INTERACTIVE COMPANION · DIGIT BENCH · METHODS

The Digit Bench

A live model accompanying “When the First Digit Test Works and When It Accuses the Innocent”

Two instruments sit below, both running the same test statistics the article uses: Pearson's chi-squared on nine digit bins at 8 degrees of freedom, and Nigrini's mean absolute deviation against fixed bands. Nothing here calls a server. The eight datasets in the bench are the exact digit counts analysis/benford-real-data.py printed from the cached files under analysis/data/, typed in once and never touched again, and the simulations use a small seeded generator so a reload gives the same digit histograms every time.

Model 1. The digit bench

Pick a dataset. The bars are the digits it actually produced; the dashed step is what Benford's law expects at this sample size. The statistics update live and are computed the same way the article computes them, so you can check any of the eight rows in §3 against what appears here.

Dataset Start on D1. Its chi-squared p-value (0.4726) says "unremarkable" while its MAD (0.0152) just crosses Nigrini's nonconformity line, the exact tension §4 walks through by hand. Then try D3 for a clean pass and D6 for the most dramatic failure in the whole study.

n: – χ² (8 d.f.): – p: – MAD: – Cohen's w: –

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Model 2. Spread and the false-accusation counter

The left slider builds a log-uniform window S decades wide, starting at zero, and shows its exact first-digit law, no random sampling involved, the same closed-form calculation behind Figure 2 of the article. Watch the MAD readout touch zero at every whole decade and rise in between. The right side is a genuine Monte Carlo: it draws honest datasets, exactly Benford or a chosen lognormal spread, of a size you pick, and tallies how often the standard test convicts them anyway. Every one of those datasets is innocent by construction.

Decades spanned, S 1.00 S = 1.00 is the default and it conforms exactly (MAD = 0), because a window exactly one decade wide covers every mantissa once. Drag to 0.40 to see D6's earthquake-magnitude problem rebuilt from scratch, or to 1.45 to find the local worst case between one whole decade and the next.

window: – MAD of this window: – verdict: –

Sample size per honest dataset, n 100 Positions map to n = 100, 500, 1000, 5000, 20000. At n=100 the MAD rule should settle near 91.8%, the article's own headline number for this exact n. Move to n=1000 and watch it collapse toward 0%, the same collapse §7's table shows. Honest generating law Switch to σ=0.30 at n=1000 and both rules climb toward 100%, the chi-squared line as well as the MAD rule: this honest law is simply too narrow to ever look Benford at that sample size, matching Figure 5's LOGN030 curve. Test rule Toggle rules on the same data stream to see the MAD rule and the chi-squared test disagree, exactly as they did for D1 in the article.

0honest datasets tested

0flagged by the rule

–running false-accusation rate

Counter is running. Switch tabs or reduce motion to pause it.

Three numbers here are worth checking by hand against the article, and the defaults are chosen so a fresh page load already shows them. D1 on the bench opens at χ² = 7.608, p = 0.4726 and MAD = 0.01521, the exact three numbers §4 derives by hand from the same nine counts. The spread model opens at S = 1.00, where the closed-form MAD is exactly 0, the mechanism §5 rests on. The counter, left running at its default n=100 with the MAD rule, will settle near the article's measured 91.8% false-accusation rate within a few hundred draws.