Examples of Back of the Envelope (BotE) Reasoning
Praveen Kumar Paritosh  January 10, 2000

Introduction
The back of the envelope (BotE) is fundamentally about scale, and the different levels of relevance and importance of things. The higher vantage point that these provide, in blurring away the details, allows us to discern connections and patterns that give us a better understanding of the world, to see the forest from the trees. Here we look at some examples of these, and how we can abstract the basic strategies which go into BotE. We need a strong grasp of domain (sometimes completely lateral thinking, and figuring out an analogical piece of knowledge from a completely different domain can be useful), intuition about what parts of the problem are important and imagination to dream up shortcuts.
Are the skills in doing BotE tied up with the domains where they have been used? In the examples discussed, and from our experience, it seems that one who is smart at coming up with good estimates, can usually always use this skill in problems from completely different domains, besides, by its nature it is usually an inter-disciplinary expertise. There is certainly some specific ways that a physicist would estimate that would be different from the way a physician diagnoses, but the focus here is the general principles that can be employed to answer questions from any domain. The goal is to write a computer program that can use these to either automatically use a knowledge base (like the CYC, specific domain theories or the Internet) to come up with such answers, or guide a user through the process of arriving at these estimates by making explicit the strategies that go into it. The distinct parts of the process of BotE are– 
1.	Modeling – Given a scenario and a query, coming up with the right set of principles to apply. For example, in a question from mechanics, one has to decide whether it can be described by the work energy theorem, conservation of momentum, or any other principles of mechanics. This is not only valid for technical domains, even when we talk of something like when will one turn up for a party, we use the model that we have built up of the person (by experience at seeing him turn up late for parties, or be fussy about punctuality) to come up with an answer.
2.	Parameter estimation – Once we have the model, to make it solvable; we might need to make estimates of parameters in the model in presence of incomplete information. In case of no direct information, we might use analogy, previous examples of similar problems, etc. 
3.	Simulation – After we have instantiated the model built in 1 by parameters and facts in the scenario, both quantitative and qualitative simulation may be used to reach to the answer. To make complicated quantitative simulation tractable, techniques like linearising the model, lumping parameters together, etc can be used. For the mechanics problem (and a lot of technical problems) it is the “algebra” and “arithmetic” part, solving differential equations, and plugging in the numbers. This is usually where the computers outperform people in the precision and speed, but it is an interesting that how people use smart tricks and thumb rules to come up with approximate answers, for example the “Rule of 72”, which says that if you spend a sum of money for y years at r percent interest rate, and if r x y = 72, your money will double, so one thousand dollars at six percent interest rate takes twelve years to double, and at eight percent it will take nine years (calculations yield 2012, and 1999 dollars respectively).

A question of Relevance
Consider the following problem – 
Carl Lewis’ Atlanta Olympic 1996 long jump that got him the gold was 8.50 m. Question: How long was he in the air, and how high did he go during the jump?
By itself, the question could be very complicated as one works out the aerodynamics of his jump, which will depend upon the wind and weather conditions at Atlanta, besides the exact manner in which Carl Lewis performs long jumps (does he land straight, spin while jumping, the mechanics of how he propels himself), the mass distribution of Carl’s body at that point of time, the elasticity and the coefficient of friction of the floor-material, and so on. But it doesn’t take a rocket scientist to discern that you could get a reasonable first-estimate by assuming that Carl is a point mass hurled through as a projectile, whose range is provided. The simple equations of projectile learnt in college physics could then be applied. One would need to make an assumption on the horizontal velocity of Carl on the take off, and then some “algebra” yields the answer. In this simple example, although the problem statement doesn’t talk about it, to successfully solve the problem, one has to figure out what is relevant and what is not. What will be relevant, depends upon the query, and the level of analysis  (at quantum mechanical levels of analysis lot of things could become important!). If it was Carl’s coach analyzing his jumping technique, then a lot more things become relevant to answering the query – his mass distribution, the way he takes off, aerodynamics, to name a few.
Irrelevant	Relevant Assumptions	
Carl Lewis	World Class athlete, so horizontal velocity  10 m/s
(since short run-up, use 100m world record data)
[Parameter Estimation]	
Atlanta, Olympics, 1996, gold	Point mass, projectile motion [MODEL]	
To be able to successfully use some piece of knowledge, we need the meta-knowledge, or some kind of a control knowledge that tells us the relevance of that knowledge to specific problem solving goals.
Fermi Problems
“… the estimation of rough but quantitative answers to (unexpected) questions about many aspects of natural world. The method was the common and frequently amusing practice of Enrico Fermi, perhaps the most widely creative physicist of our times. Fermi delighted to think up and at once to discuss and to answer questions which drew upon deep understanding of the world, upon everyday experience, and upon the ability to make rough approximations, inspired guesses, and statistical estimates from very little data.” (Morrison, 1963)

Fermi problems are an interesting study as they epitomize the art of BotE. As these questions have a large part of information missing or “unsaid,” coming up with a reasonable model becomes even more difficult. The emphasis is on the process of looking out for relevant information, and making reasonable assumptions when it is not available. It illustrates the difference between estimation and guessing. The fabled canonical Fermi problem is the question “how many piano tuners are there in Chicago?”. Faced with such a question without warning during a physics lecture, one would say that he/she does not know and cannot know, and if forced will produce a random guess that seems “plausible.” The problem with such an answer is that one does not know how much confidence to place in the answer, because the constrains which it follows have not been clearly identified, and so even if estimations have been made at an intuitive level (of “plausibility”), one cannot place the bounds on the possible error in it. 
As earlier said, a lot of information required to come up with an answer may not be available, and this is where reasonable assumptions become very important, the key thing is that these must be explicitly made – so that one may go back and refine them. This is important, as with bad data/guesses/estimates, even valid processing might yield poor results. It is crucial to be honest, and have the meta-knowledge of what we do not know. In the days of the Tacoma bridge, the advanced theories of turbulence required to model the aerodynamic lift on suspension bridges were not yet discovered, and it is interesting to learn from Jon Roeblings, the designer of the Brooklyn bridge which is the only one from that era that still stands. He realized that the design has a lot of assumptions that are not really valid, and he designed the stiffness of truss on the Brooklyn bridge about six times what the normal design predicted – overdesign for ignorance. In the following sections we will look at some of Fermi (and similar) problems. 
Q1 How many piano tuners are there in Chicago?
Lets first look at how to go about trying to find an answer to this question, and then do an analysis of the process. 
Approximately how many people are there in Chicago?  	3,000,000	
Does every individual own a piano?
ASSUME: individuals don’t own pianos; families do	No	
About how many families are there in Chicago?
ASSUME: five people make a family	600,000	
How many pianos are there in Chicago?
ASSUME: one out of ten families owns a piano	60,000	
How many tunings would these 60,000 pianos need in an year?
ASSUME: one tuning per year	60,000	
How many pianos can one tuner tune in an year?
ASSUME: one piano a day, for 200 working days	200	
How many piano tuners would be needed?	60,000/200 = 300
≈ 102 	

The first thing one realizes as he/she starts thinking is that this question asks for the number of a certain type of professionals in a city. Since professionals are people, the uppermost bound is the population of the city. Experience tells us that not everybody is a piano tuner, in fact very few are. What do we know about this profession – it falls under the general category of repair/maintenance/troubleshoot of a particular object/device, here the piano. Obviously, the number of these professionals will be strongly dependent on the number of such devices (pianos) that are to be found in Chicago. As pianos are found with people (besides people, if we were doing a detailed analysis, we would include schools, universities, and concert halls), we go back to the population of Chicago. Now, what could be the relation between the number of people and number of pianos – does one person own one or ten pianos? No, there is usually a piano in the family. The number of families is a complicated question, for a start we assume that one-fifths of the population gives us the number of families. Now, we search our experience to find out the number of pianos that these six hundred thousand families will have – do we recall seeing a piano in every family? Definitely not, but then we see pianos once in a while – maybe one in ten to one in hundred families have pianos. These are the order of magnitude bounds, we could calculate the bounds for piano tuners, depending upon this critical factor, the piano-per-family statistic of a city. This statistic will have a direct affect on our end result, and this is a good example where the best that you can do for a crucial piece of information is guess. But the bounds mentioned appear reasonable. Once, the number of pianos is found, we want to get back to tuning. Most of us (who have no experience with a piano, or maintaining it; according to above estimate those who were in the nine-tenths of families) have the faintest of ideas regarding what goes into tuning of a piano, but we have come across some or the other musical device that requires to be “tuned” as the mechanical properties of the strings, or other parts change by use, wear, or a change of weather. At homes, one usually doesn’t really bother about perfection so much, and play it so often, so at some homes once in years, and at others where its more actively used, tuned a couple of times in an year. At concert halls, the pianos would be tuned after every concert, to guarantee the quality of music or even to change the tuning to suit the character of the performance. So on an average once a year seems to be reasonable. One important observation here is that we implicitly chose a basis of one year as that makes things easier. 
	Now, that we have arrived at the number of tunings per year required in Chicago, the law of demand and supply (which is a form of a conservation law) tells us that ideally, we should have as many piano tuners in business as can do this. For our basis of one year, lets see how many pianos could a tuner tune. This is another crucial place in this problem, where we can just make a guess. Piano tuning is a rather obscure profession, I don’t have friends who are piano tuners – do they work full time on piano tuning (which does not seem to be a job where there is a consistent regularity, maybe if nobody is bothered during summer, they cant go jobless). A better model of what piano tuning entails, and the nature of the profession would have been helpful. So, we make another guess – a piano tuning, on an average per day, so two hundred working days would mean two hundred piano tunings, which gives us the number of tuners needed to be 300. As there are lot of places where we have made order of magnitude assumption, we could be sure only to an order of magnitude, so 102 is the answer. The advantage of making our assumptions explicit is that we can go back, and refine them to get a more accurate estimate.
Of Pianos and Tuners
	I did some research on piano tuning, and the profession to see how much off the mark the above estimate is. The piano is a complex blending of many diverse and costly raw materials like top quality wood of many species, iron, steel, copper, brass, plastics, wool, cotton, various adhesives, etc. There are more than nine thousand parts in the key and action combination alone. The piano strings are made of the highest grade steel, and are called as the “Blue Chip” of the steel industry, the combined tension of two hundred of which present in a piano exerts a pull of greater than eighteen tons. These strings bear upon the sounding board by means of wooden bridges and a system of reverse bearings which practically lock the strings and board together. Each of these strings must be kept at the proper designed tension or it will be off pitch and produce an inharmonious tone, which is when the piano is said to be out of tune.
	Many piano dealers employ their own staff of tuner-technicians who are capable of doing the tuning. However, majority of the piano tuners are independent professionals. Piano tuning does not entail knowing how exactly to play the piano, it requires a practiced ear to recognize the correct notes. A standard tuning fork sets the standard for entire piano, and through various mechanical devices, it is the tuners job to get the correct frequencies by adjusting the tensions in the string. 
There are about four thousand piano tuners in the whole of United States. The average cost of a piano tuning is about one hundred dollars, and it takes between one to three hours. There is a piano tuners guild, and various piano tuning schools. Most of this information is on the Internet (one good source is www.piano-tuning.com). The population of Chicago City as per 1990 Census data was 2,783,726, and the number of families were 631,894, giving a ratio of 4.45 people per family (1990 US Census Data, Database C90STF1A).
People, when asked this Fermi problem, and given the freedom to ask for any information they feel is required, usually would not want to know the nitty-gritty of piano tuning or tuners, the knowledge presented above. But they easily do figure out that population of the city in concern is important, whereas, if one looks at the question, there is less reference to population than piano tuning. All of the above knowledge would have helped one to come up with a better assumption for the number of pianos that a tuner does in an year – people’s one piano a day encapsulates this model that they have. And there our assumption seems to be not very correct, maybe two tunings a day would be a better average. 
Q2 How many jelly beans fill a one-liter bottle?
As with the previous example, lets try to work out a solution, and then try to analyze what went in coming up with solution. 
Solution 1
What is the approximate volume of a jelly bean?
ASSUME: shape cylindrical, diameter, d = 1.5 cm, length, l = 2cm	Volume = π (d/2)2l
= 3.5 cm3	
Can unitary method be used to compute the number of jelly beans, now that the volume of one is known?	No, they are not tightly packed.	
Number of beans?
ASSUME: A packing density of 0.8, which means that 20% of the space is empty (the void fraction)	N = 0.8 liter/ V
= 226 
≈ 200 to 300	

Solution 2
Construct, or visualize a paper cube that measures 1 cubic inch. 
How many jelly beans will fit this cube?	
Approximately 4	
How many cubic inches in a liter? 	1 inch = 2.54 cm
=> 62 cubic inches	
Number of jelly beans? 	N = 62 x 4 = 248
≈ 200 to 300	

This is also a “how many” question, that asks us to come up with a number for a certain thing. Let us try to abstract from these two examples the general strategies for “how many” questions. The objective in these questions is to find the cardinality of a set, the number of a certain species subject to certain constraints. So, the general form of these questions is – 

How many/much x are there (in C)? 
where x is an object, and C represents a container set for objects like x. Objects that exist can be arranged in hierarchies. The general strategies learnt from above examples are – 
1.	Find bounds for the number of x, n(x), as n(y) < n(x) < n(z), where z is something up in the hierarchy, and y below x. So, there are lesser piano tuners than the population of the city.
2.	Is there any other related set C’ for which n(x) can be more easily calculated? For example, the set of families in the piano examples makes more sense than set of individuals. Once such a set is known, we should also be able to relate C and C’. So, from the population, the number of families can be found once we figure out that there are five people to a family.
3.	Proceed from different directions, so that “conservation laws” can be used to relate your results to figure the missing parameters. 
4.	Find out the property that relates the number of x to C, for example, volume is the property in jelly bean example. 
a.	Find value of property for a single x.
b.	Find value of property for the set C.
c.	Use unitary method, apply correction factor if not directly applicable.
5.	If it is difficult to come up the number of x in C, and a property for one unit of x, consider an easier to imagine, more familiar chunk of x, and see how many such chunks could be found in C. For example, if we have to find the number of hair on our head, its difficult to count all of it, or find out the area that one strand of hair consumes on the skull – an easier task would be to count the number of hairs on, say 1 cm2 of skull (similar to solution 2 in jelly bean problem)
We could attempt to answer questions like – 
1.	How many car repair shops are there in Evanston? (In fact the procedure above developed would exactly apply to any repair/troubleshoot/maintain profession)
2.	How many 8.5x11 inch sheets were used by Northwestern undergraduate students in the last quarter?
a.	How many undergraduates are there?
b.	What relates the number of students to sheets of paper used – courses, assignments, lecture notes?
c.	Assume the number of courses, number of pages of lecture notes the student prints out on an average, number of assignments through the quarter, and add for junk.
Q3 When it rains, water would accumulate on the roofs of flat topped buildings if there were no drains. Estimate the total force the roof of ILS would have to support if there was a heavy rainfall and the roof drains were plugged.
The force that the roof will have to support will be equal to the weight of water, which can be found out once we know the volume of water accumulated. For volume we need to know the surface area of the roof, and the amount of rainfall. 
(Visual recall) ILS has a L shape with both it arms approximately rectangular (one of the arms is longer than the other), shown below. Now that the shape is known, lets try to place numbers. For width; I recall space near my office, which I think is 4-5m wide, then the copier and walk space slightly more than a meter, and then the printers, till Jeff’s room will be 4-5 more meters, and then Jeff’s room is slightly smaller than mine, about 3-4 meters. So total width about 15m. The length of the arm I estimate about (less than) 50m. Once we have this, the surface area is quite easy. (These are all overestimates, and thus we’ll have an upper bound.)

15m

50m

5m

zoo

Copier, printer, etc

Figure 1 Rough sketch of ILS roof

Force

Mass

g

Volume

density

area

height

Shape and size

Figure 2 Dimensional tree of unknowns/subproblems

*

*

*

What is the surface area of the rooftop?
ASSUME: above (geometry, etc)	S = 50*15 + 35*15 = 1275m2	
What is the volume of water collected on a heavy rain?
ASSUME: 1 inch of rainfall (can be refined from more accurate data about Chicago weather)	V = S*h 
= 1275*0.025 m3
=31.875 m3	
Mass of this water
KNOWN: density (water) = 1000 Kg/m3	M = 31.875*1000
= 31875 Kg	
Force exerted
KNOWN: F = m*g	F = 31875*9.8
300,000N= 300 KN	

Q4 Estimate the amount of work a person does shoveling the walk after a snowstorm. 
Net work done in shoveling the entire walkway is equal to that in one act of shoveling multiplied by the total number of shovelings required. The shovel is used to gather the snow, and throw it to side. The work done in this will be (mainly) equal to the sum of 
•	the work done in lifting the snow up the shovel to a certain height (W1), and 
•	the kinetic energy imparted to the snow when it is thrown to side (W2)		.
For the work done in one shoveling, we’ll need the mass of snow that is gathered in one shovel, and the height to which it is raised, and the velocity with which it is thrown to the side. For the number of shovelings required, we’ll need the length/breadth of the walk, and the thickness of the snow bed.

Work done in shoveling the walk

Work in one shovel

Number of shovelings

Work in lifting

Work in throwing

m

g

h

+

*

length

breadth

height

density

Mass of shovel

*

*

m

v

*

*

*

+

0.5 *

*

2

X distance

Y distance

PROJECTILE

Number of shovelings

Total amount of snow

m

Length of walk

Depth of snow

Width of walk

*

*

Figure 3 Dimensional tree for Q4

What is the volume of snow gathered in one go?
ASSUME: Shovel area 1ft by 1ft, can hold snow about 0.5 ft high 	V = 1*1*0.5 = 0.5 ft2
= 0.0142 m3	
What is the mass of this snow?
ASSUME: Moderately packed, density 1/5th water 200 Kg/m3	M = 0.0142 * 200 
= 2.84 Kg	
What is the work done in lifting this snow up (W1)?
ASSUME: Shovel lifted 0.5m high; weight of shovel 1Kg
KNOWN: Work done = m*g*h	W1 = (2.84+1)*9.8*0.5
= 18.82 J	
What is the kinetic energy imparted to snow? Need velocity.
ASSUME: Walkway is 2m wide, and man in center throws it out of the way; initial velocity (u) is horizontal.
KNOWN: vertical distance from ground, h = 0.5m; horizontal distance to travel, x = 1m; motion = projectile. 	x = u*t …. (I)
h = 0.5*g*t2 … (II)
0.5 = 0.5*9.8*u-2
u = (9.8)0.5 
= 3.13m/s
KE =W2 = 0.5*m*u2
= 18.82 J	
Net work done in one shovel 	W = W1+W2
= 37.62 J	
Total amount of snow in the walk (S)
ASSUME 1. Length of house from street 10m
2. Snow fell to a depth of four inches	S = volume of snow * density
= 10*2*(4*0.0254) *200 = 406.4 Kg	
Total work done (TW)
KNOWN: TW = Number of shovelings (n) *W	n = 406.4/2.84 = 143
TW = 143*37.62
 5000 J	
Q3 A suggestion for putting satellites into orbit cheaply without using rockets is to build a HUGE tower containing an elevator. One would put payload in the elevator, lift it to top, and just step out into the orbit. Estimate the weight of such a tower. 
This is a question where there are just too many unknowns, and all that we may come up with might be a very primitive estimate, as the designs of the trusses, etc that will be able to support such a tower, and the architecture, are things which will require a lot of domain knowledge of structural engineering, etc. Lets see if we can come up with some estimate. 
The tower has to be about 300Km tall, so that we can straightaway put things in orbit. What should be the area of the base? What should be the basic structure of a tower like this? Some knowledge about tall buildings/towers is required at this stage. Some of the tall buildings that we know are the Sears tower, Petronas, etc; but this problems asks for a tower, not a residential complex. One of the tallest towers is the Eiffel tower, Paris. Maybe we’ll need more re-inforcing than the Eiffel as it is much more big, and has to support the powerful elevator/payload, etc as well; but that gives us a (maybe) lower bound. Its quite easy to find precise information about Eiffel – (from http://www.greatbuildings.com/cgi-bin/glk?http://www.endex.com/gf/buildings/eiffel/eiffel.html)
Height: 300.51 meters (986 feet) (+/- 15 cm depending on temperature)
Weight: 7,000 tons (1,000 tons removed during 1990's renovation) 
Base: 2.5 acres Foundation

What is the volume of such a tower?
ASSUME: Conical shape, with base radius = 30Km, height = 300Km. 
KNOWN: V=1/3*r2h	V = 1/3 **30*30*300
= 282600 Km3	
How much steel would go into the above tower? As not all of it is steel, and the exact amount of steel will depend upon the design, lets compute the effective density of steel for Eiffel. 
ASSUME: Conical shape
KNOWN: Base area = 2.5 acres = 2.5*4047 m2 =10117.5m2
Height = 300 m,  Mass = 7000 tons	V = 1/3*10117.5*300
= 1011750 m3 
= 1.01175 Km3
Density = 7000/1.01175
 7000 tons/Km3	
Mass of the tower, using volume and density estimated earlier. 	M = V*d = 7000*282600
= 1978200000
 2 x 109 tons	

Important BotE Strategies
	The following is a set of general strategies that I came up with in trying to study the process of BotE. A lot of them are general, though there is a focus on Physics problems.

Approximations
1.	Numerical approximations: Throw away constants, area of circle is r2, g is 10 m/s2, do not worry about factors such as 3/5, sin θ = θ = tan θ, and cos θ = 1, as long as θ is small, and in radians.
2.	Know typical/reasonable numbers – Typical density of materials, specific heats, energy value (calories in food, calorific value of fuels, etc), if a new problem comes up, see if it can be bound with existing knowledge. Worst case assumptions – so if you have to chose a density for the Earth, what will it be?
3.	Tinker with geometry – Assume things to be spherical, flattened, or oblong as the case may be. Mountains can be square blocks, or if more precise pyramids. Use geometry to simplify, and use concepts like the equivalent radius, to use an analysis that assumes a spherical object over one which is not. Assume correction factors (like packing fraction in the jelly bean problem) to encapsulate complicated geometry, etc.
4.	Use discrete functions over smoothly varying continuous functions, use average fixed values rather than continuously varying parameters.
Principles
5.	Basic principles of conservation and continuity of important physical parameters, “What goes in, must come out.” – in problems from physics, these will be the balances for force, mass, energy and momentum. This is also a principle borrowed from the engineering domain, where the first thing one learns in Fluid Mechanics is the Navier-Stokes equations, which are the most general formulations of conservation laws, in cartesian, spherical, and cylindrical co-ordinate systems. Equipped with this tool, in specific problems, one just strikes out those details (terms in the equations) which are not relevant, to come up with a basic model of the system. 
6.	Basic ideas of the unitary method, flows, residence times, gradients, fluxes, the way different physical quantities can be related to each other. For example, Little’s law is a very useful concept which says that “The average number of things in the system is a product of the average rate at which things leave the system, and the average time each one spends in the system.”
7.	Figure out the boundary conditions for the system – how does it behave at extremes? Check for correctness of form of results – a parameter that should go to zero, as another is zero, should not be connected with the other in an equation that has a constant term hanging. Check if the method gives a correct solution in a case where the answer is known.
8.	Dimensional checks – You can only add similar dimensional quantities, and the product of two physical quantities gives rise to one with the dimensions as the product of the two. In case of incomplete knowledge, models may be built on just the dimensional knowledge. Dimensional analysis can be very useful in figuring out what variables are relevant, and what one could compute from the given information (In fact, the dimensions hide in them the statements of physical laws, as they contain the information found in the equations) – so one could build a dimensional chain of variables involved. The basic dimensions are the mass (M), length (L), time (T), current (I), and temperature (θ). Dimensional analysis knowledge can be very helpful in sifting out the relevant from the huge space of knowledge. Also, de-dimensionalizing physical equations is one of the common ways in engineering to come up with notions of reasonable values. Commonly used for scaleup, where the dimensionless groups (like Reynolds number) in a scaled-down pilot version are employed in thinking about full-blown scaled up versions, whose pilot models, and otherwise simulation would be difficult. As human beings we do it – one of the simplest and quite powerful tools of the physicist and the engineer.
Conclusions
Back of the envelope reasoning is not only useful, but is a method to reason at different levels of approximation. In case of incomplete knowledge, it is robust, and degrades gracefully. The meta-knowledge that we have, which is awareness of the applicability and limitations of the knowledge that we possess, is crucial in figuring out the (ir)relevant knowledge during an analysis. Besides relevance, there are other basic, naïve principles of how different physical parameters interact is important in generating on-the-fly models for phenomenon of interest. The lessons learnt from these examples are -
1.	Problems can be broken down into a hierarchical tree of sub-problems, until they become solvable. Usually this breakup is guided by dimensional relationships between the unknowns, conservation laws, unitary method, etc.
2.	At each levels, if exact models/parameter values are not available, assumptions; guided by analogy, or a knowledge base of reasonable assumptions (along with their applicability) should be used. These assumptions should be made explicit, so that the result can be made more accurate by refining these assumptions. 
3.	Most of the parameter estimation that we do is sensory recall (experience) - have seen it/heard/felt/used it. e.g. - How tall is the building? How far is the grocery store? How much time will it take to cook rice for six people? From some benchmarks, we learn to meter our senses. Since I have run the 100m tracks, 400m tracks, cycled to school which I knew was 11Km, travelled in cars at 40Km/h (the average speed in India), seen a 1m rod, used the 6inch ruler for school geometry (from which I got an idea of milimeters, centimeters, etc), know that I am 1.7m tall, etc; I have these experiential sensory landmarks/meters for length. All these landmarks may not be internally described as length, for example I know it took me 40 mins to cycle to school, so when I cycle I might use time to keep track of distance. Now, if I have to estimate length of a building, or the distance to grocery store; since I have some sensory data on these things by experience, I then try to reformulate the problem so that I could think in terms of one of the landmarks that I have. One important property of this reasoning is the ability to smoothly extra/intrapolate between the few discrete landmarks that we have. 
4.	What is relevant, and what is not is one of the most important concerns. Relevance might be suggested by dimensions of quantities involved, and equations (models) that contain the physical quantities being observed and the desired result; or other similar questions.