What Is a Back-Of-The-Envelope Calculation?

A back-of-the-envelope calculation is an informal mathematical computation, often performed on a scrap of paper, such as an envelope. A back-of-the-envelope calculation uses estimated or rounded numbers to develop a ballpark figure quickly. The result should be more accurate than a guess, as it involves putting thought to paper, but it will be less accurate than a formal calculation performed using precise numbers and a spreadsheet or calculator.

Key Takeaways

  • A back-of-the-envelope calculation is a quick and informal mathematical computation.
  • This type of calculation is used as an estimate to arrive at a ballpark figure when needed quickly.
  • Back-of-the-envelope calculations are more accurate than a guess but not as refined and accurate as a formal analysis.
  • A back-of-the-envelope calculation does not require any research but rather relies on the estimator's current knowledge.
  • These types of calculations are also referred to as Fermi problems.

Understanding a Back-Of-The-Envelope Calculation

A back-of-the-envelope calculation might be used to determine whether further research and more detailed calculations are warranted. In the STEM disciplines (science, technology, engineering, math) such calculations are commonplace for those who suddenly become inspired by an idea or want to perform a quick exercise to estimate a number.

The fact that a back-of-the-envelope calculation references writing out an idea or calculation on an envelope is meant to demonstrate how little preparation goes into the analysis. The individual is either so hurried or not bothered, that they will find whatever material is nearby that they can write on.

For mundane thinkers sitting at a coffee shop who just want to estimate the number of cars that pass through a bridge toll booth, the number of customers that visit a fast-casual restaurant during lunch hour, or earnings per share (EPS) of a company in five years, back-of-the-envelope calculations are as useful in framing these quantitative concepts.

Inspired scribbling on the back of an envelope throughout history has led to significant discoveries by great men and women, and it helps ordinary people with business ideas or investors with trading ideas to get a start.

The beauty of a back-of-the-envelope calculation is that it does not require any research and is based on the estimator's current knowledge, not requiring any further information that is not currently known in the moment. Usually, the general details or accurate inputs aren't as important as arriving at some sort of clearer picture of the problem at hand.

Fermi Problem

Back-of-the-envelope calculations are sometimes referred to as Fermi problems, named after physicist Enrico Fermi. Fermi was known for being able to make close approximations to problems with the thinnest of data, and sometimes with no data at all. Enrico Fermi won the Nobel Prize in physics in 1938.

Real World Example

At the 2017 World Government Summit, Elon Musk explained how aliens from outer space could reach Earth: "I'll give you some back-of-the-envelope calculations. Any advanced alien civilization that was at all interested in populating the galaxy, even without exceeding the speed of light, if you're only moving at about 10% or 20% the speed of light, you could populate the entire galaxy in, let's say, 10 million years, maybe 20 million, max."

Here, Musk did not use any detailed analysis nor did he perform any research. He came up with the numbers and the calculation while speaking, based on his own knowledge, to arrive at an estimate to the problem he was discussing.