence

$$P(\text{at least one})=1-(1-p{)}^{n}P(\backslash text\{at\; least\; one\})\; =\; 1\; -\; (1\; -\; p)^n$$

Probability distributions

Binomial probability: at least k successes

$$P(X\ge k)=\sum _{j=k}^n\left(\genfrac{}{}{0px}{}{n}{j}\right)p^j(1-p{)}^{n-j}P(X\; \backslash ge\; k)\; =\; \backslash sum\_\{j=k\}^\{n\}\; \backslash binom\{n\}\{j\}\; p^j\; (1-p)^\{n-j\}$$

Binomial probability: at most k successes

$$P(X\le k)=\sum _{j=0}^k\left(\genfrac{}{}{0px}{}{n}{j}\right)p^j(1-p{)}^{n-j}P(X\; \backslash le\; k)\; =\; \backslash sum\_\{j=0\}^\{k\}\; \backslash binom\{n\}\{j\}\; p^j\; (1-p)^\{n-j\}$$

Binomial probability: exactly k successes

$$P(X=k)=\left(\genfrac{}{}{0px}{}{n}{k}\right)p^k(1-p{)}^{n-k}P(X\; =\; k)\; =\; \backslash binom\{n\}\{k\}\; p^k\; (1-p)^\{n-k\}$$

Expected value and variance of a discrete random variable

$$E[X]=\sum x_i{p}_i,\mathrm{Var}(X)=\sum (x_i-\mu {)}^2{p}_{i}E[X]\; =\; \backslash sum\; x\_i\; p\_i,\backslash quad\; \backslash operatorname\{Var\}(X)\; =\; \backslash sum\; (x\_i\; -\; \backslash mu)^2\; p\_i$$

Exponential probability above a value

$$P(X>x)=e^{-\lambda x}P(X\; >\; x)\; =\; e^\{-\backslash lambda\; x\}$$

Exponential probability below a value

Geometric distribution: first success on trial k

$$P(X=k)=(1-p{)}^{k-1}p,P(X\le k)=1-(1-p{)}^{k}P(X\; =\; k)\; =\; (1-p)^\{k-1\}p,\backslash quad\; P(X\; \backslash le\; k)\; =\; 1\; -\; (1-p)^k$$

Hypergeometric probability (sampling without replacement)

$$P(X=k)=\frac{\left(\genfrac{}{}{0px}{}{K}{k}\right)\left(\genfrac{}{}{0px}{}{N-K}{n-k}\right)}{\left(\genfrac{}{}{0px}{}{N}{n}\right)}P(X\; =\; k)\; =\; \backslash frac\{\backslash binom\{K\}\{k\}\backslash binom\{N-K\}\{n-k\}\}\{\backslash binom\{N\}\{n\}\}$$

Inverse normal (normal percentile)

$$x=\mu +\sigma {\mathrm{\Phi }}^{-1}(p)x\; =\; \backslash mu\; +\; \backslash sigma\backslash ,\backslash Phi^\{-1\}(p)$$

Mean and standard deviation of a binomial distribution

$$\mu =np,\sigma =\sqrt{np(1-p)}\backslash mu\; =\; np,\backslash quad\; \backslash sigma\; =\; \backslash sqrt\{np(1-p)\}$$

Negative binomial: r-th success on trial n

$$P(N=n)=\left(\genfrac{}{}{0px}{}{n-1}{r-1}\right)p^r(1-p{)}^{n-r}P(N\; =\; n)\; =\; \backslash binom\{n-1\}\{r-1\}\; p^r\; (1-p)^\{n-r\}$$

Normal probability above a value

$$P(X>x)=1-\mathrm{\Phi }\left(\frac{x-\mu }{\sigma }\right)P(X\; >\; x)\; =\; 1\; -\; \backslash Phi\backslash left(\backslash frac\{x\; -\; \backslash mu\}\{\backslash sigma\}\backslash right)$$

Normal probability below a value

Normal probability between two values

Poisson probability: at least k events

$$P(X\ge k)=1-\sum _{j=0}^{k-1}\frac{e^{-\lambda }{\lambda }^j}{j!}P(X\; \backslash ge\; k)\; =\; 1\; -\; \backslash sum\_\{j=0\}^\{k-1\}\; \backslash frac\{e^\{-\backslash lambda\}\backslash lambda^j\}\{j!\}$$

Poisson probability: at most k events

$$P(X\le k)=\sum _{j=0}^k\frac{e^{-\lambda }{\lambda }^j}{j!}P(X\; \backslash le\; k)\; =\; \backslash sum\_\{j=0\}^\{k\}\; \backslash frac\{e^\{-\backslash lambda\}\backslash lambda^j\}\{j!\}$$

Poisson probability: exactly k events

$$P(X=k)=\frac{e^{-\lambda }{\lambda }^k}{k!}P(X\; =\; k)\; =\; \backslash frac\{e^\{-\backslash lambda\}\backslash lambda^k\}\{k!\}$$

Quality control

Individuals and moving range (I-MR) chart limits

$$UC{L}_x=\bar{x}+2.66\overline{MR}UCL\_x\; =\; \backslash bar\{x\}\; +\; 2.66\backslash ,\backslash overline\{MR\}$$

Process capability (Cp, Cpk) and expected nonconforming

$$C_p=\frac{USL-LSL}{6\sigma },C_{pk}=\min (\frac{USL-\mu }{3\sigma },\frac{\mu -LSL}{3\sigma })C\_p\; =\; \backslash frac\{USL\; -\; LSL\}\{6\backslash sigma\},\backslash quad\; C\_\{pk\}\; =\; \backslash min\backslash left(\backslash frac\{USL\; -\; \backslash mu\}\{3\backslash sigma\},\; \backslash frac\{\backslash mu\; -\; LSL\}\{3\backslash sigma\}\backslash right)$$

X-bar and R control chart limits

$$UC{L}_{\bar{x}}=\overline{\stackrel{ˉ}{x}}+A_2\bar{R},UC{L}_R=D_4\bar{R}UCL\_\{\backslash bar\{x\}\}\; =\; \backslash bar\{\backslash bar\{x\}\}\; +\; A\_2\backslash bar\{R\},\backslash quad\; UCL\_R\; =\; D\_4\backslash bar\{R\}$$

X-bar and S control chart limits

$$UC{L}_{\bar{x}}=\overline{\stackrel{ˉ}{x}}+A_3\bar{s},UC{L}_s=B_4\bar{s}UCL\_\{\backslash bar\{x\}\}\; =\; \backslash bar\{\backslash bar\{x\}\}\; +\; A\_3\backslash bar\{s\},\backslash quad\; UCL\_s\; =\; B\_4\backslash bar\{s\}$$

c chart limits (defects per unit)

$$\bar{c}\pm 3\sqrt{\bar{c}}\backslash bar\{c\}\; \backslash pm\; 3\backslash sqrt\{\backslash bar\{c\}\}$$

p chart limits (fraction defective)

$$\bar{p}\pm 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}\backslash bar\{p\}\; \backslash pm\; 3\backslash sqrt\{\backslash frac\{\backslash bar\{p\}(1-\backslash bar\{p\})\}\{n\}\}$$

Regression and correlation

Confidence interval for a correlation (Fisher z)

$$\tanh (\mathrm{atanh}(r)\pm \frac{z_{\alpha \mathrm{/}2}}{\sqrt{n-3}})\backslash tanh\backslash left(\backslash operatorname\{atanh\}(r)\; \backslash pm\; \backslash frac\{z\_\{\backslash alpha/2\}\}\{\backslash sqrt\{n-3\}\}\backslash right)$$

Covariance

$$s_{xy}=\frac{1}{n-1}\sum (x_i-\bar{x})(y_i-\bar{y})s\_\{xy\}\; =\; \backslash frac\{1\}\{n-1\}\backslash sum\; (x\_i\; -\; \backslash bar\{x\})(y\_i\; -\; \backslash bar\{y\})$$

Durbin-Watson statistic

$$d=\frac{\sum _{t=2}^n(e_t-e_{t-1}{)}^2}{\sum _{t=1}^n{e}_t^2}d\; =\; \backslash frac\{\backslash sum\_\{t=2\}^\{n\}\; (e\_t\; -\; e\_\{t-1\})^2\}\{\backslash sum\_\{t=1\}^\{n\}\; e\_t^2\}$$

Kendall's tau-b rank correlation

$${\tau }_b=\frac{C-D}{\sqrt{(n_0-n_1)(n_0-n_2)}}\backslash tau\_b\; =\; \backslash frac\{C\; -\; D\}\{\backslash sqrt\{(n\_0\; -\; n\_1)(n\_0\; -\; n\_2)\}\}$$

Multiple linear regression

$$\widehat{\beta }=(X^{\mathrm{\top }}X{)}^{-1}{X}^{\mathrm{\top }}y,R^2=1-\frac{SSE}{SST}\backslash hat\{\backslash beta\}\; =\; (X^\backslash top\; X)^\{-1\}X^\backslash top\; y,\backslash quad\; R^2\; =\; 1\; -\; \backslash frac\{SSE\}\{SST\}$$

Pearson correlation coefficient

$$r=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum (x_i-\bar{x}{)}^2\sum (y_i-\bar{y}{)}^2}}r\; =\; \backslash frac\{\backslash sum\; (x\_i\; -\; \backslash bar\{x\})(y\_i\; -\; \backslash bar\{y\})\}\{\backslash sqrt\{\backslash sum\; (x\_i\; -\; \backslash bar\{x\})^2\; \backslash sum\; (y\_i\; -\; \backslash bar\{y\})^2\}\}$$

Prediction from a least squares line

$${\widehat{y}}_0=b_0+b_1{x}_0\backslash hat\{y\}\_0\; =\; b\_0\; +\; b\_1\; x\_0$$

Significance test for a correlation coefficient

$$t=r\sqrt{\frac{n-2}{1-r^2}}t\; =\; r\backslash sqrt\{\backslash frac\{n-2\}\{1-r^2\}\}$$

Simple linear regression (least squares line)

$$\widehat{y}=b_0+b_{1}x,b_1=\frac{S_{xy}}{S_{xx}},b_0=\bar{y}-b_1\bar{x}\backslash hat\{y\}\; =\; b\_0\; +\; b\_1\; x,\backslash quad\; b\_1\; =\; \backslash frac\{S\_\{xy\}\}\{S\_\{xx\}\},\backslash quad\; b\_0\; =\; \backslash bar\{y\}\; -\; b\_1\backslash bar\{x\}$$

Spearman rank correlation

$${\rho }_s=r(\mathrm{rank}(x),\mathrm{rank}(y))\backslash rho\_s\; =\; r\backslash left(\backslash operatorname\{rank\}(x),\; \backslash operatorname\{rank\}(y)\backslash right)$$

Reliability and survival

Exponential reliability (constant failure rate)

$$R(t)=e^{-\lambda t},MTTF=1\mathrm{/}\lambda R(t)\; =\; e^\{-\backslash lambda\; t\},\backslash quad\; MTTF\; =\; 1/\backslash lambda$$

Kaplan-Meier survival estimate

$$\widehat{S}(t)=\prod _{t_i\le t}(1-\frac{d_i}{n_i})\backslash hat\{S\}(t)\; =\; \backslash prod\_\{t\_i\; \backslash le\; t\}\backslash left(1\; -\; \backslash frac\{d\_i\}\{n\_i\}\backslash right)$$

Log-rank test comparing two survival curves

$${\chi }^2=\frac{{(\sum (d_{1i}-e_{1i}))}^2}{\sum v_i}\backslash chi^2\; =\; \backslash frac\{\backslash left(\backslash sum\; (d\_\{1i\}\; -\; e\_\{1i\})\backslash right)^2\}\{\backslash sum\; v\_i\}$$

Reliability of a parallel system

$$R_p=1-\prod (1-R_i)R\_p\; =\; 1\; -\; \backslash prod\; (1\; -\; R\_i)$$

Reliability of a series system

$$R_s=\prod R_{i}R\_s\; =\; \backslash prod\; R\_i$$

Weibull reliability, hazard rate and MTTF

$$R(t)=e^{-(t\mathrm{/}\eta {)}^{\beta }},h(t)=\frac{\beta }{\eta }{\left(\frac{t}{\eta }\right)}^{\beta -1}R(t)\; =\; e^\{-(t/\backslash eta)^\backslash beta\},\backslash quad\; h(t)\; =\; \backslash frac\{\backslash beta\}\{\backslash eta\}\backslash left(\backslash frac\{t\}\{\backslash eta\}\backslash right)^\{\backslash beta-1\}$$

k-out-of-n system reliability

$$R=\sum _{j=k}^n\left(\genfrac{}{}{0px}{}{n}{j}\right)r^j(1-r{)}^{n-j}R\; =\; \backslash sum\_\{j=k\}^\{n\}\backslash binom\{n\}\{j\}\; r^j\; (1-r)^\{n-j\}$$

Sampling distributions

Probability that a sample mean is above a value

Probability that a sample mean is below a value

Out of scope

• Open-ended explanation, essay writing or tutoring dialogue: it is a solver, not a chatbot.
• Questions outside its certified statistics capabilities, and any question whose stated numbers it cannot all use — it refuses with the reason instead.
• High-stakes engineering, medical or legal decisions without independent review.

Limitations

• It answers what its certified capabilities cover (descriptive statistics, probability distributions, confidence intervals, hypothesis tests, regression and correlation, experimental design, quality control and reliability); anything else is refused with a reason.
• ask() reads documented question forms and common phrasings; unusual wording may be refused — use solve() for guaranteed results.
• Results are as correct as the certified relations and their stated conditions; a certified relation is a checked model, not a claim about every physical situation.

Citation

@software{statistics_ai999x_1_0_3,
  title  = {statistics-ai999x: an offline, verified statistics solver},
  author = {Farabi, Ibnul Jaif}, organization = {AI999X},
  year   = {2026}, version = {1.0.3},
  url    = {https://huggingface.co/AI999X/statistics-ai999x}
}

Contact

Questions, bug reports and requests: open a discussion on https://huggingface.co/AI999X/statistics-ai999x/discussions or visit https://ai999x.com.

Support the project

statistics-ai999x is free and open source. If it helps you, you can support further development on Buy Me a Coffee.

Created and engineered by Ibnul Jaif Farabi · AI999X · ai999x.com