Grab the banana-stand receipts — every price, every count — and look only at the first digit of each. You’d guess each digit 1–9 shows up equally, about 11% each. Wrong: the digit 1 leads about 30% of the time, and 9 barely 5%. This lopsided curve haunts populations, river lengths, stock prices — and when someone cooks the books, their fake numbers don’t follow it. Tax auditors use this. Slide to sweep the ledger.
Benford’s Law: in many naturally occurring datasets that span several orders of magnitude, the leading digit d appears with probability P(d) = log₁₀(1 + 1/d). So 1 leads ~30.1% of the time, 2 ~17.6%, down to 9 at ~4.6%. It arises because such data are roughly uniform on a LOGARITHMIC scale — equal multiplicative steps — making it scale-invariant. Because humans fabricating numbers tend to spread first digits evenly, deviation from Benford is a red flag used in forensic accounting and election-fraud screening. A fail-loud self-check throws unless P(1) ≈ 0.301 and P(1) > P(9). ◆ real statistics, node-verified.
Benford applies to data spanning several orders of magnitude with no imposed scale — NOT to bounded data (adult heights, dice). It flags anomalies for investigation; a deviation is a signal to look closer, not proof of fraud on its own.