Benford’s Law

Benford’s Law is one of the fundamental laws of mathematics that states, “In a large dataset, approximately 30% of numbers start with a 1 while less than 5% start with a 9.”

In this article, we have covered, Benford’s Law definition, formula, examples and others in detail.

Benford’s Law states that the leading digit is usually small in many real-world cases. It is seen that the leading digit is usually either 1 or 2. So as the value of the digit increases the chance of utilizing that very digit becomes small. It is seen that the chance of number 1 appearing as the leading digit is about 30.1% whereas for 9 it is about 4.6%.

Before writing about the law, let us have a quick look at some situations. We use numbers to keep count of entities. For instance, consider the number of followers on X(formerly known as Twitter). Some have 120000 followers, some have 3000 followers. Here in this, the leading digits are 1 and 3. Also, we all know that each number carries equal weight and nine digits can be used as leading digits. So the chance or Probability of occurrence of each digit is 1/9 or 11.11% for the leading digit. However, this is not applicable in real-world scenarios.

Let us have a look at Benford’s law Curve,

From the curve, we can see that in many real-world scenarios, number 1 has the maximum chance of occurring. The probability is 30.1%. Number 9 has the least chance of occurrence. So it is a decreasing curve. It is to be noted that the graph is logarithmic and follows a non-uniform distribution.

Probability of the occurrence of any particular digit d in the first place of the number is given by the formula:

P(d) = log₁₀(1 + 1/d)

Benford’s Law is widely used. Some of the applications are as follows:

Fraud Detection

Benford’s Law is widely used in fraud detection. As we all know Benford’s law curve is logarithmic. So fraudsters make use of arbitrary numbers to commit fraud. So if the leading digits of the numbers do not follow the above curve, we can conclude that there are chances that a fraud might have occurred.

Stock Prices Analysis

Benford’s Law is widely used by analysts to analyze stock market prices. It helps to check the authenticity of the stock market data. If there are some deviations then the analysts can look for the deviations and analyze accordingly.

Some of the disadvantages of Benford’s Law are:

Large Dataset

This law is applicable when we have a large dataset. This law does not work for small datasets. So this curve requires lots of data points to analyze and create a curve which will check whether they are similar or not.

Not Ultimate Law

This law does not guarantee 100% results. It is just a tool to measure the uncertainty in data. For example, if stock prices deviate from the pattern, it does not mean that fraud has occurred. It can also mean that some other errors can also affect the actual value.

Assumes that Values are Independent

This law pre assumes that the values present in the dataset are independent. But this is not so. It might happen that a particular dataset is following a specific pattern.

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Q1. Calculate the probability of the occurrence of 2 as the leading digit using Benford’s law

Solution:

Here, value of digit d is 2

Using Benford’s law formula we get

P(2) = log10​(1+1/2)

= log_10​(1+0.5)

= log10​(1.5)

= 0.17609

≈ 17.61%

Q2. Suppose we have 5000 records related to stock. How many records we can expect the starting digit to be 8?

Solution:

Using Benford’s Law we need to calculate probability of occurrence of number 8 as leading digit

P(8) = log_10​(1+1/8)

= log₁₀​(1+0.125)

≈ 0.0512

Expected number of records with leading digit 5

= 5000×0.0512

= 256

Expected Number of records with leading digit 5 is 256.

Q3. Calculate the probability of occurrence of 1 or 4 as the leading digit in the stock market.

Solution:

Using Benford’s Law we need to calculate probability of occurrence of number 1 as leading digit,

P(1) = log₁₀​(1+1/1)

= log₁₀​(1+1)

≈ 0.301

Similarly for 4

P(4) = log_10​(1+1/4)

= log_10​(1+0.25)

≈ 0.0969

Therefore combining two probabilities we get

P(1 or 4) = 0.301 + 0.0969

= 0.3979

= 39.79%

Probability of occurrence of 1 or 4 as leading digit is 39.79%

Q4. Calculate the probability of the occurrence of 5 as the leading digit using Benford’s law.

Solution:

Here value of d is 5

P(2) = log_10​(1+1/5)

= log_10​(1+0.2)

= log₁₀​(1.2)

= 0.0792

≈ 7.92%

What is Benford’s Law?

Benford’s Law or the first digit law states that the leading digit of the values in the dataset is small. As the leading digit increases the chance of occurrence also decreases.

Does Benford’s Law provide 100% accurate results?

It must be noted that Benford’s Law does not guarantee 100% results. It provides a base to analyze the results.

Why does Benford’s Law not work for small datasets?

Benford’s Law is a probabilistic law. To determine the underlying pattern, this law requires large amount of data so as to avoid biasedness. It also makes use of large amount of data as it uses logarithms to detect the leading digit.

Can we use Chi-Square Test to test the Acceptability of Benford’s Law?

Yes we can make use of Chi Square test to check whether the data follows the Benford’s Law or not.

What are some Advantages of Benford’s Law?

Benford’s Law has many advantages. Some of them are as follows:

• Used in Fraud Detection.
• Used in Analysis of Data.
• Used for Preliminary Screening Tests.