Estimation: Back of the envelope calculations

It is very useful to be able to make quick order of magnitude estimates using back-of-the-envelope calculations.

Microsoft uses such questions to determine how "intelligent" you are, as far as practical problem solving goes.

In the area of information systems and computer technology, a productivity paradox has been identified.

Delivered computing power in the U.S. economy has increased by more than two orders of magnitude since 1970, yet productivity, especially in the service sector, seems to have stagnated. ... No, computers do not boost productivity, at least not most of the time.

Communications of the ACM

Two orders of magnitude is a factor of 100, or a difference of 100 to 1.

What range is within an order of magnitude of 50?

The range 5 to 500 is within an order of magnitude of 50.

If an estimate x is within a factor of 4, then the actual value is somewhere in the range from 0.25*x to 4.0*x.

If the lowest possible value is 10 and the highest possible value is 90, then an estimate of 30 is within a factor of 3.

Your answer to a real world problem should always match reality. One of my undergraduate physics professors, also the head of the department, always stressed, "If you work out a problem and your result is that a baseball pitcher is throwing the baseball 100,000 miles per hour, then something should tell you that you did something wrong". But what? Suppose that your answer does not match reality. If your answer is off by more than an order of magnitude (a factor of 10), you might want to consider the following possible errors in the computation (the computation is used to determine the logical implication(s) of the model).

A back-of-the-envelope calculation is a simple model of a problem whose calculation can provide useful information to be used in making a decision.

The term supposedly came about during World War II during an atomic bomb test. Right before the blast, a scientist tore up some paper, threw it in the air when the blast went off, then approximated the distance that the paper pieces traveled, did a quick calculation on the back of an envelope, and determined the yield of the explosion within an order of magnitude, days before the actual calculations confirmed his estimate. (related by Jon Bentley in one of his books).

Let us look at the problem of estimating an academic computing budget snyder12.. One could list every piece of equipment needed for every possible need for the next 5 years. By the time this is done, the information is probably out of date. A back-of-the-envelope calculation might assume, for example, that a workstation lasts about 5 years, costs about $5000 over its lifetime (initial cost, printing, software, upgrades, network connection, etc.) or about $1000 per workstation per year. So if you need 120 workstations on a continuing basis, that amounts to about $120,000 per year on a continuing basis. Anything less will result in a net degradation in performance.

This type of problem solving is often called a "back of the envelope" calculation.

You have 2 minutes, or 120 seconds, to answer this question.

You must stop work at that time.

No one is to write anything after that time.

How many ping pong balls will fit inside the room you are in?

How would you modify your answer/technique if I gave you an additional 5 minutes?

How much more accurate would your answer be?

Answers given by 11 students in CSCI 476 (Fall, 1999).

900,000,000,000,000
600,000,000,000
100,000,000
10,000,000
3,500,000
1,500,000
1,350,000 (used square feet rather than cubic feet)
1,000,003
162,431
1

Here is one quick estimate.

The size of a ping pong ball is about 1 square inch.

A cubic foot is 12 inches * 12 inches * 12 inches, which is about 1500 to 2000.

The number of cubic feet in the room is about 30 feet wide * 30 feet high * 12 feet high, or about 10,000 cubic feet.

So, about 15,000,000 to 20,000,000 ping pong balls would fit in the room. Let us say, 18,000,000.

Then the range that is within an order of magnitude is from 1,800,000 to 180,000,000.

Problems arise when you are far from the real answer.

Answers given by 6 students in CSCI 621 (Spring, 2003)

2,964,000,000,000,000,000,000,000,000,000,000,000,000 (used power notation)
10,000,000
10,000,000
1,000,000
50,000
10

A cubic foot is 12 inches * 12 inches * 12 inches, which is about 1500 to 2000. In your head, you can quickly decide that it is more than 10*10*10 = 1000 by about 50% or so, for about 1500 or so ping pong balls per cubic foot. (The actual value of 12*12*12 is 1728).

The room is about 24 feet wide by 32 feet long by 14 feet high. This is about 30 (rounding 24 up) by 30 (rounding 32 down) by 10 (rounding 10 down), or 9000 cubic feet. (The actual value of 24*32*14 is 10,752).

Quick estimate: 1500*9000 = 13,500,000

More elaborate computation: 12*12*12*24*32*14 = 18,579,456

Of course, ping pong balls may not pack exactly as 1 per cubic inch.

You can do the multiplication easily by hand in two minutes, but may have to go with the quick estimate if you had, say 30 seconds to make the decision.

Taking 18,000,000 as a 2-minute estimate, any answer from 1,800,000 to 180,000,000 would be within an order of magnitude of the actual answer.

In this case, we would probably never attempt to fill a room with ping pong balls and then count them. But, the air in the room is filled with molecules, and properties of those molecules may be important in certain circumstances.

In any event, the ability to quickly make accurate estimates can be extremely useful when decisions may be based on many factors, each of which has to be estimated quickly.