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Fermi estimation explained

Fermi estimation is a method you can use to make estimates about the data you encounter day-to-day. In this article, Dr Humphries shows you how.

‘$17 million was wasted due to food stamp fraud in the US.’

How do you feel about the headline above? Does it feel like it includes a misleading statistic? If you were in the US, would you be enraged?

This section is going to help you unpack statements like this to find the true impact. You will learn how to use Fermi estimation to look past the emotion the headline is trying to make you feel and dig down into the truth.

What is Fermi estimation?

In today’s fast-paced world, quick and practical problem-solving methods are essential for tackling complex issues efficiently. One such method that has garnered attention is Fermi estimation. Named after the renowned physicist Enrico Fermi, this technique enables individuals to swiftly arrive at reasonable approximations for intricate problems by using orders of magnitude. Orders of magnitude, in the context of Fermi estimation, refer to approximating quantities to the nearest power of 10. This kind of simplification is useful when you need to make a rapid estimate without precise data.

Fermi estimation empowers us to make informed decisions and verify claims in a time-efficient way, by breaking down these problems into manageable parts. In this article, we’ll explore the concept of Fermi estimation and its application through real-world examples.

Unpacking Fermi estimation

At its core, Fermi estimation is a problem-solving approach that leverages rough but educated guesses to simplify complex problems. This technique is particularly useful when exact figures are unavailable or unnecessary, yet a general understanding or rough approximation is needed. The process involves breaking a problem into simpler components, making educated estimations of their values, and then performing straightforward calculations to obtain an approximate solution.

Illustrating with an example

To grasp the practicality of Fermi estimation, let’s consider a seemingly whimsical scenario: determining the number of tennis balls that could fit into an Olympic-sized swimming pool. This exercise involves estimating the volume of both the tennis balls and the pool itself.

Assuming the tennis balls have a diameter of 4 centimetres, we can approximate their volume by envisioning them as cubes with 4 centimetre sides, giving a volume of:

4 x 4 x 4 = 64 cubic centimetres = 0.000064 cubic metres.

Graphic showing one tennis ball inside a cube outline with each cube side measuring 4 centimetres each. The right side of the image shows the tennis ball as one part of a larger cube comprised of 4 x 4 tennis balls in cube outlines, demonstrating how a volume calculation should work.Select this link to expand the image (you will need to use the ‘back’ button in your browser to return to this page).

Shifting our attention to the Olympic-sized swimming pool, boasting dimensions of 50 metres in length, 25 metres in width, and a depth of 2 metres, we calculate its volume to be:

50 x 25 x 2 = 2,500 cubic metres.

By dividing the pool’s volume by that of a single tennis ball, we estimate that the Olympic-sized swimming pool can fit 39,062,500 tennis balls:

2500/0.000064 = 39,062,500 tennis balls

This seemingly playful exercise showcases the effectiveness of Fermi estimation in breaking down complex geometrical calculations into manageable approximations.

Harnessing the power of orders of magnitude

Another aspect of Fermi estimation involves harnessing orders of magnitude. This involves rounding values to the nearest power of ten, to simplify calculations without changing the nature of the problem. This technique is helpful when exact numbers aren’t required.

Let’s revisit our headline.

It claimed that $17 million was wasted due to food stamp fraud. We want to express that as a percentage of the food stamp budget in the US. This is because we know that a typical loss in the retail sector is about 1.5-3%, so we want to compare that with the food stamp loss of $17 million to determine whether the situation is as severe as the headline suggests.

To do this, let’s suppose that:

  • around 12.5% of Americans received food stamps which we approximate to 10% (1)
  • food stamps have an annual value of around $2,190 which we round down to $2,000 (2)
  • in 2016, there were approximately 324 million people living in the US, but to make our calculations simple, we round this number down to 300 million people (3).

Using these values, we can quickly estimate the US’s annual food stamp budget to be approximately $60 billion:

0.1 x 300 million x 2000 = $60 billion.

With this estimation, we deduce that roughly 17 million/60 billion = 0.000283 or approximately 0.02% of the food stamp budget is used fraudulently.

If the headline had been ‘0.02% of Food Stamp money is wasted due to fraud,’ would that have made you feel the same emotions as the one that was published?

This Fermi estimation provides a ballpark insight into the scale of the issue without the need for exhaustive calculations. Although it is not as precise as it could be, it does help us quickly understand the impact of the claim made in the headline: that the loss made in the food stamp program is substantially less than anything expected in the retail sector.

Fermi estimation is a useful way to solve problems when quick approximations will suffice. It works by breaking down complex problems, making educated guesses, and doing simple calculations to help you find reasonable answers quickly. Using orders of magnitude makes it even more practical, keeping important information while simplifying the calculations.

Reference

1. Desilver, D. (2023). What the data says about food stamps in the U.S. | Pew Research Center [Internet]. 2023 July [cited 2023 September 22]. Available from: https://www.pewresearch.org/short-reads/2023/07/19/what-the-data-says-about-food-stamps-in-the-u-s/#:~:text=That%20translates%20to%2012.5%25%20of,Guam%20and%20the%20Virgin%20Islands.

2. Luhby, T. (2023). Here’s why millions of Americans will lose $3 billion in monthly food stamp benefits starting in March. | CNN Politics [Internet]. 2023 March [cited 2023 September 22]. Available from: https://edition.cnn.com/2023/03/01/politics/food-stamps-pandemic-emergency-allotments/index.html.

3. Jordan, J. (2016). Census Bureau projects U.S. and world populations on New Year’s Day. | United States Census Bureau [Internet]. 2016 December [cited 2023 September 22]. Available from: https://www.census.gov/newsroom/press-releases/2016/cb16-tps158.html.

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Critical Evaluation in Data Science: Data, the World, and You

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