Understanding statistical thinking through real-world models.
Interactive visual explanations of probability, risk, and statistics (Poisson frequency, loss severity, Monte Carlo, mortality, credit risk, portfolio optimization, statistical learning, AI and LLM risk, game theory, decision theory) for actuaries, FRM, CFA, and data professionals.
Beyond Averages turns the bread-and-butter mathematics of probability into clickable, sliderable demos: nineteen interactive modules covering dice and card probability, claim frequency, loss severity, expected loss, Monte Carlo simulation, correlation and diversification, mortality and life-assurance pricing, worked actuarial examples, a logistic-regression risk scorer, statistical learning fitted live (credit scoring with ROC curves, fraud detection with decision trees, customer segmentation with k-means), the statistics of AI itself (LLM token sampling, validating AI error rates, conformal prediction, AI monoculture risk), game-theoretic equilibria, and decision theory under uncertainty. Built for actuaries, risk and finance professionals, FRM and CFA candidates, and students who want intuition before formulas.
Interactive modules
- Dice & convolution
Interactive 3D dice simulator. Roll up to five dice and watch the empirical sum histogram converge to the exact convolution probabilities: uniform to triangular to bell-shaped, the central limit theorem made visible.
- Cards & hypergeometric
Interactive card-dealing simulator. Deal hands from a 3D deck and compare empirical frequencies of aces, pairs and colors against exact hypergeometric probabilities, and see why a deck's memory matters.
- Frequency
Interactive Poisson distribution explorer for insurance claim counts and hospital admissions. Move the rate λ and watch P(X=k), the mode, and the probability of zero or extreme counts update live.
- Severity
Interactive loss severity explorer comparing Log-Normal, Pareto (heavy-tail) and Gamma distributions. Adjust parameters and see the mean, median, 95th and 99th percentile claim sizes, and the share of total cost from the largest 5% of claims, update in real time.
- Expected loss
Interactive aggregate loss model for a personal motor book: Poisson claim frequency with Log-Normal severity across a portfolio of policies. Compute expected loss and run a 20,000-year Monte Carlo to see the aggregate loss distribution and its 95th, 99th and 99.5th percentiles.
- Monte Carlo
Interactive Monte Carlo simulator using geometric Brownian motion. Set drift, volatility and horizon to simulate 2,000 paths and read the 5th–95th percentile fan chart for an investment portfolio or an insurer's invested reserve fund.
- Correlation
Interactive bivariate correlation explorer. Slide ρ from −1 to +1, see simulated scatter plots, compare standalone versus portfolio volatility for stocks and bonds or two insurers in one region, and switch to a t-copula to see the joint crashes a single correlation hides.
- Mortality
Interactive Gompertz-Makeham mortality explorer. Change age, term and discount rate, switch between an illustrative basis and the Standard Ultimate Survival Model, and see the survival curve, annual death probabilities q(x), life expectancy and net premiums for term and whole-of-life assurance.
- Applied models
Six interactive worked examples built from the site's models: pricing a motor portfolio, planning a 10-year investment reserve, hospital surge capacity, a two-asset efficient frontier (CFA), loan-portfolio expected loss versus Basel IRB capital (FRM), and routing a delivery fleet.
- ML claim scorer
Interactive logistic regression demo that scores a driver's annual claim probability from age, licence years, prior claims, vehicle value, mileage and urban location, with a feature-contribution chart explaining the score.
- Credit scoring
Interactive credit scoring demo: a logistic regression fitted live on 2,000 synthetic loan applicants, with ROC curve, AUC, Gini and KS, a confusion matrix at any threshold, adverse-action reasons for a declined applicant, a profit-maximising approval cutoff, and a calibration plot.
- Fraud detection
Interactive fraud detection demo on 1,600 synthetic insurance claims: compare logistic regression, a CART decision tree and a random forest by their decision boundaries, watch deeper trees overfit, and judge models by precision and recall at your investigators' capacity.
- Customer segmentation
Interactive k-means clustering demo for segmenting policyholders and investors: step through Lloyd's algorithm from k-means++ seeds, see why features must be standardised, choose k with the elbow and silhouette charts, test it on data with no real segments, and read a segment profile table.
- LLM sampling
Interactive demo of how large language models pick the next token: softmax probabilities, temperature, top-k and nucleus (top-p) sampling, repeated asks, and majority-vote self-consistency, framed for claims triage, financial summaries and number extraction.
- Conformal prediction
Interactive conformal prediction demo: wrap any model in prediction intervals or prediction sets with a distribution-free coverage guarantee. Compare constant and adaptive bands for claim costs, use 'not sure' sets to route AI triage to a human, and see what breaks under regime change.
- Validating AI
Interactive guide to validating an AI or LLM tool's error rate: exact Clopper-Pearson and Wilson intervals, the rule of three, the number of test cases needed to rule out a target error rate, and the probability that a flawed model passes your validation.
- AI monoculture
Interactive model of AI concentration risk: 20–50 firms with identically accurate AI models, a dominant vendor or shared foundation model, and the resulting distribution of simultaneous failures versus independence, with policy levers for vendor diversity and human challenge.
- Decision theory
Four interactive investment decisions: invest now or wait (real options and the hurdle above NPV), build vs buy under fat-tailed cost overruns, pilot-first with the expected value of information, and the sunk-cost kill decision.
- Game theory
Interactive Nash equilibrium explorer with three financial scenarios: Cournot competition between two firms, a Diamond-Dybvig-style bank run with its tipping point, and the repeated game behind insurance underwriting cycles.
From theory to decisions
- Poisson distribution: claim frequency, hospital admissions, customer arrivals
- Log-Normal, Pareto, Gamma: loss severity, loss tails, portfolio drawdowns
- Frequency × Severity: expected loss, premium pricing
- Monte Carlo simulation: investment uncertainty, reserve adequacy
- Correlation: portfolio diversification, contagion risk
- Gompertz-Makeham survival: mortality, term and whole-of-life pricing
- Logistic regression: risk scoring, claim probability
- Convolution & hypergeometric: dice, cards, sampling without replacement
- Nash equilibrium: price wars, bank runs, underwriting cycles
- Real options & value of information: invest or wait, pilots, sunk costs