Across street addresses, populations, invoice totals and physical constants, the leading digit isn't evenly spread โ a 1 shows up about 30% of the time, a 9 under 5%. That lopsided fingerprint is Benford's lawBenford's law: in data spanning several orders of magnitude, the first digit d appears with probability log10(1 + 1/d). It holds for naturally multiplying quantities, not for capped or made-up ones., and numbers that don't follow it โ invented figures, capped values โ stand out. Load an example below or paste your own, then test it.
Your numbers
Should fitEach term is the sum of the prior two, so the run grows by a near-constant ratio (โ1.618ร). Multiplicative growth is the engine of Benford's law.
Parsed 200 numbers with a leading digit.
First-digit distribution
123456789
observedBenford expected
0.7ฯยฒ (df 8)Pearson's chi-square: the summed, expectation-weighted gap between observed and expected counts. Bigger means a worse fit. The 5% threshold at 8 degrees of freedom is 15.5.
0.9996p-valueThe chance of a gap this large (or larger) if the data really did follow Benford. Below 0.05 is the usual flag that it doesn't.
0.004MADMean absolute deviation: the average gap between an observed and expected digit share. Auditors read it against fixed bands (first digit: โฒ0.006 close, โฒ0.012 acceptable), but those assume thousands of values โ on a small sample lean on the p-value instead.
โ ConsistentThe leading digit track Benford's curve closely (p = 0.9996). Nothing here looks off โ this is what naturally growing numbers do.
Digit by digit
Digit
Expected
Observed
Count
1
30.1%
30%
60
2
17.6%
18%
36
3
12.5%
12.5%
25
4
9.7%
9%
18
5
7.9%
8.5%
17
6
6.7%
6%
12
7
5.8%
5.5%
11
8
5.1%
6%
12
9
4.6%
4.5%
9
Benford holds for quantities that span several orders of magnitude and grow by multiplying โ revenues, river lengths, file sizes. It fails by design for capped or assigned numbers (heights in cm, phone numbers, sequential IDs), so a deviation isn't proof of anything on its own.
Benford's law ยท ฯยฒ goodness-of-fit at 8 degrees of freedom