Benford’s Law First-Digit Probability log₁₀(1 + 1/d) & Forensic Z-Score Calculator

Audit financial ledgers against Benford’s Law P(d) = log₁₀(1 + 1/d) (30.1% for digit 1 down to 4.6% for digit 9).

Benford Expected Probability for Digit 1
30.103 %
Expected Count Out of 1000 301.03 records
Observed Proportion (%) 26.5 %
Forensic Deviation Z-Statistic 2.484 Z-score
Inputs
⚡ Scenarios:
digit d
Leading non-zero digit d (1 through 9).
records
Total number of financial/ledger records audited (N).
observed
Observed count of records starting with digit d.
Results & Graph
Benford Expected Probability for Digit 1
30.103 %
Expected Count Out of 1000
301.03 records
Observed Proportion (%)
26.5 %
Forensic Deviation Z-Statistic
2.484 Z-score

Benford Expected Probability for Digit 1

Graph 1 • Benford’s Law First Digit • Domain Architecture & Live Telemetry Maps Leading Digit1to9, Total Audit Records N, Observed Digit Count through Benford’s Law First Digit Formula.
Benford Expected Probability f: 30.103 % Benford’s Law First Digit • Invariants & Metrics Benford Expected Probability 30.103 % Expected Count Out of 1000 301.03 records Observed Proportion (%) 26.5 % Forensic Deviation Z-Statist 2.484 Z-score
Graph 2 • Benford Expected Probability for Digit 1 Across Leading Digit1to9 Sweep (40% → 160%) Evaluates P(d) = log₁₀(1 + 1 / d); Z = |ObservedProp − P(d)| / √[ P(d)(1 − P(d)) across 7 operating points (current input highlighted).
0 % 16.56 % 33.11 % 30.10 0.400 30.10 0.600 30.10 0.800 30.10 1 30.10 1.20 30.10 1.40 17.61 1.60
Graph 3 • Joint Sensitivity: Leading Digit1to9 × Total Audit Records N Compares Benford Expected Probability for Digit 1 when Total Audit Records N is shifted +25% (solid) vs. −25% (dashed).
0 % 16.56 % 33.11 % --- Total Audit Records N @ 75% 30.10 -30% 30.10 -15% 30.10 +0% 30.10 +15% 30.10 +30%
Summary: Digit 1 has a Benford expected frequency of 30.1% (301 of 1000 records); observing 265 (26.5%) gives a Z-score of 2.48.
Summary: Digit 1 has a Benford expected frequency of 30.1% (301 of 1000 records); observing 265 (26.5%) gives a Z-score of 2.48.
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Formula & Mathematical Methodology: Benford’s Law First Digit Formula

P(d) = log₁₀(1 + 1 / d)
Z = |ObservedProp − P(d)| / P(d)(1 − P(d)) / N
P(d) = log₁₀(1 + 1 / d); Z = |ObservedProp − P(d)| / √[ P(d)(1 − P(d)) / N ]
  • leadingDigit1to9: Leading Digit1to9 (digit d)
  • totalAuditRecordsN: Total Audit Records N (records)
  • observedDigitCount: Observed Digit Count (observed)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: Benford’s Law First Digit

Audit financial ledgers against Benford’s Law P(d) = log₁₀(1 + 1/d) (30.1% for digit 1 down to 4.6% for digit 9).

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: P(d) = log₁₀(1 + 1 / d); Z = |ObservedProp − P(d)| / √[ P(d)(1 − P(d)) / N ].
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes Benford’s Law First Digit via P(d) = log₁₀(1 + 1 / d); Z = |ObservedProp − P(d)| / √[ P(d)(1 − P(d)) / N ].

Frequently Asked Questions

How does the Benford’s Law First-Digit Probability log₁₀(1 + 1/d) & Forensic Z-Score Calculator compute the solution?

It evaluates P(d) = log₁₀(1 + 1 / d); Z = |ObservedProp − P(d)| / √[ P(d)(1 − P(d)) / N ] directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Econometrics & Regression

Benford’s Law First-Digit Probability log₁₀(1 + 1/d) & Forensic Z-Score Calculator: inputs, method, and how to read the result

Audit financial ledgers against Benford’s Law P(d) = log₁₀(1 + 1/d) (30.1% for digit 1 down to 4.6% for digit 9).

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Leading Digit1to9: Leading non-zero digit d (1 through 9).
  • Total Audit Records N: Total number of financial/ledger records audited (N).
  • Observed Digit Count: Observed count of records starting with digit d.

What the calculation includes

With the default values shown, Benford Expected Probability for Digit 1 is 30.103 %. The calculation follows Benford’s Law First Digit Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Leading Digit1to9 from 1 digit d to 2 digit d changes Benford Expected Probability for Digit 1 (30.103 %) to Benford Expected Probability for Digit 2 (17.609 %). This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

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