Benford’s Law and Election Data

Why do first digits of votes diverge from Benford’s distribution?

There are claims of statistical “proof” of election fraud in the United States. These claims often depend on considering leading digits of vote counts. The resulting distribution does not match the Newcomb-Benford distribution.

An LBC radio host shared an article with flawed analysis, misapplying Benford’s Law.

What is Benford’s Law?

Numbers can suffer manipulation. To detect anomalies, we want an expected distribution for comparison.

Here, 1 appears as a leading digit more than 9. (Image: Significance)
(Image: Imperial College London/Adrien Jamain)

When does Benford’s Law apply?

Despite being a ‘law’, it is not universal. It is an observation about some types of data sets. William Goodman restated some guidelines for suitability towards conforming to Benford’s Law:

  • A high span of numerical values: The sample should include values across many orders of size.
  • Right-skewed distributions: Conforming sets often have origins in multiplication or combinations.
  • Non-arbitrary values: Arbitrary assignments of numbers do not exhibit these patterns.
Some data sets are close to the expected proportions. Others are not. (Image: Significance)

What about elections?

These claims of electoral fraud depend on misapplications of Benford’s Law.

Of the 2,069 precincts, 2,023 had a vote total between 100 and 1,000. (Image: R Pubs)
The Democrats have a high vote share in Chicago. (Image: R Pubs)
We would not expect conformity to the Newcomb-Benford distribution. (Image: R Pubs)

It is not simply that the Law occasionally judges a fraudulent election fair or a fair election fraudulent. Its “success rate” either way is essentially equivalent to a toss of a coin, thereby rendering it problematical at best as a forensic tool and wholly misleading at worst.

In applications like forensic accounting, non-conformity is not ‘proof’ of fraud. It is often a flag for anomalies worthy of further investigation — such as standard auditing.

This blog looks at the use of statistics in Britain and beyond. It is written by RSS Statistical Ambassador and Chartered Statistician @anthonybmasters.

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