Counting number in protest march

Every time after a protest march was held, I am always curious about the huge discrepancy between the Police's and the organizer's estimates of the number of people who participated in the march. In yesterday's protest against the extradition law1, the Police reported "22K at peak time", while the organizer claims there were 130K! So, they surely had done the counting in very different ways!

The Police's "22K at peak time" clearly implies a snapshot approach which only makes sense when the area is fixed and well defined. In yesterday's protest, however, people actually moved from Causeway Bay to Central over a period of 2 hours. So, we shouldn't be counting number at a particular time! Rather, we should be counting how many people had moved along the path of the march over a period of time. In other words, it is a flow problem rather than a static counting problem. Imagine you turn on the water tap in your kitchen and let water flow out for two hours. How much water can you collect at the end? That's the question!

Clearly, "22K at peak time" ("最高峰時有2萬2千人") is literally absurd, as far as a flow problem is concerned. Does "peak time" refer to the time when the largest number of people were present in the whole scene (from Causeway Bay to Central)? Did the Police take a snapshot at a particular instant of time and recorded a maximum of 22K people? Why does this "22K at peak time" relevant to the actual number of people who participated in the march?

In fact, the number can be quite easily worked out if it is taken as a flow problem. Let’s put the road in the "horizontal" direction for ease of referencing. (See figure below.) Suppose in roughly every S sec, a group ("vertical" line) of N people have flown through an observation point on the road. The flow rate is N/S.

Suppose three car lanes accommodate about N people lining up orthogonal to the road. News reported that people began walking from 3:40 pm, and the last group departed around 5:45 pm, i.e., a duration of 125 minutes.

The flow rate is just the total number of people flowing through the path divided by the total time, i.e.,

Now the question is how many people have moved through the observation point from 3:40 pm to 5:45 pm? A rough but reasonable estimate can be made if we take:

• Total time = 125 x 60 sec

• N = 25 , S = 2 (i.e., 25 people flew through every 2 sec)

Thus, there were 125 x 60 x N / S = 93,750 people passing through the observation point!!

If more or less people (20 < N < 30) actually moved faster or slower (1.5 < S < 3), the answer would be different. But the magnitude is still pretty much within the range 50,000 to 150,000.

I would say 90K shouldn’t be too far, after discounting early leavers. But if you include everyone who showed up, over 100K is still very probable. For this kind of protest marches, it makes no sense to talk about number at particular time, and as I said, the Police's "22K at peak time" is literally absurd. In yesterday's case, the main factor was the rather long duration of flow, more than two hours through a point along the trajectory!

April 29, 2019

_______________________1 Estimated 130,000 protesters join march against proposed extradition law that will allow transfer of fugitives from Hong Kong to mainland China — South China Morning Post, April 28, 2019.