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# The Simple Math Behind Crunching the Sizes of Crowds

How large was the crowd at the recent U.S. presidential inauguration? Or the inauguration 8 years ago? Or at last Saturday’s Women’s March in Washington D.C.? Keep on reading to find out how crowd sizes are estimated.

By
Jason Marshall, PhD
Episode #298

There are over 7 billion people on Earth today, and occasionally a bunch of them decide to converge for one reason or another. We’re talking thousands, tens of thousands, hundreds of thousands, or even millions of people in the same place at the same time. As those of us in the United States (and no doubt around the world) have recently witnessed, such events include things like U.S. presidential inaugurations and political marches.

In case you haven’t heard, there’s been a bit of a fracas brewing over the exact sizes of crowds at certain events held in the U.S. this past week. While I don’t want to get into the political aspects of these issues (nor the very reasonable questions about why we’re having these sideshow conversations at all), I feel that it’s important to note that estimating crowd sizes is a solved problem that’s actually pretty straightforward. And it’s relevant to us math fans because it’s really nothing more than a simple exercise in basic math.

So, how do crowd estimate experts estimate crowds? Let’s find out.

## Counting Crowds

The most reliable method for estimating the size of a crowd is to actually count the size of the crowd. In truth, I’d say this method provides a measurement of the crowd size rather than an estimate since it’s just a matter of counting up people—there’s no other math involved. The beauty of this method is that the uncertainty in your measurement should be extremely low, which means you can be confident about the count’s reliability. Such a direct count is easy to do when crowd sizes are relatively small or when people have to pass through doors or turnstiles, but it’s hard (or even impossible) to do when crowds are large and spread out.

The most reliable method for estimating the size of a crowd is to actually count the size of the crowd.

In such cases, we have to rely upon our math and reasoning skills. In particular, we have to change tactics from performing a count to performing an estimation. As with any estimate, there will be uncertainty in the final tally since the whole process relies upon a set of assumptions which each are accompanied by some uncertainty. But the beauty of math and statistics is that it provides us with a framework within which we can not only estimate the size of a crowd, but also estimate the quality of our estimate. Which means we can accurately calculate the range of the crowd size which, with very high probability, contains the actual number of people.

## Low-Tech Crowd Size Estimates

For medium to large crowds spread out over medium to large areas, your best-bet low-tech solution for obtaining accurate crowd counts is to use the method developed by UC Berkeley professor Herbert Jacobs in the 1960s. While observing crowds of protestors gathering in the plaza below his office window, Jacobs realized that he could take advantage of a geometrical/architectural feature of the plaza to come up with crowd size estimates. Since the concrete of the plaza was poured in a grid pattern, Jacobs started by performing an accurate count of the number of people in a few typical grids, and then obtained his total tally by multiplying this average count per grid by the total number of grids. As I said, this isn’t exactly rocket science.

We can get a bit fancier and obtain a more complete understanding of crowd sizes by incorporating uncertainty. For example, imagine you are in charge of counting a crowd. The first thing you do is find a vantage point that gives you a good overview of the scene. Even if you’re not so lucky as to have a grid pattern poured in concrete to work with, you can still mentally divide up the scene into some sort of grid. Suppose you count the people in several squares and conclude that the average square contains between 20 and 25 people (since the density of all crowds does naturally vary from place-to-place). To come up with your estimate of the crowd size, you can then multiply the low and high estimates of the number of people per square by the total number of squares. If there are 300 squares, you would conclude that there are between 20•300=6,000 and 25•300=7,500 people in total.