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Evaluating Benford’s Law

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Author: Megan Doyle

Grade Level: 9th to 12th; Type: Mathematics/Computer Science

Objective:

This experiment will investigate the applicability of Benford’s Law to many sets of everyday data.

Research Questions:

  • For which types of data is Benford’s Law valid?
  • Are there certain types of data sets that do not conform to Benford’s Law?

Benford's Law, also called the first-digit law, describes how in lists of numbers from many everyday sources of data, the leading digit is distributed in a specific, predictable way. This experiment will evaluate many sources of data to see if Benford’s Law holds true.

Materials:

  • Many real-life sources of data
  • Computer
  • Graphing software
  • Notebook for analyzing results

Experimental Procedure:

  1. Gather many large datasets. Examples of data could include: A list of every country and its population; income tax returns;utility bills; distance of stars from the Earth in light years.
  2. Tally the leading digit of every number you encounter in a dataset. This can be done on a computer using Microsoft Excel (see the second reference, below).
  3. Create a bar graph demonstrating the frequency of each leading digit (1 to 9) in the dataset.
  4. Analyze your results. Do the numbers in your data set conform to Benford’s Law? The expected percentage of numbers starting with each digit is: 1: 30.1%; 2: 17.6%; 3: 12.5%; 4: 9.7%; 5: 7.9%; 6: 6.7%; 7: 5.8%; 8: 5.1%; 9: 4.6%
  5. Repeat your analysis for other sets of data. Do all of your datasets appear to conform to Benford’s Law? What types of datasets do not follow Benford’s Law?

Terms/Concepts: Benford’s Law

References:

  • Benford, F. 1938. The law of anomalous numbers. Proceedings of the American Philosophical Society 78:551.
  • Lynch, A. and Xiaoyuan, Z. “Putting Benford’s Law to Work.”
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