First Digit Probability and Benford's Law

TL;DR

This paper derives the first digit probabilities for various functions, explains their relation to real-life data, and demonstrates how Benford's Law can be applied to analyze the distribution of leading digits in different contexts.

Contribution

It provides a new method for calculating first digit probabilities for continuous and discrete functions, including derivation of Benford's Law without external sources.

Findings

Derived first digit probabilities for basic algebraic functions

Connected probabilities to real-life data scenarios

Presented a method for analyzing digit distribution in approximate data

Abstract

The following work is written in easy language for college level students. It shows how the first digit probabilities of a group of continuous real-valued functions can be calculated. Thus, examples explaining how the probabilities are related to specific real-life situations, as well as the summary for all basic algebraic functions, were brought to the reader's attention. Besides, the Benford's formula was derived with no use of any additional guiding sources. Moreover, a comprehensive analysis of a group of certain discrete functions was performed by approximating the functions to the above-mentioned continuous ones, taking limits, and other methods. The work can be applied for calculating the first digit probabilities of more advanced functions as well while using the same approach. Furthermore, the technique can be useful while dealing with a large set of highly approximate numbers and a conclusion about their nature needs to be made.