## Back of the Envelope Calculations

Whether one leans toward the qualitative or the quantitative, a critical skill for the entrepreneur, policy analyst, innovator, natural or social scientist, and citizen is to be able to "size things up," either in one's head or on the back of the proverbial envelope. It's one of the most basic skills of critical thinking — before we engage in complex and time consuming analyses, we need to have a ballpark idea of what the numbers are likely to be. In this module participants will learn the importance of along with the basics of how to do "back of the envelope calculations"

We know from studies of real world behavior that we are subject to bounded rationality — we rarely have all the information we'd like, the time or techniques to fully analyze the information we have, the luxury of complete rationality. But we this does not give us an excuse to put our heads in the sand or just to "wing it." Back of the envelope (BOTE) or Fermi calculations are a solution to this problem. They are not a natural talent, they are a learned skill.

Ultimately, this topic is about making optimal use of the time and information available to produce the best possible guestimate.

## Precision, Accuracy, Validity, and Reliability

Recall from research methods that there measurements have these four characteristics. A measurement can be more or less precise - how fine a resolution or how many decimal places? A measurement can be accurate (read "near the true value") or inaccurate (aka "wrong"). A measurement can be valid - it measures what we say it measures - or not. And a measurement can be reliable - if you measure again using the same instrument you will get more or less the same result.

## The Elements of Rough Calculations

## Numbers to Know

Basic numbers we should know (in their most basic form)

Ten basic numbers a person in policy should have an intuitive feel for

WorldAbout 7 billion and U.S.About 310 million

^{1}population; US legislature435 congress + 100 senators and California legislatureAssembly 80, Senate 40^{2}; number of California counties58; US GDP$13 trillion (thousand billion)^{3}; diameter of earth8,000 miles; median household income in the US$50,000^{4}Basic sizes. Width of a finger or hand. Ceilings. Doors. Tables. Width of a car.

[http://www.census.gov/ipc/www/popclockworld.html

Orders of magnitude

Communicating Relative Magnitudes

Getting a feel for magnitudes

How to approximate a number. Eschew meaningless precision.

Round

Order of magnitude

Scientific notation

Call it something you can remember

If the universe were a baseball diamond type analogies.

Dimensional analysis

Guesstimate

Heuristic

Rule of thumb

Sanity test

How to carry worst/best case values through a calculation

## Exercises

How many movies have you seen? How many bananas have your peeled in your lifetime?

How many hours did you spend on facebook last year?

How many golf balls will fit in this room?

About how much area would we need for a 250 car parking lot?

If the world energy consumption is about 5 x 10

(1)^{20}joules^{5}and US is about 1 x 10^{20}joules^{6}, then what is the ratio of US per capita usage to that of the world as a whole?Announce rather than discard inconvenient uncertainty

^{7}## References

Mitchell N Charity <ude.tim.scl|ytirahcm#ude.tim.scl|ytirahcm> A View from the Back of the Envelope

Wikipedia. "Back of the envelope calculation"

Louisiana Lessons. 1996. Classic Fermi Questions with annotated solutions

Wikipedia. Fermi Problem

mittechtv. 2010. BLOSSOMS - The Art of Approximation in Science and Engineering: How to Whip Out Answers Quickly

Lori Ann White The art of back-of-the-envelope calculations: Students estimate their way through pop culture problems to learn a life skill. Symmetry Online Magazine

## Back of the Envelope Calculations