Application of the m* and d* Statistics for Assessing the Quality of Data in Psychological Research Using Benford's Law (Illustrated with Reaction Time Measurements)



The objective of the proposed study was to examine the properties of statistics used to assess the conformity of the distribution of the first significant digit to Benford's Law, m* and d*, with relatively modest sample sizes (10≤n ≤70). A simulation study was conducted to achieve this goal. Data were simulated following a log-normal distribution with parameters that mimic the distribution of reaction time measurements. The distribution of the first significant digit was examined in standardized values raised to the power of γ; 5≤γ≤100. It was found that the statistic m* does not depend on the power of the number, unlike d*. Critical values were established for samples ranging from 10 to 70 observations with an increment of h=10. It turned out that for small n, the critical values of the statistic d* are close to asymptotic, while the critical values of the statistic m* are significantly larger. The functionality of the established critical values was tested within the framework of an experimental study: one respondent performed the Stroop cognitive test in accordance with the instructions (control case), while another violated them (experimental case). It was discovered that the statistic d* does not allow for differentiation in the behavior of subjects. Conversely, m* proved sensitive to changes in respondent behavior, and in the experimental case, it significantly more often allowed for the rejection of the null hypothesis regarding the conformity of the distribution of the first significant digit of the standardized value of reaction time to Benford's Law compared to the control. Thus, a preliminary conclusion is made that the statistic m* is more functional compared to d* in studying the quality of data on reaction time with small n.

General Information

Keywords: Benford's Law, reaction time, m* statistics, d* statistics

Journal rubric: Data Analysis

Article type: scientific article


Received: 13.03.2024


For citation: Kolachev N.I. Application of the m* and d* Statistics for Assessing the Quality of Data in Psychological Research Using Benford's Law (Illustrated with Reaction Time Measurements). Modelirovanie i analiz dannikh = Modelling and Data Analysis, 2024. Vol. 14, no. 2, pp. 23–44. DOI: 10.17759/mda.2024140202. (In Russ., аbstr. in Engl.)


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Information About the Authors

Nikita I. Kolachev, PhD in Psychology, Senior Lecturer, Department of Psychology, National Research University Higher School of Economics, Moscow, Russia, ORCID:, e-mail:



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