Making Estimates

Problems :  Hairs on Human Head, Atoms in a Grain, Piano Tuners in CU
McDonalds franchises in the US, Storage Capacity of CDs


The ability to estimate – to order of magnitude or so – the size or probability is useful in many endeavors:

-  To provide a rough check of more exact calculations
-  To provide a rough check of research results or hypotheses
-  To obtain estimates of quantities when other resources aren’t available
-  To obtain estimates of quantities that are difficult to measure precisely
-  To obtain estimates of quantities for which no firm theoretical prediction exists

⇒ very important in interdisciplinary sciences, soft matter, astrophysics
To provide bounds for possible design alternatives
e.g., how many grains of sand are there on earth’s beaches?
         how many piano tuners are there in Chicago?
         how many atoms are in your body?

These are sometimes referred to as Fermi problems, who was famous for (among
other things) posing and solving such problems.

Getting started:
(1). Don’t panic when you see the problem
(2). Write down any fact you do know related to the question
(3). Outline one or more possible procedures for determining the answer
(5). Keep track of your assumptions
(4). List the things you’ll need to know to answer

Make everything as simple as possible!
(1). Round numbers to “convenient values” 
       e.g., π ≈ 3; 8.4 ≈10; etc.
(2). Choose convenient geometries when modeling
       e.g., a spherical cow, a cubic grain of sand, etc.
(3). Make “educated” guesses, 
       keep track of them, as they'll set bounds on estimate
estimate's fidelity 
 4). Use ratios when possible, and dimensionless parameters
(5). If possible exploit plausible scaling behavior of some quantity,
       i.e., estimate unknowns by assuming it scales linearly with some parameter

Checking your estimates:
(1). Make sure results are dimensionally correct! ⇒ a very powerful tool!
(2). Check estimate's plausibility , if possible
(3). Check estimate's plausibility using alternate method
(4). Perform “reality check” on estimate based number & size of approximations made
(5). More quantitatively - place “bounds” on your estimate:
       for “upper bound” – in equations, put largest/smallest values numerator/denominator 

       for “lower bound” – the reverse

http://physics.illinois.edu/undergrad/SeniorThesis/EstimatesResearch.pdf