Fluctuation Depth Scans

Variance Numerator Quotient Scan

A variance-numerator quotient scan divides the weighted squared-deviation sum by the stated total weight, turning the fluctuation numerator into the variance row by row. The arithmetic is exact only for the stated counts or probabilities, squared deviations, numerators, quotients, 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.

The variance quotient is the weighted squared-deviation numerator over total weight. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The numerator is the sum of weighted squared terms

The three contributions are 4, 0, and 4. Their sum is 8.

Nσ=4+0+4=8N_{\sigma}=4+0+4=8
Variance numeratorThe numerator is assembled from the contribution rows.xwdsqterm01-24422000412444 + 0 + 4 gives numerator 8

The denominator is total weight

The weights are 1, 2, and 1. Their total is 4.

W=1+2+1=4W=1+2+1=4
Variance denominatorThe denominator is the same weight ledger.xwdsqterm01-24422000412441 + 2 + 1 gives total weight 4

Variance is numerator over total weight

Variance divides numerator 8 by total weight 4, giving 2.

σ2=84=2\sigma^{2}=\frac{8}{4}=2
Variance quotientThe checked fluctuation ledger pins the quotient.xwdsqterm01-24422000412448 over 4 gives variance 2