Assessing Conformance with Benford’s Law: Goodness-Of-Fit Tests and Simultaneous Confidence Intervals
Bibliographic record
Abstract
Benford's Law is a probability distribution for the first significant digits of numbers, for example, the first significant digits of the numbers 871 and 0.22 are 8 and 2 respectively. The law is particularly remarkable because many types of data are considered to be consistent with Benford's Law and scientists and investigators have applied it in diverse areas, for example, diagnostic tests for mathematical models in Biology, Genomics, Neuroscience, image analysis and fraud detection. In this article we present and compare statistically sound methods for assessing conformance of data with Benford's Law, including discrete versions of Cramér-von Mises (CvM) statistical tests and simultaneous confidence intervals. We demonstrate that the common use of many binomial confidence intervals leads to rejection of Benford too often for truly Benford data. Based on our investigation, we recommend that the CvM statistic Ud(2), Pearson's chi-square statistic and 100(1 - α)% Goodman's simultaneous confidence intervals be computed when assessing conformance with Benford's Law. Visual inspection of the data with simultaneous confidence intervals is useful for understanding departures from Benford and the influence of sample size.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".