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.
Modern High-Tech Crowd Size Estimates
With really big events such as presidential inaugurations or political marches, modern technology offers some assistance to the crowd estimating experts of the world. Today you can launch a balloon or drone-borne camera system to hover over an event and take high resolution images of the crowd. But even with these technological advances, the best crowd estimates still ultimately come down to measuring the average density of people in a representative region of the crowd and then multiplying this value by the total area over which those people are spread out.
Modern technology offers some assistance to the crowd estimating experts of the world.
There are a few additional modern twists on the simple grid pattern technique in use today. For example, computers (and the humans that write the software they are running) are pretty good at looking at images and classifying different regions of those images based upon their densities. So instead of breaking up an image into a regular grid pattern, a computer can figure out the different amoeba-shaped regions in an image that share similar densities. Once you have such a map of the different regions, you can turn it into a more precise crowd count by calculating the actual density of humans in each region and performing the arithmetic problem just as before. The idea is the same, but the technique does provide more precise results.
The Certainty and Uncertainty of Crowd Sizes
Even though we have the mathematical know-how to reliably estimate crowd sizes, you’ll still find that different groups can come up with very different estimates. Most of the time these differences don’t point to a problem with the math but are instead traceable to the biases of the person (or people) performing the estimate. For example, police, media, and event organizers often report different numbers. Why? Well, it’s fairly obvious that different groups might benefit by having larger or smaller numbers reported. Event organizers typically want to bolster their message, so they might be inclined to be “generous” with the assumptions they make in calculating their values.
While it’s good to keep all of this in mind, the real takeaway message here is that calculating crowd sizes isn’t all that complicated. The bottom line is that pictures don’t lie … and with the help of a bit of simple math, they provide all the information needed to accurately estimate crowd sizes.
Wrap Up
Okay, that’s all the math we have time for today.
For more fun with math, please check out my book,The Math Dude’s Quick and Dirty Guide to Algebra. Also, remember to become a fan of The Math Dude on Facebook and to follow me on Twitter.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Crowd image from Shutterstock.
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