Wednesday, 26 September 2012

making estimates

What happens whenever you encounter a problem you can't solve, or a question you can't answer? Rather than simply giving up, you try to estimate a right answer or a solution; you try to come as close as you can to the result you are looking for. This process of estimating something you don't know can range from a simple "educated guess" to systematic theories developed in sciences like Statistics and Numerical Analysis.
Before beginning the understanding of how some estimation methods work, try to make a few "educated guesses" in order to answer the following questions (unless, of course, you already know the answers)!
  1. How many water bottles do you need to fill-up your local swimming pool?
  2. How many grains of brown sugar are there in a teaspoon?
  3. What is the area under the graph of the curve y=x2 between the points 0 and 1?
  4. How many letters are there in the novel you read last summer?
Hopefully, your answers were not completely random, but they involved some sort of intuition, or even systematic thinking. In the paragraphs that follow, I will try to help you organize your thoughts, so that the estimates you will be making from now on, will be based on solid facts and arguments, rather than pure intuition and guesses.

Think of what you know

The first thing you need to do when encountering a problem you get stuck on, is to figure out what you do know about the problem. It is very unlikely that you will be presented with a task you know absolutely nothing about. So, organize the things that you know and try to extract results from them.
For example, what do you know about your local swimming pool? To calculate its volume, you need to know its length, its width and its depth. You definitely know its length - it is 25 meters. As far as its depth is concerned, you are not sure - but you do know that its deep end is 2 meters deep and that its shallow end is 1.20 meters deep. What about its width? You know the swimming pool has eight lanes. How wide is each one of them?
For another example, you don't know yet how to calculate areas of strange shapes. However, you must certainly know how to calculate the area under the curve y=x and also the area under the curve y=12x. These are simply triangles. How do these relate to our original problem of finding the area under y=x2, though?

Understand what you don't know

All of the above problems involve at least one quantity which is unknown to you, and prevents you from finding your answer. In the swimming pool example, you don't know the width of each lane. In the example with the sugar, you don't know the volume of a brown sugar grain, even though you might remember that a teaspoon holds about 5ml. In the example with the book, you may remember that there are 300 pages, but you don't know how many letters there are in a page.
However, once you decide what a suitable value for all those unknown parameters might be, you will be very close to finding a very sensible answer to your question.

Approximating what you don't know

I will now give you a few ideas on what you could do, if you need to make a reasonable approximation. Of course, every problem is different, and other approaches than these might work, but still they are a good way to start.
1. Use inequalities
Even if you don't know what the value of a particular quantity is, you might know that it certainly will not be greater than a specific value nor will it be less than another value.
For example, plot y=xy=x2 and y=12x on the same axes. You should see that the area under the graph of y=x2 is certainly less than the area under the graph of y=x. But it is also more than the area under the graph of y=12x. So, the area A that we are looking for, satisfies the inequality

This is definitely a decent approximation, and a little mathematical intuition might lead us into picking the value inside the interval [14,12], which corresponds to the right result...
2. Work recursively
Sometimes answering a hard question is only a matter of answering a series of easier questions. The point is to identify exactly what you are looking for in each problem, so that you know which are these easier questions.
Your first hard question is: How many letters there are in a book? Since you know how many pages there are in a book, you try to answer how many letters there are in one page.
But instead of answering that, try to find how many lines there are in one page. That shouldn't be hard. Guess about 25 (that's how many lines there are in a word document).
So, how many letters are there in one line? Still a little hard - so think about how many words there are in one line. About 10 sounds reasonable. Now, the average english word has roughly 4.5 letters.
Therefore, you have broken down calculating the number of letters in a book to calculating the number of letters in a word, then a line, then a page and therefore the whole book. This is how recursive thinking helps break down a complicated question to a few straightforward tasks.
3. Compare to quantities you know
You want to make an estimate about the volume of a grain of sugar. This is likely to be around a milimetre, or even less in fact. To compare the size of a grain of sugar to a milimetre, all you need to do is draw a square with side 1mm, using your ruler. Does this look pretty much like the size of a grain? It really does, so this is your estimate.
4. How many "units" fit in the whole
A typical idea in approximating is to see how many "small" things fit into a "larger". That way, given that we know the dimensions of the "small" thing, we can obtain an estimate for the dimensions of the larger one (and vice versa, of course).
To estimate the width of the pool, which you don't know, how can you work? You can start by estimating the width of a lane. Think that if there are four people resting at the end of the lane, they fit even if they are quite crammed. This means that since each person occupies 50 cm (in width), the lane needs to be about 2 meters wide. Therefore, a pool with eight lanes, should be about 16 meters wide.
But you can also work the other way too. If somebody asks you, for example, how much a staple weighs, all you need to do is estimate how much a box (with 5000 staples) weighs, and then divide!

Get your answer!

1. You found that the pool is 25 metres long, 16 metres wide and (on average) 1.6 metres deep. So, its volume is

So, if we use water bottles of 500 mL, we will be needing 1,280,000 of them in order to fill up our swimming pool!

2. We estimate that a grain of sugar occupies a volume of 1 cubic milimetre. Now, a teaspoon has a capacity of 5mL, so it will contain about 5,000 grains of sugar.

3. If you decide to stop with this approximation of the area under the curve, there is not much more to do than simply guess a value between 14 and 12. Some might go for the average of these two, that is 38. Somebody who is more lucky, might guess the correct value of 13 (which is not completely unlikely - it is just the second most natural thought when somebody asks you to pick a number between a quarter and a half).

4. A simple multiplication is all you need to do. In total, we estimate about 300×25×10×4.5=337,500 letters. How many of these do you think are a's?

Think about your error

This is probably the most important aspect of an approximation. In some cases, you may be able to specify this precisely - for example in the calculation of the area under y=x2 you know that whatever value you choose in the interval [14,12], your error cannot be more than 14.
In other cases, you need to consider how much each estimation you made affects your final calculation. For example, if the volume of a grain of sugar is 1±0.2 cubic milimetres, then your approximation  for the number of the grains should be between 4167 and 6250.
Another way of describing an error, which is encountered in advanced statistics, is thinking about confidence intervals. So, you estimate that the quantity you are looking for has a value which lies with high probability in a specific interval.
In general, the more precise you can be about the error of your approximation, the more sense this will make, and the more useful it will be. This is why there are entire branches of mathematics are devoted to developing techniques for error control and optimization of approximations.

Final Thoughts...

The point of this article was to show you how to make estimations of things you normally can't find. A good approximation is in many cases as useful as a proper solution, so you shouldn't underestimate its value.
Most of the time, you will be encountering problems which, even though you might not know how to solve, will have aspects which you could more or less estimate. And the best way to estimate is not to guess, but to gradually analyze each problem and discover how you can make the most out of what you already know - in a similar fashion to what we did above.
So, use the above methods, or improvise, and remember that if no one can find the answer to a problem, your goal is to arrive as close to it as you can!

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