Benford's Law: Textbook Exercises and Multiple-choice Testbanks

TL;DR

This paper explores how physics and chemistry textbook exercises follow Benford's Law and examines whether this pattern can be exploited to cheat on multiple-choice tests, finding limited practical vulnerability.

Contribution

It demonstrates textbook exercise answers conform to Benford's Law and assesses the potential for using this pattern to improve guessing strategies in tests.

Findings

Textbook answers follow Benford's Law

Testbank answers conform to Benford's Law

Testbank is secure against Benford-based guessing

Abstract

Benford's Law describes the finding that the distribution of leading (or leftmost) digits of innumerable datasets follows a well-defined logarithmic trend, rather than an intuitive uniformity. In practice this means that the most common leading digit is 1, with an expected frequency of 30.1%, and the least common is 9, with an expected frequency of 4.6%. The history and development of Benford's Law is inexorably linked to physics, yet there has been a dearth of physics-related Benford datasets reported in the literature. Currently, the most common application of Benford's Law is in detecting number invention and tampering such as found in accounting-, tax-, and voter-fraud. We demonstrate that answers to end-of-chapter exercises in physics and chemistry textbooks conform to Benford's Law. Subsequently, we investigate whether this fact can be used to gain advantage over random guessing in multiple-choice tests, and find that while testbank answers in introductory physics closely conform to Benford's Law, the testbank is nonetheless secure against such a Benford's attack for banal reasons.