The Benford law is used worldwide to detect non-conformance or data fraud in numerical data. In its weak form, it says that the first non-zero digit of a data item from a universe is not uniformly distributed, but logarithmically distributed. In particular, the first non-zero digit is One, with a probability of approximately 0.3. In the present paper, we suggest a new class of tests, the Ones Scaling tests, which are motivated by the scale-invariance property of Benford's law. Various scaling factors are chosen, and then the probability is tested that the product of the original observation with the scaling factors has the first significant digit One. Two distance measures of empirical and Benford probabilities are considered: the Euclidean and Mahalanobis distances. All our tests are illustrated by real and simulated data and are compared by competitive statistical tests. The analysis of specifically selected and designed simulated manipulations shows that this class of tests is a useful alternative for detecting special data fraud.
Kössler et al. (2026) studied this question.