The Weighing Table

Every value has a weight. Pile up the weighted distances from center — then square-root the pile to fold it back into the spread.

Stand at the center. Measure how far each value sits from you. Square it. Weight it by probability. Pile it up. The pile tells you how spread the world is around you. Take its square root and you get σ — that spread as a single step you can pace out from the center.

The Spread

Distance from Center

Each value's weighted squared distance from μ.

Connected Spells

Drainage counts the ways. Probability House builds the mass. Weighing Table measures the spread and the co-movement.

E[X] needs probabilities from the Probability House. Counting the ways to land on each value may need Drainage.

Formula Reference — Kap 5 + 6

E[X] — Expected ValueKap 5

Σ x_i · P(x_i) = μ

The center. Weighted average of all values. Every other formula measures from here. (E[ ] is the expectation operator — “the average of” — not the number e.)

Var[X] — VarianceKap 5

Σ (x_i − μ)² · P(x_i)

Average squared distance from center, weighted by probability. Lives in squared-units — it is the pile, not the final distance.

σ — Standard DeviationKap 5

σ = √Var[X]

Undo the squaring. Variance floats off the line as an area; its root folds it back onto the axis. σ is the typical distance a value sits from the center μ, in the same units as the values. For Var = 0.49, σ = √0.49 = 0.7.

Cov(X,Y) — CovarianceKap 6

ΣΣ (x_i − μ_X)(y_j − μ_Y) · P(x_i, y_j)

Do X and Y move together? Walk every cell, check direction, weight, pile up. (Y is just a second variable — we use Y, not E, so it never clashes with Euler’s e.)

ρ — CorrelationKap 6

ρ = Cov(X,Y) / (σ_X · σ_Y)

Covariance scaled to [−1, 1]. Removes unit dependency. Pure direction signal.