--- name: nist-stat-handbook description: "Knowledge base from the NIST/SEMATECH e-Handbook of Statistical Methods (NIST HB 151) — the practical statistics reference for engineering, metrology, and quality. Use for: exploratory data analysis and the four univariate assumptions (4-plot); measurement process characterization including bias/precision, calibration designs, gauge R&R, and ISO/GUM uncertainty budgets; production process characterization (stability vs. capability); process modeling and regression (LS/WLS/NLS/LOESS); design of experiments (screening, fractional/factorial, response-surface, Taguchi); statistical process control (Shewhart/CUSUM/EWMA charts, capability indices, acceptance sampling); product/process comparisons (hypothesis tests and confidence intervals for 1/2/3+ groups, ANOVA, multiple comparisons); and reliability (lifetime & repair-rate models, accelerated testing, reliability growth). Scope limits: this is applied frequentist statistics for measurement and quality — it does NOT reproduce the per-distribution formula galleries, worked case studies, datasets, plot images, or Dataplot/R code of the original web Handbook (those are described, not copied); it is thin on modern machine learning, Bayesian methods beyond conjugate reliability priors, time-series/forecasting, and Bayesian experimental design." --- # NIST/SEMATECH e-Handbook of Statistical Methods (NIST HB 151) **Source**: NIST/SEMATECH (US Government work, public domain) | **Chapters**: 8 ## When to use Use this skill when you need applied frequentist statistics for engineering, metrology, or quality work, drawing from the NIST/SEMATECH e-Handbook of Statistical Methods (NIST HB 151). It is the right pack for exploratory data analysis and the four univariate assumptions (4-plot), measurement process characterization including bias/precision, calibration designs, gauge R&R, and ISO/GUM uncertainty budgets, production process characterization covering stability vs. capability, process modeling and regression (LS/WLS/NLS/LOESS), design of experiments (screening, fractional/factorial, response-surface, Taguchi), statistical process control (Shewhart/CUSUM/EWMA charts, capability indices, acceptance sampling), product/process comparisons (hypothesis tests and confidence intervals for one, two, or more groups, ANOVA, multiple comparisons), and reliability (lifetime and repair-rate models, accelerated testing, reliability growth). **Prerequisites:** none, plain Markdown; no MCP server, API key, or licence tier needed at runtime. ## How to Use This Skill - **Without arguments** — load the Core Frameworks below: the EDA mindset, the characterization→modeling→improvement→control→comparison→reliability arc, and the routing rules for picking a method. - **With a statistical task** — route to the right chapter and pattern. Examples: "is my data in control?" → ch01 (4-plot); "state the uncertainty of this result" → ch02 (GUM budget); "design an experiment to optimize yield" → ch05 (DOE); "which control chart for small drifts?" → ch06 (CUSUM/EWMA); "compare three machines" → ch07 (ANOVA + multiple comparisons); "fit a life distribution" → ch08 (Weibull). - **With a chapter** — `ch01` EDA · `ch02` measurement/uncertainty · `ch03` process characterization · `ch04` regression · `ch05` DOE · `ch06` SPC · `ch07` comparisons · `ch08` reliability. Supporting files: `glossary.md`, `patterns.md`, `cheatsheet.md`. **Prerequisites:** none — plain Markdown; no MCP server, API key, or runtime licence needed. > **What this Handbook is.** A practitioner's reference for *engineering statistics* — > measurement, process, and quality work — not a mathematical-statistics textbook. Its > through-line is the **engineering scientific method**: characterize a process, model it, > improve it by experiment, hold the gains with control charts, compare alternatives with > the right test, and predict how long it will last. The signature attitude (from ch01) > is **look at the data before imposing a model.** ## Core Frameworks & Mental Models ### The EDA stance — let the data reveal the model (ch01) Classical analysis imposes a model (normality, linearity), then estimates and tests it. **Exploratory Data Analysis (EDA)** holds that decision back: plot the data first and let its structure surface. The unifying baseline is the **univariate model** `Y = c + ε` and its **four assumptions** — the data have (1) fixed location, (2) fixed scale, (3) randomness (no autocorrelation), and (4) a fixed distribution. The **4-Plot** (run-sequence, lag, histogram, normal-probability) tests all four at a glance. Pass all four ⇒ the process is "in statistical control" and predictable. A failed panel is a *diagnostic* pointing at a physical cause. **Residuals** are the universal diagnostic: an adequate model leaves structureless residuals. ### Characterizing a measurement process — goodness decomposed (ch02) Treat measurements as the output of a process and ask how *good* they are. Goodness splits into **bias** (quantitative offset from the reference/true value), **short-term variability / precision** (level-1), **long-term / day-to-day variability** (level-2, often *larger* than precision for very precise instruments), and **uncertainty**. The engine is the **check standard** — a database of repeated measurements on a stable surrogate — monitored by an **individuals Shewhart chart** (and **EWMA** for small shifts). **Calibration designs** are redundant least-squares intercomparison schedules (anchored by a **restraint** = the known reference value) that assign values while cancelling constant **left-right bias** and linear **drift**. **Gauge R&R** nested designs separate repeatability, reproducibility, and stability. Uncertainty follows the **ISO/GUM** framework: every component a standard deviation, grouped by **Type A** (from statistical analysis) vs. **Type B** (other means), combined by **root-sum-squares** with **sensitivity coefficients**, expanded by a coverage factor `U = k·u` (k≈2 ⇒ ~95%), with **Welch-Satterthwaite** for unknown degrees of freedom. ### Characterizing a production process — stability before capability (ch03) Frame the process as a **black box** (inputs / controlled factors / uncontrolled noise / output), then expand drivers with a **fishbone** diagram. Distribution = **location + spread + shape**. Models span the **Continuous Linear Model (CLM)**, **ANOVA**, and **discrete (contingency/chi-square)** forms. Two rules dominate: **add variances, never standard deviations** when combining error; and **establish stability before claiming capability** — Cp/Cpk on an out-of-control process are meaningless. Interpret graphically, then confirm numerically. The most expensive mistake is skimping on planning. ### Modeling a process — function plus error (ch04) The general statistical model is `y = f(x; β) + ε`: a deterministic function of the factors plus random error. Four method families build it — **linear least squares**, **nonlinear least squares**, **weighted least squares (WLS)** for non-constant variance, and **LOESS** for local, assumption-light smoothing. A model is used to **estimate, predict, calibrate, and optimize**. Six standard assumptions (random errors, mean zero, constant variance, known distribution, independence, correct functional form) are checked through **residual analysis** — the same look-at-the-residuals discipline as EDA. ### Improving by experiment — sequential DOE (ch05) **Design of Experiments** varies factors *together* to estimate effects efficiently; its four objective classes are **comparative, screening, response-surface, and regression**. The strategy is a **staircase**, not one big experiment: screen many factors cheaply with **fractional factorial / Plackett-Burman** designs, test curvature with **center points**, **foldover** to de-alias, then spend three-level **response-surface** runs (**CCD**, **Box-Behnken**) near the optimum. **Resolution** (III/IV/V) is the confounding dial; the **defining relation** and **generators** set the **aliasing**. **One-factor-at-a-time is a trap** — it misses interactions. **Taguchi** robust design (parameter design via inner/outer arrays, then tolerance design) minimizes sensitivity to noise. Always randomize; finish with **confirmation runs**. ### Holding the gains — statistical process control (ch06) **SPC/SQC** uses control charts to separate common-cause from special-cause variation. **Phase I** builds and validates limits from history; **Phase II** monitors new data. Chart families: **Shewhart** (X-bar & R/s, individuals) for large shifts; **CUSUM** (V-mask) and **EWMA** (memory depth λ) for small sustained drifts; **attributes** charts for counts; **Hotelling's T²** for correlated multivariate data. **Control limits ≠ specification limits** (voice of the process vs. voice of the customer). Compare charts by **Average Run Length (ARL)**: sensitivity is bought with false alarms (**WECO rules**). Decide the response in advance with an **OCAP**. **Acceptance sampling** (OC curve, AQL/LTPD, producer's/ consumer's risk, AOQ/AOQL) governs lot accept/reject decisions. ### Comparing alternatives — the right test (ch07) Route by **how many processes**: one (t/z, chi-square, exact-binomial/Wilson), two (two-sample/paired t, F-test, two-proportion z / Fisher exact), or three-plus (**ANOVA** then a **multiple-comparison** procedure — Tukey, Scheffé, Bonferroni, Tukey-Kramer, Marascuilo; **Kruskal-Wallis** if non-normal). The spine is **test ⇔ interval duality**: reject exactly when the hypothesized value falls outside the matching confidence interval. ANOVA **analyzes variances to draw conclusions about means**; fixed vs. random effects changes the inferential target. Use **exact** methods for small samples, **approximations** (with their validity gates) for large; **"failure to reject" is not proof of equality.** ### Predicting how long it lasts — reliability (ch08) **Reliability is quality over time** — R(t) = probability of surviving past t. The first fork is **non-repairable vs. repairable**. Non-repairable units get a **lifetime distribution** (Weibull, lognormal, exponential, gamma, extreme value, Birnbaum-Saunders), reasoned about via the **hazard rate** h(t) (a *speedometer*) and the **bathtub curve**; data are **censored** (Type I by time, Type II by count) and information scales with the number of **failures**. Repairable systems get a **repair-rate** model (HPP; **NHPP Power Law / Duane-AMSAA** for **reliability growth** under **TAAF**) reasoned about via N(t), M(t), m(t) — they have *no single failure rate*. **System reliability** is built **bottom-up** from series/parallel/k-of-n/standby blocks (series = product of part reliabilities). **Accelerated testing** needs a justified **acceleration model** (Arrhenius, Eyring, Coffin-Manson) that rescales time. --- ## Chapter Index | # | Chapter file | Topic | Key content | |---|---|---|---| | 1 | [ch01](chapters/ch01-nist-stat-handbook-exploratory-data-analysis.md) | Exploratory Data Analysis | EDA philosophy, univariate model, four assumptions, the 4-plot, residuals, problem categories | | 2 | [ch02](chapters/ch02-nist-stat-handbook-measurement-process-characterization.md) | Measurement Process Characterization | Bias/precision/uncertainty, check standards, Level-1/2/3 model, calibration designs, gauge R&R, **ISO/GUM uncertainty budgets** | | 3 | [ch03](chapters/ch03-nist-stat-handbook-production-process-characterization.md) | Production Process Characterization | Black-box & fishbone framing, CLM/ANOVA/discrete models, error propagation, stability vs. capability | | 4 | [ch04](chapters/ch04-nist-stat-handbook-process-modeling.md) | Process Modeling | `y = f(x;β)+ε`, LS/WLS/NLS/LOESS, four uses of a model, six assumptions, residual analysis | | 5 | [ch05](chapters/ch05-nist-stat-handbook-process-improvement-doe.md) | Process Improvement (DOE) | Design objective classes, factorial/fractional/Plackett-Burman, resolution & confounding, response surface (CCD/Box-Behnken), Taguchi | | 6 | [ch06](chapters/ch06-nist-stat-handbook-process-monitoring-and-control.md) | Process Monitoring & Control | SPC, Shewhart/CUSUM/EWMA/attributes/multivariate charts, Phase I/II, ARL, capability, acceptance sampling | | 7 | [ch07](chapters/ch07-nist-stat-handbook-product-and-process-comparisons.md) | Product & Process Comparisons | Hypothesis tests & intervals for 1/2/3+ groups, test/interval duality, ANOVA, multiple comparisons, goodness-of-fit | | 8 | [ch08](chapters/ch08-nist-stat-handbook-assessing-product-reliability.md) | Assessing Product Reliability | Lifetime & repair-rate models, bathtub curve, censoring, acceleration models, system reliability, reliability growth, Bayesian | ## Topic Index - **4-Plot / four univariate assumptions** → ch01, cheatsheet - **Acceleration models (Arrhenius/Eyring/Coffin-Manson)** → ch08 - **Acceptance sampling / OC curve / AQL / LTPD** → ch06, cheatsheet - **ANOVA (one-way / two-way / fixed-random effects)** → ch07, ch03, ch05 - **Average Run Length (ARL)** → ch06 - **Bathtub curve / hazard rate** → ch08 - **Bias / accuracy (measurement)** → ch02 - **Black-box & fishbone (cause-and-effect) diagram** → ch03 - **Calibration designs / restraint / drift & left-right bias** → ch02 - **Capability (Cp, Cpk) / stability vs. capability** → ch06, ch03, cheatsheet - **Censoring (Type I / Type II)** → ch08 - **Check standard** → ch02, patterns - **Confidence intervals / test–interval duality** → ch07, cheatsheet - **Confounding / aliasing / resolution (III/IV/V)** → ch05 - **Control charts (Shewhart / CUSUM / EWMA / attributes / T²)** → ch06, cheatsheet - **Design of Experiments (DOE) / objective classes** → ch05, patterns - **EDA (exploratory data analysis) philosophy** → ch01 - **Error propagation / add variances / sensitivity coefficients** → ch02, ch03, patterns - **Factorial & fractional factorial / Plackett-Burman** → ch05 - **Foldover / center points** → ch05 - **Gauge R&R (repeatability / reproducibility / stability)** → ch02 - **Goodness-of-fit (Anderson-Darling, K-S, chi-square, Shapiro-Wilk)** → ch07, ch08 - **Hotelling's T² (multivariate SPC)** → ch06 - **Hypothesis tests by group count (1/2/3+)** → ch07, cheatsheet - **ISO/GUM uncertainty / Type A & Type B / expanded uncertainty** → ch02, patterns, cheatsheet - **Lifetime distributions (Weibull / lognormal / exponential / gamma)** → ch08, glossary - **LOESS / local regression** → ch04 - **Multiple-comparison procedures (Tukey / Scheffé / Bonferroni)** → ch07 - **Out-of-Control Action Plan (OCAP)** → ch06 - **Phase I vs. Phase II (SPC)** → ch06 - **Process model `y = f(x;β)+ε` / regression (LS/WLS/NLS)** → ch04 - **Randomized block / completely randomized designs** → ch05 - **Reliability R(t) / non-repairable vs. repairable** → ch08, patterns - **Reliability growth / Duane / AMSAA / TAAF** → ch08 - **Residual analysis** → ch04, ch01 - **Response surface (CCD / Box-Behnken / RSM)** → ch05 - **Run-sequence / lag / histogram / normal-probability plots** → ch01 - **Sampling / stratification / sample size** → ch03 - **Sequential experimentation strategy** → ch05, patterns - **Series / parallel / k-of-n / standby system models** → ch08 - **Statistical Process Control (SPC / SQC)** → ch06 - **Taguchi / parameter & tolerance design** → ch05 - **Three-level standard-deviation model (Level 1/2/3)** → ch02 - **Uncertainty budgets / Welch-Satterthwaite** → ch02, cheatsheet - **WECO rules / control vs. specification limits** → ch06 ## Supporting Files - [glossary.md](glossary.md) — key statistical, metrology, DOE, SPC, and reliability terms, alphabetical, with chapter references - [patterns.md](patterns.md) — ten reusable techniques (4-plot gate, black-box framing, GUM uncertainty budget, stability-before-capability, chart selection, sequential DOE, comparison-test routing, reliability fork, check-standard control) with When/How/Trade-offs - [cheatsheet.md](cheatsheet.md) — chapter map, the four assumptions, comparison-test selector, control-chart selector, DOE & reliability decision tables, uncertainty-in-one-screen, and tells & smells --- ## Scope & Limits **Covers**: applied (frequentist) statistics for engineering, metrology, and quality across the eight Handbook chapters — exploratory data analysis; measurement-process characterization and ISO/GUM uncertainty; production-process characterization; process modeling and regression; design of experiments; statistical process control and acceptance sampling; hypothesis testing, confidence intervals, ANOVA and multiple comparisons; and product reliability (lifetime models, accelerated testing, reliability growth, basic Bayesian reliability). **Does not cover / thin on**: the original web Handbook is an HTML/web-native reference of many short interlinked pages with extensive **per-distribution formula galleries, worked case studies, datasets, plot/figure images, and Dataplot/R code blocks** — this pack **synthesizes the concepts and method-selection logic in original words and deliberately does not reproduce** those galleries, case studies, datasets, images, or code (they are described, not copied — a quality and licence-safety choice). It is also **thin on**: modern machine learning / statistical learning; Bayesian methods beyond conjugate reliability priors; time-series analysis and forecasting; Bayesian/optimal experimental design; survey/causal-inference methodology; and software-specific how-tos. For SE-process context see `sebok`, `dau-se-guidebook`, `nasa-npr-7123`; this pack is the quantitative-methods companion to those. **Source version**: NIST/SEMATECH e-Handbook of Statistical Methods, **NIST Handbook 151** (2012 edition, last updated October 2022). **Public domain** — a work of the U.S. Government (17 U.S.C. 105); free to reproduce, transform, and redistribute, including commercially. SEMATECH co-developed the content; attribution to NIST/SEMATECH is a courtesy, not an obligation, and neither endorses this pack.