Tens of Thousands? Try Hundreds of Thousands: The Math of Crowd Estimation

Frustrated and unconvinced about the number of protesters that is being reported by nearly every media source for Saturday January 27th March in Washington DC, I decided to work out the math involved. I was having trouble grasping some frame of reference as to how many people tens of thousands looks like compared to half a million. Obviously the difference is enormous but I kept asking myself what frame of reference I have that could some how make sense of these estimates as they have been presented.

The DC Police Department no longer releases official statements on crowd sizes but they never fail to provide an “unofficial” number drastically smaller than others that have been put forward (this was equally true during the march on Washington in September 2005). United for Peace and Justice (UFPJ - http://www.unitedforpeace.org), who organized the march, lists 500,000 as their unofficial estimate. As much as I would be interested in the way UFPJ came up with this estimate, I would be more interested in how nearly all media sources have come up with tens of thousands. Tens of thousands has been repeated so often that it is now generally accepted as fact. It is under this context that I decided to take a run at a fact based estimation that involves some simple geometry and algebra which all media sources seem incapable of performing.

There are two simple components to figuring out an estimate of protesters that day: area and density. If you take the overall area which was occupied by protesters divided by the average density of people you come up with an estimate that is relatively close to how many individuals were actually there. With this formula in mind I see several ways to do this with some degree of accuracy and we’ll go through the numbers taking various approaches for comparison purposes.

First we need to compute the area. To do this I am going to use the GMap Pedometer (http://www.gmap-pedometer.com/) a handy little tool powered by Google Maps that will help us measure the lengths and widths of the spaces in which events took place on Saturday.

Measuring the Rally

One way to calculate the area is using the rally on the National Mall. We can figure out an estimated width and length of the rally based on the actual dimensions. Of course, I can’t be certain of these dimensions without satellite images but somehow I doubt the Defense Department will send me these images considering the individuals in charge (as US Rep. John Conyers http://johnconyers.com/ pointed out at the rally on Saturday, “George Bush has a habbit of firing military leaders who tell him the Iraq war is failing”). The rally extended the length of two to three blocks of the National Mall at a minimum (3 blocks would be .47 miles or 2,500 feet and 2 blocks would be .367 miles or 2000 feet). I am estimating that the rally on the Mall was about .12 miles or 633 feet wide. The low estimate shows that the total area of the rally was around 1,266,000 sq feet with a high estimate around = 1,582,500 sq feet.

Next we need to estimate a crowd density. This is the harder of the two to variables to estimate but I will consider my experience there and once again compare high and low estimates. During the rally there was an extremely high density of people gathering around the two stages becoming increasingly sparse further away from these areas. The most conservative estimate I can imagine would be 1 person per 10 sq feet (this is really an unreasonably low density but serves to prove the point that there is no way Saturday’s event was in the tens of thousands) and I would argue that a more likely number is 1 person per 5 sq feet. So on the unreasonably low end, taking the low area estimate with the low density estimate we see a minimum of 1,266,00 / 10 = 126,600 people. Then taking what I would say is still a relatively conservative view 1,582,500 / 5 = 316,500 people.

Measuring The March

Then there is the march route, consisting of a loop around the Capitol building roughly two miles long. To give a nice round number let’s call the two mile route 10,600 feet. Let’s say each lane of the road was 15 feet across (this is probably less than the average lane but again gives us a nice round number and it is harder to argue with). For about half the march they blocked off two of the four lanes in the road, but there were plenty sections that had all lanes open. We’ll say that the march covered three lanes on average for a total of 45 feet. This gives the march a total area of 477,000 sq feet.

We know from first hand accounts, that as the frontline of the march completed the route, they could see the tail of the march still making their way out of the National Mall. This means that we can consider the total area of the march as being occupied (this is actually an exciting thought since it means that peace supporters had the nation’s capitol completely surrounded for a period of time that afternoon). Starting out, we were almost unable to move, advancing only a few feet a minute. After the first hour or so we were actually marching along at a reasonable pace. Averaging this out, I would estimate around 3 sq feet per person. So here we see that at a minimum 477,000 / 3 = 159,000 people who participating in the march.

But then I am left questioning whether the front of the march actually saw the tail or whether they were finishing just as some in the middle of the march just starting. This question makes more sense when considering the timing involved. From where I was in the march, it seems likely that I was somewhere in the anonymous middle. I couldn’t see the begin and I couldn’t see the end. It was an endless stretch of people either way I looked. After the first hour of marching, my section rounded the first corner onto Constitution Ave. With this in mind, consider how long it would take to walk two miles if no one were in front of you to slow you down. At a painstakingly slow pace it might be 45 minutes and if you really stretched the imagination we could say an hour. So if we consider my section of the march the middle (at least not the end), and we allow that the front of the march took an hour to complete the loop this would mean that they would still have been seeing the middle rounding the first corner as they finished and that we could safely consider the crowd more dense than it seemed considering the calculated area. 477,000 / 2 = 238,500 thousand.

Conclusions

Prior to the march, UFPJ estimated that around 100,000 protesters would participate in this event. Considering the calculated estimates I am putting forward here, I am very confident that the original estimate was very low. UFPJ’s estimate during the event was 500,000 which is significantly higher and puts this event up with the most widely attended rallies on the National Mall in history. While half a million is less realistic than I would like to admit, not even breaking 100,000 is even less likely in my opinion.

I have to commend The Washington Post for their avoidance in repeating the tens of thousands claim and I admit that having done these estimates, UFPJ has some explanation in justifying their numbers. But given my calculations (and my participation in the event) I feel safe quoting 200,000-300,000 as a realistic number. Given the January weather and the seemingly last minute mobilizing efforts, my final estimate is that the next rally will easily reach the 500,000 mark. Congress will have to act to end this war soon or they will face an increasingly determined number of citizens acting against this war. If they fail to reverse the complacency and misguided action of the past they will fail to uphold the overwhelming referendum for peace that brought the Democrats to the majority power last November.


Posted at Tuesday, January 30, 2007
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