Benford's Law Calculator (Expected First-Digit Frequencies)
How often a number should start with each digit in real-world data, the expected count in your data, and a z-test of whether your own count is too far off.
Results
The dashed line marks the value you entered.
What this tool does
In many collections of figures that stretch across several orders of magnitude — invoices, populations, river lengths, share prices — a 1 leads about 30.1 % of the time and a 9 only 4.6 %, following P(d) = log₁₀(1 + 1/d). Auditors use the pattern to spot invented or tampered numbers, which often show fewer leading ones than expected. It does not apply to figures confined to a narrow range (adult heights, ages) or assigned ones (phone numbers). The z-test here compares a single digit, with the usual continuity correction; a result above 1.96 differs more than chance would explain at the 5 % level.
Formula
P(d) = log₁₀(1 + 1 ÷ d) · P(d₁d₂) = log₁₀(1 + 1 ÷ (10d₁ + d₂)) · z = (|p̂ − p| − 1 ÷ 2n) ÷ √(p(1 − p) ÷ n)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
d1 | First digit (1 to 9) | — |
d2 | Second digit (0 to 9) | — |
nn | Number of figures in your data | — |
ob | How many start with that first digit (optional) | — |
P1 | Expected share starting with that digit | % |
P12 | Expected share starting with both digits | % |
CU | Expected share starting with that digit or a lower one | % |
EX | Expected count in your data | — |
ZS | z-statistic of your count (|z|) | — |
VD | Does it fit Benford's law? (5 % level, one digit) | — |
Worked example
- First digit (1 to 9)1
- Second digit (0 to 9)0
- Number of figures in your data1000
- How many start with that first digit (optional)310
- Expected share starting with that digit30.103 %
- Expected share starting with both digits4.139 %
- Expected share starting with that digit or a lower one30.10 %
- Expected count in your data301.0
- z-statistic of your count (|z|)0.58
- Does it fit Benford's law? (5 % level, one digit)Consistent with Benford's law
Limitations
- The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.
- The default values are typical reference figures, not measurements of your situation. Replace them with your own data whenever you have it.