The Science of Crowd Counting & 3 Ways to Teach Kids Estimation

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Estimating crowd size, such as what we’ve been seeing in the news, is a skill kids will enjoy learning. President Obama’s inaugural address, January 2009. DoD photo by SMSgt Thomas Meneguin, U.S. Air Force

First and foremost, I’m not turning this into a political post. This is like several of my other posts I’ve written over the years about how to turn something into a fun lesson for the kids.

In this case, the “something” is a topic that’s been pretty dominant in the news: the question of how to estimate crowd sizes based on a photograph. But we can use other simple techniques to show kids how to do this.

My sons, ages 12 and 14, watch CNN Student News several times per week in class, and this week they saw a story about the disputes in the January 20th inauguration crowd size based on the now-infamous photograph comparisons between 2009 and 2017. My sons and I briefly discussed how similar estimation techniques can be used to win counting contests. As my sons disappeared to do homework, I was left pondering whether any of those “Count the Candy in the Jar” techniques were used with the overhead photos of the inaugurations.

Estimating jellybeans in a jar is a classic skill that every youngster should understand. Image credit: Flickr user Michael Miller, via CC BY 2.0.

What am I talking about? You know: you visit your local grocery store or office supply store and there’s a jar filled with candy, pencil erasers, or keyrings, with an announcement to guess the number of those items in the jar to win a prize. It turns out there are numerous mathematical techniques that can help with those estimations.

Counting Crowds Is Truly a Science, Complete with Peer-Reviewed Journal Articles

Fellow GeekDad writer Dave Banks shared a Popular Mechanics article from 2011 that applies quite a bit of math and science to how crowd sizes are estimated (the article also delves into the politics of why estimates might vary so much, but I will stick to the math and science). I was intrigued with the history of why we started caring about crowd sizes: Vietnam War protests at the University of California, Berkeley were the impetus for the modern version of crowd-size counting. Modern methods now employ the use of satellites, pixel-counting, and a 360-degree network of drone-enabled cameras flying over the crowds.

The 2011 Popular Mechanics article was announcing a peer-reviewed journal article by Paul Yip and Ray Watson published in the statistics journal Significance with new techniques to estimate crowd sizes based on satellite and aerial photography. Yip and Watson introduced the term “mosh-pit density,” which certainly caught my attention. “Mosh-pit density” was defined in the article as one person per 2.5 square feet… which is approximately one person in a 1.5-foot x 1.5-foot area. Considering I’m about 1.5 feet across at my shoulders with my arms tucked in, that’s pretty dense. So it’s as simple as calculating the area of a space and multiplying it by the established density. Read the article (it’s relatively straightforward reading, which was refreshing) to learn more about some of the difficult variables authorities encounter when putting these methods into practice.

But let’s get back to our kids. Here are some techniques to help introduce large-number estimation, with the backdrop of the “counting the candy in the jar” types of exercises.

How Many Jellybeans Are in the Jar?

Method 1: Recall those equations from geometry. If the vessel is a standard cylindrical or rectangular shape, use a volume calculation to estimate. Use units of “jellybeans” instead of inches or centimeters to measure the heights and diameters. Then the answer to the calculation will be in “jellybeans!” tells us

The volume formula for a cylinder is V = pi x r^2 x h. One suggestion for using this formula involves rounding pi to 3 and counting the radius as half of the jar’s diameter in beans. For example, if a jar is 10 beans in diameter and 20 beans in height, the volume is three times 5 squared times 20, or 3 times 25 times 20. The answer is approximately 1,500 beans.

Method 2: Carry around some trivia. Did you know about 930 jellybeans fit in a one-gallon vessel? I didn’t know that either, until I researched some estimation methods. Be able to divide this estimate based on whether the vessel is near a cup, pint, or quart size.

Another way to estimate the number of jellybeans in a jar is to make a guess based on the volume of the jar. There are approximately 930 jellybeans in a gallon. Look at the jar to estimate its volume, and make a guess accordingly. For instance, if the jar is about the size of a quart, there would be approximately 233 jellybeans in the jar. This is because a quart is 1/4 gallon, and 930 jellybeans divided by four is roughly 233.

Method 3: Carry around more trivia. According to a cursory Google search, it appears that one jellybean is about 3.37 cubic centimeters. If you know a vessel’s volume in cubic centimeters, it’s simple to divide the vessel’s volume by the jellybean’s volume. Don’t forget to account for the spacing between jellybeans, which this author assesses to be about 20%. So take 80% of whatever answer you get.

The approximate volume of one jelly bean can be thought of as a small cylinder 2 cm long and 1.5 cm in diameter (Precisely articulated as: Volume of 1 Jelly Bean = h(pi)(d/2)^2 = 2cm x 3 (1.5cm/2)^2 = 3.375 or 27/8 cubic centimeters).

Due to the jelly bean’s shape and irregularities, there is considerable airspace in the container, along with the jelly beans.  It can be assumed that 20% of a given volume is air rather than jellybeans (though for very small or irregularly shaped containers, this figure might be slightly more… never estimate more than 25% air by volume.  Really 20% is the best value to use for n > 100).

So, to get your answer, you will want to determine the number of cubic centimeters in the container volume and multiply that number by can simply use a calculator to divide the volume of the container in cubic centimeters by 2.7 (which is 3.375 * .8 to allow for air space).  For example, search for “cubic centimeters per gallon” and Google returns “1 US gallon = 3 785.4118 cubic centimeters”.  You can then calculate your answer.

Many of these techniques involve carrying some seemingly “useless” trivia in your back pocket. And what if you are faced with counting something with unusual shapes, such as Goldfish crackers, Hershey’s Kisses*, or Valentine’s Day conversation hearts? I am a subscriber to Method 1, myself, and it will work well with any of those unusually-shaped items.

*Folks have modeled the shape of a Hershey’s Kiss to calculate its volume, which makes Method 3 a possibility. Check out this forum discussion!

If your kids are seeing discussions about crowd sizes at inaugurations, be sure to take a minute to discuss how to estimate values.