See the mathematics move

Set P(A), P(B) and their overlap; watch the union, the complement and the feasible range of the joint move.

A probability tree and a 1,000-person grid: see how prevalence drives the meaning of a positive test.

Transform X into aX + b and watch the mean shift and the spread scale by |a|.

Seven distributions, their exact shape, moments, and a histogram of simulated draws.

Drag to rotate the joint density; change the correlation and see the contours and the regression line.

Two groups, their means and spreads: split total variance into within and between parts.

Draw thousands of sample means from skewed or bimodal parents and watch the normal shape appear.

Mean, median and a shrinkage estimator compete across repeated samples.

One hundred intervals from one hundred samples: count how many catch the true mean.

The null and alternative sampling distributions, the rejection region, α, β and power.

Move the parameter along a Bernoulli log-likelihood; see the peak, its curvature and the LR test.

Push a normal estimate through ln, √, 1/x or x² and compare the delta-method normal with the truth.

Prior × likelihood = posterior. Change prior strength and data and watch the posterior move.

MCAR, MAR and MNAR on an education–wage scatter: complete-case bias, IPW repair and slope attenuation.

Change who gets which treatment and watch the pooled comparison reverse the stratified one.

Drag two vectors and watch the dot product, the angle and the projection; the geometry behind correlation and slopes.

Edit a 2×2 matrix and see the plane stretch, rotate, shear or collapse; the determinant is the area of the image square.

Raise the correlation between two regressors and watch the cloud of coefficient estimates stretch along a line.

Two equations as two lines: as the determinant shrinks, the solution becomes fragile and then disappears.

Move a candidate line; only the OLS line makes the residuals orthogonal to the regressors and the SSR minimal.

A symmetric matrix turns the unit circle into an ellipse whose axes are its eigenvectors.

Correlated draws built with a Cholesky factor, their covariance ellipses, and the variance of any linear combination.

Approximate eˣ, ln(1+x), sin x or the logistic function by polynomials of order 1–4 around any point.

Click anywhere on the SSR surface to see −∇S = 2X′(y − Xβ) point downhill toward the OLS estimate.

Race the two algorithms on an ill-conditioned bowl and on a curved valley; tune the step size until it breaks.

Utility maximisation on a budget line: tangency, the Lagrange multiplier and the cost of a non-optimal bundle.

Simulated data with X′X, (X′X)⁻¹, β̂, s² and standard errors computed live from the matrix formulas.

A confounded scatter beside its added-variable plot: the residual-on-residual slope equals the multiple-regression coefficient.

Fan-shaped errors make classical intervals under-cover; robust (sandwich) intervals restore 95%.

Estimate a Poisson rate from two moments, compare identity and optimal weighting, and watch the J test catch overdispersion.

Twelve workers with both potential outcomes: hide the missing half, randomise, and watch estimates centre on the ATE.

Direct and spillover effects of deworming-style treatment; why within-village comparisons miss the total effect.

A heterogeneous effect τ(x) averaged over three different populations — and when they coincide.

Workers enrol when their gain beats the cost: decompose the naive gap into ATE, selection bias and heterogeneity bias.

Thousands of re-assignments: baseline imbalance and estimates under complete, Bernoulli and self-selected assignment.

The true design variance versus the Neyman estimator as effect heterogeneity grows; coverage of the 95% interval.

Enumerate every possible assignment under the sharp null and read off an exact p-value — for means or ranks.

Complete versus within-district randomisation of a schooling programme: the variance gain from blocking.

Re-randomise a training experiment and compare the spread of three estimators when slopes differ by arm.

Minimum detectable effect against the number of schools for any ICC, cluster size and baseline R².

An insurance lottery with compliers, always-takers and never-takers: only the Wald ratio recovers the CACE.

Mirrored propensity-score histograms, extreme weights, effective sample size and what trimming costs.

Stratify on the estimated score: covariate imbalance vanishes within strata and the bias with it.

Monte Carlo of two IPW estimators; shift the outcome level and only the normalised (Hájek) version stays stable.

Switch the outcome and propensity models between right and wrong: AIPW survives as long as one is right.