pybenford 0.1.1

Professional-grade Benford's Law analysis toolkit for forensic accounting, auditing, and fraud detection

pybenford

Professional-grade Benford's Law analysis toolkit for forensic accounting, auditing, and fraud detection.

Why pybenford?

Existing Benford's Law packages on PyPI cover first-digit chi-square and not much else. Most are unmaintained. pybenford implements the complete Nigrini forensic accounting workflow from Benford's Law: Applications for Forensic Accounting, Auditing, and Fraud Detection (Nigrini, 2012):

• MAD conformity classification with Nigrini's empirical thresholds (close, acceptable, marginally acceptable, nonconformity)
• Distortion factor model for detecting overstatement vs. understatement
• Second-order test on differences of sorted values
• Summation test with uniform 1/90 expectation
• Mantissa arc test (Alexander, 2009) with L-squared statistic
• Number duplication analysis
• All standard digit tests: first, second, third, first-two, first-three, last-two
• Z-statistic with Fleiss continuity correction, chi-square, Kolmogorov-Smirnov
• Publication-quality matplotlib visualizations
• Pure NumPy internals, no pandas dependency

Installation

pip install pybenford

Quick Start

from pybenford import BenfordAnalysis

analysis = BenfordAnalysis(data)  # list, numpy array, or pandas/polars Series
result = analysis.first_digit()
print(result)

Every result object has a formatted print() output. Three lines from raw data to a conformity report:

=======================================================
  First Digit Test  (n=3,195  alpha=0.05)
=======================================================
 Digit   Count   Observed   Expected   Z-Score   Sig
     1    956   29.92%    30.10%      0.20
     2    595   18.62%    17.61%      1.48
     3    389   12.18%    12.49%      0.52
     4    299    9.36%     9.69%      0.61
     5    255    7.98%     7.92%      0.10
     6    197    6.17%     6.69%      1.16
     7    180    5.63%     5.80%      0.36
     8    171    5.35%     5.12%      0.57
     9    153    4.79%     4.58%      0.53
-------------------------------------------------------
 MAD:        0.0034 — Close Conformity
 Chi-Square: 4.6922  (critical: 15.5073) — Pass
 KS:         0.0083  (critical: 0.0240)  — Pass
=======================================================

For tests with many digit bins (first-two, first-three), the display shows only flagged digits instead of all 90 or 900 rows:

=======================================================
  First Two Digits Test  (n=3,195  alpha=0.05)
=======================================================
 Flagged Digits (7 of 90):
 Digit   Count   Observed   Expected   Z-Score
    35     24    0.75%     1.22%      2.35  *
    49     16    0.50%     0.88%      2.19  *
    66     33    1.03%     0.65%      2.56  *
    70     29    0.91%     0.62%      1.99  *
    75     28    0.88%     0.58%      2.13  *
    76      9    0.28%     0.57%      2.03  *
    77      9    0.28%     0.56%      1.99  *
-------------------------------------------------------
 MAD:        0.0015 — Acceptable Conformity
 Chi-Square: 104.9157  (critical: 112.0220) — Pass
 KS:         0.0102  (critical: 0.0240)  — Pass
=======================================================

All results are also accessible programmatically:

result = analysis.first_digit()

result.mad                     # 0.0034
result.mad_conformity          # "close"
result.chi_square              # 4.6922
result.chi_square_significant  # False
result.ks_statistic            # 0.0083
result.ks_critical             # 0.0240
result.z_scores                # array of per-digit Z-scores
result.significant_flags       # bool array of flagged digits
result.observed                # array of observed proportions
result.expected                # array of expected Benford proportions
result.digits                  # array of digit labels
result.counts                  # array of raw counts
result.n                       # number of records analyzed
result.alpha                   # significance level used
result.test_name               # e.g. "First Digit Test"

Data Preparation

analysis = BenfordAnalysis(
    data,                        # list, array, or Series of numbers
    sign_filter="positive",      # "all", "positive", or "negative"
    min_abs_value=10.0,          # exclude small values (optional)
    drop_zero=True,              # exclude zeros (default: True)
)

print(analysis.profile)          # data profile per Nigrini Ch. 4

sign_filter separates income from expense items for independent analysis. min_abs_value excludes values below a minimum magnitude, since very small numbers distort digit distributions.

Visualization

Plot functions return (Figure, Axes) with no side effects.

from pybenford.visualization import plot_digit_test, plot_mantissa_arc

result = analysis.first_two_digits()
fig, ax = plot_digit_test(result, show_confidence=True)
fig.savefig("first_two_digits.png", dpi=150)

arc = analysis.mantissa_arc()
fig, ax = plot_mantissa_arc(arc, analysis.clean_data)

from pybenford.visualization import plot_z_scores
fig, ax = plot_z_scores(result, critical_value=1.96)

Available Tests

Method	Description	Reference
first_digit()	First significant digit (1-9)	Nigrini Ch. 5
second_digit()	Second significant digit (0-9)	Nigrini Ch. 5
third_digit()	Third significant digit (0-9)	Nigrini Ch. 5
first_two_digits()	First two digits (10-99)	Nigrini Ch. 5
first_three_digits()	First three digits (100-999)	Nigrini Ch. 5
last_two_digits()	Last two digits (00-99), uniform expected	Nigrini Ch. 5
second_order()	Digit test on sorted differences	Nigrini Ch. 6
summation()	Sum proportions vs. uniform 1/90	Nigrini Ch. 5
distortion_factor()	Overstatement/understatement detection	Nigrini Ch. 6
mantissa_arc()	Uniformity of mantissas on unit circle	Nigrini Ch. 7
number_duplication()	Most frequently duplicated values	Nigrini Ch. 5

Statistical Measures

Each digit test result includes:

• Z-statistic per digit bin (Fleiss continuity correction)
• Chi-square goodness-of-fit with critical value
• Kolmogorov-Smirnov statistic with critical value
• MAD (Mean Absolute Deviation) with Nigrini's conformity classification
• Per-bin significance flags at configurable alpha

MAD Conformity Thresholds

MAD is the preferred conformity measure because chi-square and KS become overly sensitive with large datasets (N > 25,000), rejecting near-perfect conformity. MAD is sample-size independent.

Test	Close	Acceptable	Marginal	Nonconformity
First digit	< 0.006	< 0.012	< 0.015	>= 0.015
Second digit	< 0.008	< 0.010	< 0.012	>= 0.012
First two digits	< 0.0012	< 0.0018	< 0.0022	>= 0.0022
First three digits	< 0.00036	< 0.00044	< 0.00050	>= 0.00050

References

• Nigrini, M.J. (2012). Benford's Law: Applications for Forensic Accounting, Auditing, and Fraud Detection. Wiley.
• Miller, S.J. (2015). Benford's Law: Theory and Applications. Princeton University Press.
• Kossovsky, A.E. (2014). Benford's Law: Theory, the General Law of Relative Quantities, and Forensic Fraud Detection Applications. World Scientific.

License

MIT License. See LICENSE for details.