Signed Deviation Ledger Cross-Scan - Thermodynamics and Statistical Mechanics Step by Step

A signed-deviation ledger cross-scan lists each state's difference from the mean, keeping the sign so above-mean and below-mean contributions can cancel in the first check. The arithmetic is exact only for the stated states, probabilities or counts, deviations, and rows in the ideal ensemble model. Real systems add preparation bias, finite samples, correlations, detector/counting calibration, thermalization limits, finite precision, and measurement uncertainty.

Variance starts with signed distances from the mean, not with a black-box square. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Deviation row 1

Value 0 minus mean 2 gives signed deviation -2.

x-\mu=0\mathbin{-}2=-2

Deviation row 2

Value 2 minus mean 2 gives signed deviation 0.

x-\mu=2\mathbin{-}2=0

Deviation row 3

Value 4 minus mean 2 gives signed deviation 2.

x-\mu=4\mathbin{-}2=2

Signs disappear only after the squaring step

The three signed deviations are negative two, zero, and positive two. This page keeps the sign visible before variance squares it.

\begin{array}{c|c|c}x&\mu&x-\mu\\0&2&-2\\2&2&0\\4&2&2\\\end{array}