Back of the Envelope Skills

Back of the Envelope Calculations

  1. Class I : Ballpark Guesses are not Guesses

A common rhetorical ploy in policy-related public relations is the "Every seven minutes something happens" gambit. It is rather effective since it cuts right through any bias the hearer might have toward thinking a particular phenomenon is unusual just because she does not often encounter it.

While the technique is mathematically sound, it is interpretively flawed because it is completely dependent on the size of the area under consideration. A murder occurs almost never in this classroom, but in Oakland one happens every three or four days and in California about every four hours while in the US as a whole,

Example: Homicide

Intentional homicide, rate per 100,000 population, by sub-region, 2004



Whether one leans toward the qualitative or the quantitative, a critical skill for the policy analyst, 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 a basic skill 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. 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 don't want to be winging it either.

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 are four

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 million1 population; US legislature435 congress + 100 senators and California legislatureAssembly 80, Senate 402; number of California counties58; US GDP$13 trillion (thousand billion)3; diameter of earth8,000 miles; median household income in the US$50,0004

Basic sizes. Width of a finger or hand. Ceilings. Doors. Tables. Width of a car.


Orders of magnitude

Communicating Relative Magnitudes

Getting a feel for magnitudes

How to approximate a number. Eschew meaningless precision.
Order of magnitude
Scientific notation
Call it something you can remember

If the universe were a baseball diamond type analogies.

Dimensional analysis
Rule of thumb
Sanity test

How to carry worst/best case values through a calculation


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 1020 joules5 and US is about 1 x 1020 joules6, then what is the ratio of US per capita usage to that of the world as a whole?

\begin{align} USPC = \frac{1 \times 10^{20} joules} {3 \times 10^8 people} \end{align}
\begin{align} WPC = \frac{5 \times 10^{20} joules} {7 \times 10^9 people} \end{align}
\begin{align} \frac{USPC} {WPC} = \frac{\frac{1 \times 10^{20}} {3 \times 10^8}} {\frac{5 \times 10^{20}} {7 \times 10^9}} \end{align}
\begin{align} \frac{USPC} {WPC} = \frac{\frac{1 \times 10^{12}} {3}} {\frac{5 \times 10^{11}} {7}} \end{align}
\begin{align} \frac{USPC} {WPC} = \frac{7 \times 1 \times 10^{12}} {3 \times 5 \times 10^{11}} \end{align}
\begin{align} \frac{USPC} {WPC} = \frac{7 \times 10} {3 \times 5} = \frac{70} {15} = \frac{14} {3} \simeq 5 \end{align}

Announce rather than discard inconvenient uncertainty7


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