Page Summary
This page is about techniques for doing quick and dirty calculations that let us have a feel for the size or extent of things even in the absense of detailed information.

Related Pages

Possible New Material for 2012

Data Analysis with Open Source Tools by Philipp K. Janert. Ch 7 141-161 is "Guesstimation and Back of the Envelope"


Wikipedia. "Back of the envelope calculation"
Mitchell N Charity A View from the Back of the Envelope
Mitchell N Charity Why Approximate
Wikipedia. Fermi Problem
Louisiana Lessons. 1996. Classic Fermi Questions with annotated solutions
mittechtv. 2010. BLOSSOMS - The Art of Approximation in Science and Engineering: How to Whip Out Answers Quickly


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 ways of thinking about the "quality" of a number or measurement. The utility of these categories for the present purpose is that they allow us to relax our concerns in appropriate ways.

Precision is "how many decimal places." We routinely relax precision concerns when we round.
Accuracy is whether a measure is correct or not, how far it is from the true value. In BOTEC we are concerned with what side of the "real" value our guestimates of various component values are. In other words, we will often know that we don't know a number accurately, but we can say "the highest it's likely to be is X and the lowest Y" and proceed from there.

Validity is whether or not a measurement measures what we think it measures. In BOTEC we tend to assume the validity of the numbers we are working with.

Reliability refers to whether a measurement or observation technique provides the same result repeatedly. Generally not an issue in BOTEC.

An Example

About how many 8th graders in California?

Suppose I know that there are about 37 million people in California.1 Let's assume the life expectancy in the US is 74 years (chosen because I have a feeling it's about right and it is $2 \times 37$.2.

Next let's assume that the age distribution is flat3 which is to say that there are just as many people who are 3 as 19 as 45 years old, etc. Spread over 74 years the 37 million would leave about a half a million in each age year.

So, I guess there are about 500,000 eighth graders in California.

The actual number appears to be between 515,000 and 580,00042,571,022

The Elements of Rough Calculations

Back of the envelope calculations begin with a number that's the easiest to find out, the most reliable to guess, most straightforward to observe. Thus, one part of developing a facility for BOTEC is to have at your "finger tips" a handful of important numbers and measures.

Numbers to Know

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

A few numbers a person in policy should have an intuitive feel for

WorldAbout 7 billion and U.S.About 310 million5 and California37 million Oakland390,000 — call it 400,0006population;
US legislature435 congress + 100 senators and California legislatureAssembly 80, Senate 407; number of California counties58; US GDP$13 trillion (thousand billion)8; diameter of earth8,000 miles; median household income in the US$50,0009

Basic sizes. Width of a finger (3/4") or hand (4"). Ceilings (most 8'). Doors (standard is 2'6" wide, 6'6" high). Tables (30" height standard). Parking space (9' x 20')10

Orders of magnitude

First of all, you need to understand exponents. AB means A times itself B times. Don't continue until you get that.

So, what is $(A \times A) \times (A \times A \times A)$? Well, we can write it as

\begin{align} A \times A \times A \times A \times A = {A}^{5} \end{align}

but we can also write

\begin{align} {A}^{2} \times {A}^{3} \end{align}

So, we can state the rule:

\begin{align} {A}^{B} \times {A}^{C} = {A}^{B+C} \end{align}


  1. What is ${10}^{3} \times {10}^{4}$?
  2. What is ${10}^{0} \times {10}^{0}$?
  3. What is ${2}^{2} \times {3}^{2}$?
  4. What is ${2}^{5} \times {2}^{5}$?
  5. # What is ${10}^{5} \times {10}^{-5}$?

Learn the names of the powers of ten

"One, ten, hundred, thousand, ten thousand, hundred thousand, million, ten million, hundred million, billion, …"

One 100 1 "ten to the zero"
ten 101 10 "ten to the one"
hundred 102 100 "ten to the two"
thousand 103 1,000 "ten to the three"
ten thousand 104 10,000 "ten to the four"
hundred thousand 105 100,000 "ten to the five"
million 106 1,000,000 "ten to the six"
ten million 107 10,000,000 "ten to the seven"
hundred million 108 100,000,000 "ten to the eight"
billion 109 1,000,000,000 "ten to the nine"
ten billion 1010 10,000,000,000 "ten to the ten"
hundred billion 1011 100,000,000,000 "ten to the eleven"
trillion 1012 1,000,000,000,000 "ten to the twelve"

How to do order of magnitude calculations?

What is one thirtieth of six million? We see that there's a relation here: 3 and 6. One thirtieth is one sixth times one fifth. A million is {10}^{6}.

\begin{align} \frac {1} {6} \times \frac {1} {5} \times 6 \times {10}^{5} \times 10 \end{align}
\begin{align} \frac {1} {6} \times 6 \times \frac {1} {5} \times 10 \times {10}^{5} \end{align}
\begin{align} 5 \times {10}^{5} = five \hspace {6pt} hundred \hspace {6pt} thousand \end{align}

The secret here is simply to be on the look out for numbers that cancel one another out — either by multiplying together to give approximately 10 or by canceling one another out — and by constantly separating out powers of ten for the sake of convenience.

Let's try another. How many steps would it take for me to walk home. It takes about 8 minutes by car, 2 to get to the free way, 4 on the freeway, and about 2 after getting off the freeway.

So, let's assume I drive 60 on the freeway and 15 (after taking into account lights and stop signs) on surface streets. This suggests 4 miles on the freeway and 1 mile on surface streets. There are 5280 feet in a mile so that's about 26,000 feet. I have a stride length of about 3 feet so it would be something like 9,000 steps.

Producing a High and a Low Estimate

To bracket your estimates (also known as carrying worst/best case values through a calculation) you can round all of the component values up to reach the upper estimate and down to reach the lower estimate.

Rounding Another trick is to get in the habit of rounding numbers to values that are convenient and easy to think about. How long will it take to transcribe all the interviews for your thesis? You expect to do 30 interviews lasting between 45 and 90 minutes. The first one you transcribed was a 1 hour interview that took 4.5 hours to transcribe and produced a transcript that was 23 pages long.11

To produce the high end estimate, let's say all the interviews are 90 minutes, each hour takes five to transcribe and produces 25 pages. This gives

\begin{align} 30 \times 1.5 \times 5 = \end{align}
\begin{align} 30 \times 8 = \end{align}
\begin{equation} 240 hours \end{equation}


\begin{align} 30 \times 25 = \end{align}
\begin{align} 8 \times 4 \times 25 \end{align}
\begin{equation} 800 pages \end{equation}

To produce the low end, we would use 30 interviews at 60 minutes and 4 hours per for total of 120 hours.


Getting a feel for magnitudes

How to approximate a number. Eschew meaningless precision.

Order of magnitude
Scientific notation
Call it something you can remember

Communicating Relative Magnitudes

If the universe were a baseball diamond type analogies.

Dimensional analysis

Dimensional analysis is related to "converting units" — e.g., how many seconds in a year? Both are units of time, but we need to multiply out

\begin{align} year \times \frac {365.25 days} {year} \times \frac {24 hours} {day} \times \frac {60 minutes} {hour} \times \frac {60 seconds} {minute} \end{align}

What's important here is that we remember that we are looking for a number of seconds. If we label all of our numbers with the right units and cancel them out properly, we are performing a test on whether we seem to have the right numbers in our equation.

This will come up when we work with marginal rates of return in cost benefit analysis.

See also Fun with Dimensional Analysis
Wikipedia Dimensional Analysis

Other Topics

Rule of thumb
Sanity test

Honesty, Integrity, and the Back of the Envelope

Announce rather than discard inconvenient uncertainty.12 Be alert to what you don't know for certain and try to estimate possible variation and ask how it would affect your result.

Suppose a medicine says take 2 pills if you are a normal sized adult and not to exceed 6 pills in 24 hours. But you are not a normal sized adult, you don't think — you weigh 200 pounds. If they mean 150 pounds as normal sized adult, you could probably take 8 without a problem. If they mean 180 lbs, then you probably ought not even take an extra 1.


Mitchell N Charity Why Approximate
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