# The Art of Estimation

So unless you’ve been living under a rock,  you know many people have concerns about Donald Trump being President.  This is from one of the marches.  One quick political statement, and then on to the math:  I prefer to think of them as marches, not as protests:  Trump is the President of the United States, and while I don’t like the fact, protesting it doesn’t do anything useful.  Rather, it’s a reminder…in this case, that women make up 51% of the population of the U.S., so the powers-that-be should remember this.

Back to the math.  During this event, the question came up:  How many people showed up? You could, of course, try to count the number of people, but this is difficult, since people are moving (this is the problem the Census Bureau faces…you can read a little bit about the reason why it’s important in and read quite a bit more about it in the first few chapters of my book, Constitutional Calculus).

If an exact number is important, there are a number of statistical techniques that can be used.  But often, a rough estimate is enough.  There are several ways to make such an estimate.

The first is what you’d call an order of magnitude estimate.  To get a sense of what this means, pick any whole number:  for example, you might take $n = 2$.  The powers of $n$ form a sequence of numbers:  $2^{0} = 1, 2^{1} = 2, 2^{2} = 4, 2^{3} = 8, 2^{4} = 16$, and so on.  An order of magnitude estimate occurs when we identify one of these as close to our number.

For example, in the picture, we might try to find an order of magnitude estimate for the number of people holding signs:  Is it closest to 1, 2, 4, 8, 16, 32, 64, 128 (or more)?  Don’t try to count the number of people, as this defeats the purpose of making the estimate!  (I’m thinking 16).  More commonly, we take powers of 10:  1, 10, 100, 1000, 10000, and so on.  Thus, the number of people holding signs is around 10.

The nice thing about order of magnitude estimates is that they’re easy to make if you have any experience with the so-called real world.  Thus, here’s an order of magnitude estimate for the number of marchers:  10,000.  I get this number because I know what a crowd of 100 looks like (a crowded auditorium) and have some sense of what 100,000 people look like (a stadiumful of people), and it seems that the number of marchers is somewhere between the two.

The order of magnitude estimate is fast and easy to obtain…but you get what you pay for:  it’s not very precise.  To gain more precision, you’ll need to make do a little work and deal with some facts.  (As I’ve noted elsewhere, one of the reasons mathematics is important to a free society is that it forces you to ask yourself what you really believe and why you believe it)  In this case, we have the following data points:

• The picture is from the Walkway Over the Hudson, a former railway bridge converted to a linear park.  The bridge portion is a little over a mile in length.
• The marchers occupied (in the sense of took up space, not in the sense of Wall Street) the whole length of the bridges.

The art of estimation begins by using approximations to the actual values.  This takes a little experience and a little knowledge.  In this case, the knowledge is that the bridge is a little over a mile in length, and 1 mile is 5280 feet.  We’ll estimate the bridge’s length as 6000 feet.

To estimate the number of marchers, we need to know how many people there are per foot of bridge.  An average person can squeeze into a one-foot space, which means that if packed to capacity, one linear foot of the bridge can have a lot of people across it:  imagine drawing two lines across the bridge, one foot apart.  You could probably fit a good dozen people in there.

Now you can see from the picture, the bridge isn’t that packed.  However, we can use this to provide an upper bound on the number of marchers:  the fully packed bridge could hold $12 \times 6000 = 72,000$ persons.

Why is this useful?  Suppose someone reports on the march, and says that “Over a million people attended.”  Given that a fully packed bridge would only hold around 72,000, we might suspect this number is a gross over-estimate.  Good mathematics helps us identify questionable claims.

At the same time, consider another report:  “Only a few hundred people attended the march.”  The bridge is 6000 feet long.  If there were only a few hundred people on the bridge, each person would have to be some distance from the other.  Note that you could divide 6000 by however many hundred the report claimed, but you could also do an estimate:  If each person stood 10 feet apart from the next, then the bridge would hold 600 people.  But the picture clearly shows people standing closer than 10 feet apart, so the number of marches had to be in excess of 600.

So how about an actual estimate?  Here’s what we did (several of us discussed the question):  Along 6000 feet of bridge, there was always more than one person per foot.  This gave us a minimum of 6000 people.  However, we didn’t think there were as many as twelve persons per foot of bridge, so we did the easy thing and doubled the amount for our upper estimate.   Thus we concluded that there were between 5000 and 10,000 marchers.

(Why not 6000 and 12,000?  We could, but given the nature of the numbers, we rounded to multiples of 5000…this too is part of the art of estimation)

I’ve that gives the official and unofficial estimates.

One final note:  It’s important to emphasize that thinking mathematically is a habit.  In this particular case, it’s the habit of asking questions like “How many people do you think are here?”  By answering that question, we are armed with a useful tool (an estimate of the number of marchers), which we can then use to confront the world (disparate claims over the number of marchers).

This number sense will be increasingly important in the years to come.