Statistics

Define constructs, operational measures, units, and categorical or numerical variables. Judge validity, reliability, precision, and the limits of proxy measurements.

Organize observations as rows, columns, keys, and metadata while recording where each value came from. Recognize administrative, experimental, survey, sensor, and public data sources.

Use ratios, percentages, powers, logarithms, functions, summation notation, and simple algebra in statistical formulas. Read equations and rearrange them without losing sight of what each quantity means.

Build frequency tables, cross-tabulations, grouped summaries, and rates with appropriate denominators. Detect misleading percentages and totals that hide subgroup differences.

Choose bar charts, histograms, box plots, scatterplots, and other displays that match the variables and question. Apply clear scales, labels, and comparisons while avoiding distorted graphics.

Calculate and interpret means, medians, quantiles, ranges, variance, standard deviation, and interquartile range. Compare groups using both typical values and variation.

Describe symmetry, skewness, tails, modes, and outliers, then use standard scores and transformations when useful. Connect a distribution’s shape to the process that could have produced it.

Import, inspect, filter, join, reshape, summarize, and visualize data with spreadsheets and a scripting environment such as R or Python. Keep commands in reusable scripts rather than relying on undocumented clicks.

Find invalid values, duplicate records, inconsistent categories, unit errors, and impossible combinations. Create validation checks and preserve the raw data while producing an auditable analysis table.

Trace statistics from censuses, probability, and error theory through experimental design, sampling, modern computing, and data science. Connect major ideas to the institutions and practical problems that shaped them.

Distinguish observational, cross-sectional, cohort, case-control, and prospective designs. Identify selection bias, information bias, confounding, reverse causation, and other threats to credible conclusions.

Define random experiments, outcomes, events, and sample spaces. Calculate probabilities using complements, unions, intersections, and the core probability rules.

Count ordered and unordered outcomes using permutations, combinations, and the multiplication principle. Apply counting methods to sampling, lotteries, reliability, and repeated trials.

Calculate conditional probabilities and test whether events are independent. Use Bayes’ theorem to update probabilities from evidence, including medical-test and classification examples.

Represent uncertain numerical outcomes with probability mass functions, density functions, and cumulative distribution functions. Calculate expected values, variances, and transformed outcomes.

Work with Bernoulli, binomial, geometric, negative binomial, hypergeometric, and Poisson models. Select a model by matching its assumptions to the data-generating process.

Use uniform, normal, exponential, gamma, beta, chi-square, t, and F distributions. Interpret density, tail area, quantiles, and the role of distribution parameters.

Describe two or more random variables through joint, marginal, and conditional distributions. Calculate covariance and correlation while recognizing that zero correlation need not imply independence.

See why averages stabilize and why many sampling distributions become approximately normal. Use the law of large numbers and central limit theorem while checking when their approximations are unreliable.

Describe how a statistic varies across repeated samples and calculate standard errors. Derive common sampling distributions for means, proportions, variances, and differences.

Select samples with simple random, stratified, cluster, systematic, and multistage designs. Calculate weights, account for unequal selection, and address coverage error and nonresponse.

Design randomized experiments with controls, replication, blocking, blinding, and factorial treatments. Recognize ethical limits and practical failures such as noncompliance, attrition, and interference.

Estimate unknown parameters using moments and maximum likelihood. Evaluate estimators through bias, variance, consistency, efficiency, and mean squared error.

Construct and interpret confidence intervals for common parameters. Distinguish confidence level from the probability of a fixed parameter and check the assumptions behind each interval.

Set null and alternative hypotheses, calculate test statistics and p-values, and make decisions at a chosen significance level. Separate statistical significance from practical importance.

Reason about Type I and Type II errors, effect size, statistical power, and minimum detectable effects. Plan sample sizes without treating conventional thresholds as automatic rules.

Use Monte Carlo simulation, permutation tests, and bootstrap intervals when formulas are unavailable or assumptions are doubtful. Diagnose simulation error and choose resampling schemes that preserve the data structure.

Analyze proportions, contingency tables, odds ratios, and relative risks. Apply exact, chi-square, and trend tests while checking sparse-cell and dependence problems.

Compare one or two groups using t procedures, rank tests, sign tests, and paired analyses. Choose between parametric and nonparametric methods based on the design and target quantity.

Compare several group means with one-way, factorial, repeated-measures, and blocked analysis of variance. Interpret main effects and interactions rather than relying only on an overall test.

Control family-wise error rates and false discovery rates when many claims are tested. Apply planned contrasts, post-hoc comparisons, and multiplicity adjustments without hiding the full set of analyses.

Fit a straight-line relationship between a response and one predictor using least squares. Interpret slope, intercept, residual variation, confidence intervals, and prediction intervals.

Model an outcome with several predictors, indicator variables, interactions, and transformations. Separate adjusted association from causation and interpret coefficients in their proper units.

Check linearity, residual behavior, influential observations, leverage, collinearity, and changing variance. Revise or limit a model when its assumptions do not match the data.

Estimate binary outcomes with logistic regression and interpret probabilities, log odds, odds ratios, and marginal effects. Assess discrimination, calibration, separation, and classification thresholds.

Model counts and event rates using Poisson and negative binomial regression with exposure offsets. Handle overdispersion, excess zeros, and rate comparisons.

Represent curved relationships with polynomials, splines, generalized additive models, and nonlinear functions. Control flexibility so the fitted pattern captures signal rather than noise.

Compare candidate models using holdout data, cross-validation, information criteria, and prediction error. Prevent leakage and distinguish explanatory modeling from predictive modeling.

Classify missingness mechanisms and measure how missing values change conclusions. Use principled imputation, multiple imputation, weighting, and sensitivity analysis instead of automatic row deletion.

Account for noisy predictors, imperfect outcomes, misclassification, and uncertain instruments. Use calibration data and sensitivity analysis to show how measurement error affects estimates.

Use resistant summaries, robust regression, trimming, winsorization, and influence functions when contamination or heavy tails threaten conventional methods. Compare robust results with the original analysis transparently.

Study sufficiency, completeness, information, likelihood ratios, and large-sample behavior. Derive why common estimators and tests work and where asymptotic approximations can fail.

Choose procedures by balancing losses, risks, uncertainty, and available actions. Connect hypothesis tests, estimators, and classification rules through a common decision framework.

Represent uncertainty with prior distributions, likelihoods, and posterior distributions. Build conjugate models and interpret credible intervals, posterior probabilities, and predictive distributions.

Approximate difficult posterior distributions with Markov chain Monte Carlo and related numerical methods. Check convergence, effective sample size, mixing, and sensitivity using modern probabilistic programming tools.

Build multilevel Bayesian models, run prior and posterior predictive checks, and compare models using predictive performance. Revise models through a transparent cycle of criticism and improvement.

Analyze clustered, repeated, and hierarchical observations with random effects and mixed models. Separate within-group from between-group variation and produce partial-pooling estimates.

Analyze time-to-event outcomes with censoring, survival curves, hazard functions, and proportional hazards models. Check proportionality and handle competing risks and time-dependent predictors.

Describe trend, seasonality, autocorrelation, and stationarity in ordered observations. Build and validate forecasting models while respecting temporal order and changing conditions.

Model data tied to locations using spatial dependence, variograms, kriging, and areal models. Recognize boundary effects, spatial confounding, and misleading maps.

Work with vectors, matrices, covariance matrices, and multivariate normal models. Compare several outcomes or groups while accounting for dependence among variables.

Reduce many correlated variables with principal component analysis and model hidden constructs with factor analysis. Interpret loadings, scores, explained variation, and rotation choices.

Frame prediction as supervised statistical learning and evaluate regression or classification models on unseen data. Use sound baselines, cross-validation, calibration, and task-appropriate metrics.

Fit ridge, lasso, and elastic-net models when predictors are numerous or strongly correlated. Tune penalties, interpret sparse solutions carefully, and account for selection uncertainty.

Build decision trees, random forests, and gradient-boosted models for nonlinear prediction. Tune them with validation data and explain predictions without mistaking importance scores for causal effects.

Group observations using hierarchical clustering, k-means, density methods, and finite mixture models. Evaluate stability and avoid treating every algorithmic grouping as a real population category.

Define causal effects with potential outcomes and causal diagrams. Estimate effects using randomization, adjustment, matching, weighting, instrumental variables, regression discontinuity, and difference-in-differences when their assumptions are defensible.

Use flexible models to estimate heterogeneous treatment effects and nuisance functions while protecting causal targets from overfitting. Apply double machine learning, causal forests, cross-fitting, and policy evaluation with careful validation.

Combine results across studies using fixed-effect and random-effects models. Assess heterogeneity, publication bias, study quality, and the limits of pooled evidence.

Monitor processes with control charts, capability measures, acceptance sampling, and designed improvement cycles. Distinguish ordinary variation from signals that call for investigation.

Design A/B tests and sequential studies whose data are reviewed while they run. Manage repeated looks, stopping rules, adaptive allocation, novelty effects, and guardrail metrics.

Construct prediction sets with finite-sample coverage under clear assumptions. Apply split, cross, and weighted conformal methods while checking how coverage varies across important subgroups.

Protect individuals while publishing useful summaries through access controls, aggregation, disclosure limitation, and differential privacy. Track privacy budgets and explain the accuracy–privacy tradeoff.

Create artificial records for testing, sharing, or model development without claiming they are real observations. Evaluate fidelity, privacy leakage, bias, and whether synthetic data support the intended analysis.

Organize code, data, environments, tests, version control, notebooks, and reports so another person can rerun the analysis. Build automated pipelines that record inputs, decisions, outputs, and software versions.

Use generative AI to assist with code, documentation, formula translation, and analysis planning while independently checking every result. Protect confidential data and detect fabricated methods, invalid assumptions, and plausible-looking numerical errors.

Carry a real project from question and analysis plan through data acquisition, cleaning, modeling, validation, sensitivity checks, delivery, and follow-up. Produce a complete analysis package with data documentation, executable code, results, and limitations.

Write clear methods and results, select honest graphics, and translate uncertainty for technical and nontechnical audiences. Report effect sizes, intervals, assumptions, missingness, and practical consequences rather than isolated p-values.

Handle consent, fairness, confidentiality, conflicts of interest, selective reporting, data dredging, and harmful uses of models. Apply professional guidance and document decisions that affect people or public knowledge.

Map roles in research, government, health, business, policy, and data science, along with common graduate and certification paths. Build proof of skill through reproducible projects, choose a specialty, contribute to open work, and stay current through journals, conferences, standards, and practitioner communities.