Definição de Benford

In our last lesson, we learned about logarithm scales. Now let's move onto our next topic in applications of logarithmic and exponential functions.

Benford's Law

Benford's Law is a property held by leading digits of numbers in large data sets seen in the real world.

At first, you might think that the digits $1$ through $9$ would appear as the first digit about the same number of times.

But that is not what happens.

Instead, the leading digits follow a logarithmic pattern. The number $1$ appears most often as the first digit. The number $2$ appears less often than $1$, but more often than $3$. This pattern continues all the way down to $9$, which appears the least often.

Even though it may seem surprising, this pattern shows up again and again in real-world data. According to the law, the formula for the approximate frequencies for each digit $d$ is: $$P(d)=\log\left(1+\frac{1}{d}\right)$$

So in large sets of data, $1$ appears as a leading digit approximately $30$% of the time, $2$ appears approximately $17.6$% of the time, all the way down to $9,$ which appears approximately $4.6$% of the time. The sum of all of these frequencies must be $100$%, as they encompass every possible leading digit.

$$\begin{array}{c|c} \text{Leading Digit} & \text{Probability}\\ \hline 1&30.1\%\\ 2&17.6\%\\ 3&12.5\%\\ 4&9.5\%\\ 5&7.9\%\\ 6&6.7\%\\ 7&5.8\%\\ 8&5.1\%\\ 9&4.6\% \end{array}$$

This pattern can appear in many types of data. For example, it can show up in the first digit of population numbers for different countries. It can also appear in numbers found in newspaper articles.

An interesting fact is that people are not very good at copying this pattern when they try to fake data.

Because of this, Benford’s Law is often used to help detect fraud. If the numbers do not follow the expected pattern, it may be a sign that something is wrong.

(Videos) Number 1 and Benford's Law

by Numberphile

Here's an amazing video by Numberphile on Benford's law.

Benford’s Law looks at the first digit of numbers in large data sets, especially when the numbers vary in size.

You might think that since there are $9$ possible first digits ($1$ through $9$), each one would appear about the same amount. But that is not what happens.

Benford’s Law says that about one-third of the time, a number will start with $1$. The chances slowly decrease as the digits get larger.

This means numbers are much less likely to start with $9$ than with $1$.

Benford's Law

Benford’s Law is a pattern that describes how the first digits of numbers are distributed in real-world data.

In simple terms, it tells us how often we see certain digits when we look at numbers that appear naturally, like populations, prices, or measurements.

Instead of each digit appearing the same amount, some digits show up more often than others.

This pattern may seem surprising, but it appears again and again in many different types of data.

What we can see is that the number $1$ appears about $30\%$ of the time and the rest of the numbers appear less and less as you go up the digits. This doesn't make a ton of sense when you think about it since you'd expect the numbers we see in the real world to be random, and therefore, you'd expect the digits to be evenly distributed.

What's interesting is that this pattern occurs in a wide variety of data sets, from house prices to lengths of rivers. There's no one unifying reason for why this occurs but we still have a formula for the distribution of the digits, which is: $$P(d) = \log_{10}\left(1+\frac{1}{d}\right)$$ where $P(d)$ represents the probability that a digit $d$ occurs in a data set.

Benford's Law

In 1938, Frank Benford and Simon Newcomb posited that all numbers do not serve as leading numbers equally. This law is also known as the Newcomb-Benford Law.

If we consider the world, many leading and naturally occurring numbers are not large numbers. In fact, about $30$ of all leading numbers are $1$.

Benford's Law states that in many real-life situations, the leading number is a small value. This concept seems strange since we've always associated the real world with random numbers! However, it is true that smaller numbers occur more frequently.