How can you tell how big a protest march is?

There are, broadly speaking, two ways to count a crowd.
The first is based on the fact that were not naturally all that good at guessing the size of even fairly small crowds – especially when were in them. When youre in the middle of it, even a fairly small crowd can feel quite big, and so we tend naturally to over-guess.
We can compensate for this with rules of thumb: if you think of a standard UK cinema, for example, thats about 250 people when its full. If we try to remember how big that is, and how densely-packed that is, we could try to imagine how many cinemas worth of people are in a given space. But thats hardly a satisfactory answer.
The more rigorous version of this approach borrows from ecology – from measures we use to try to sample biodiversity in a given area. If we want to know the biodiversity of a large field, we are unlikely to have time to check the whole thing. So what we can do is throw quadrants (wooden squares, for example one metre by one metre) so they land randomly, and then count everything in each, to sample the field.
If we have good aerial photos of a crowd, we can try to do the same. The crucial step is to check this randomly: if you look at any crowd, its got areas where it is much denser than others – and so if we concentrate only on these well come to a guess thats far too high. The more squares we can check, the better, and we can then use those to estimate the size of a crowd.
If there were 10 people in one 10 metre squared area, 40 in another, and 50 in a third, and the crowd overall was spread across 100 metres squared, we could estimate a crowd at around 30,000.
This is the rough approach used when police and others make “official” estimates of crowd size, and it works reasonably well – though not, of course, perfectly. A particular headache is that crowds move – especially on marches – and for long protests over numerous hours, this kind of snapshot approach would under-count.
Annoyingly, though, in London at least police dont publish these kinds of numbers any more, in part seemingly because they tend to annoy people, who suspect theyre deliberately under-counting for some nefarious purpose.
There are more rigorous measures still, some simpler and some more complex. We know the general density of crowds (and the safe density of crowds) through decades of study. So if we know the general area of a march or crowd, and have a bit of visual confirmation, its easy to generate an estimate.
Tech gives us new options too – especially for anything in a confined area with limited entrances and exits (such as a park or stadium), where its possible to actually measure people coming in or out. Soon we may be able to make better use of the fact everyone has a phone, and use those to get counts.
Sadly, we dont really tend to want any of that. What we want when were on a protest march is a big number, and there are much easier ways to get that.
That leads us to the second means of measuring crowds, which is for some bloke (its usually a bloke) to tweet out a big number. There can be lots of ways that bloke reaches his big number – it can be “from stewards, who do loads of marches, so they know what theyre talking about” (crowd counting is not part of the fairly basic training given to most stewards), or “from my mate in the police”, or some other source.
Its rarely from a systematic effort to neutrally assess crowd size based on reliable visual information. Such Twitter figures are often prone to inflation, too. For the “Put It to the People” march, people started circulating a figure of a million marchers by around lunchtime – long before anyone could have hoped to have a decent count.
By evening, that figure was boring, so someone cited an “official” figure of two million, which went viral again, topping 9,000 retweets at the time of writing.
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