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

Install · Capability guide · 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.

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.
• 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

Quickstart

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.

Command line: statistics-ai999x "<question>", statistics-ai999x --capabilities. 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.

Worked examples

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.

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