How to estimate crowd size: the Jacobs method, people per square metre and square feet per person

Crowd size is area times density

Every serious crowd estimate is the area the crowd fills multiplied by how many people stand in each unit of it. Statisticians Ray Watson and Paul Yip put it plainly in the Royal Statistical Society's magazine: estimating a static crowd "boils down to area multiplied by density" (Significance, September 2011).

They trace the method to Herbert Jacobs in 1967. Jacobs, a journalism professor at the University of California, Berkeley, as Steve Doig describes him, watched student protests in a plaza laid out in grid squares, so he could count the squares and estimate how many students stood in each one on average. His density rule, in Watson and Yip's words "still used today", is about 1 person per square metre in a loose crowd, 2 in a solid crowd and 4 in a very dense one.

The same arithmetic sizes a terrace before anyone arrives. The UK Sports Grounds Safety Authority's Worked Example A for its Guide to Safety at Sports Grounds (2018) multiplies a terrace's 244.8 m² of viewing area by an appropriate density of 4.7 spectators per m² for a holding capacity of 1,150. Estimating a crowd runs the same sum the other way, from what is in front of you.

Crowd densities in square feet and square metres

The square-feet figures quoted for loose, dense and mosh-pit crowds differ because they are rules of thumb, and the same expert has given different ones. Steve Doig, a journalism professor often quoted on crowd counts, wrote 10, 4.5 and 2.5 square feet per person in 2009, gave 10, 6 to 7 and 3 to 4 to the Associated Press in 2017, and 10 and 5 to the Columbia Journalism Review in 2019.

Converted to one unit, they agree more than they seem to. One square metre is 10.76 square feet, so "1 person per square metre" and "10 square feet per person" are the same loose crowd, and the 2.5 square feet of a mosh pit is 4.3 people per square metre, between Jacobs's 4 and Still's ceiling of 5.

Two of Still's warnings matter for anyone judging density by eye. Crowds "rarely pack in regular formation", so a sample square can hold a cluster while the next is half empty. And density "can appear higher with lower CCTV angles": a crowd photographed from head height can look solid when it is not.

Step by step: estimating a crowd that stands still

Pick the moment, measure the area, split it into zones of similar density, count sample squares in each zone, then multiply and add. Watson and Yip's conclusion adds a seventh step: say what area and density you used, so estimates can be compared.

1. Pick the moment. Area times density counts the people present at one instant, usually the peak. It is not the number who came and went during the day.
2. Measure the area from a plan, a mapping tool or paced distances. Take out the stage, barriers, fountains, tents and any roped-off space.
3. Split it into zones. Front of the stage, middle, back, edges and side streets each get their own density. Watson and Yip suggest stratifying into high, medium and low density regions.
4. Count sample squares. From an overhead or high-angle photo, mark squares of known ground size, such as 5 m by 5 m, spread through each zone. Count the heads in each with a tally counter, one tap a head.
5. Work out each zone's density as heads counted divided by the area of squares counted.
6. Multiply and add: zone area times zone density, for every zone.
7. Report a range with the area and density used, so anyone can check it.

Counting a sample and multiplying is not special to crowds of people. Birders estimate a flock too large to count one by one in the same way, counting a block of ten to twenty-five birds and judging how many such blocks the flock fills, the block method for counting a flock of birds.

Worked example: a rally in a town square

A rally fills a plaza measured from a plan at 120 m by 80 m, 9,600 m². The stage and a fountain take 600 m², leaving 9,000 m² the crowd could stand on. A photo from a building at the back shows the crowd thick at the front, steady in the middle, thin at the back and absent along one edge.

For the middle zone, eight 5 m × 5 m squares (25 m² each) are marked across the photo and counted:

452 heads in 200 m² is 2.26 people per m², a solid crowd. The squares range from 1.76 to 2.80, which is why one square is never enough. The other zones are counted the same way.

The shortcut answers show why zones matter. The whole 9,000 m² at Doig's 2009 "tight" density of 4.5 sq ft (2.39 per m²) gives 21,500. At mosh-pit density, 4.31 per m², it gives 38,800, more than twice the zoned figure, from the same photo.

How wide the range really is

An estimate is only as good as the less certain of its two inputs, and the errors combine. Watson and Yip give the relative standard error of the total as the square root of the sum of the squared relative errors of area and density, and put what a moderate photograph allows at about 10%.

In the worked example the area is from a plan, so 5% is fair, but density varies from square to square. The eight middle squares have a standard error of their mean of 0.12 per m², 5.4% of 2.26, before allowing for the front and back zones, which were judged on fewer squares. A fair report would be "about 17,000, likely between 14,000 and 20,000, from 9,000 m² at an average of 1.9 people per m²".

How to check a claimed crowd size

Divide the claimed number by the area and see whether the density it needs is believable. A claim that needs more than about 4 to 5 people per square metre across the whole area cannot be right, since that is a packed crowd from edge to edge.

Watson and Yip ran this check on the 2011 candlelight vigil in Hong Kong's Victoria Park. Major news agencies reported the organisers' 150,000; the police said 77,000. The area was known, 42,000 m², and their density estimate was slightly under 2 per m², close to the police figure. 150,000 ÷ 42,000 = 3.6 per m², which would have needed the whole park packed close to mosh-pit density. It was not.

The best-known dispute shows how far apart the figures can be. For the 1995 Million Man March in Washington, Watson and Yip record organiser estimates of 1.5 to 2 million, a US Park Police estimate of 400,000, and 870,000 from Boston University's remote sensing unit with a margin of error of about 25%. After the row, the Park Police stopped giving official crowd estimates.

Marches and crowds that move

Area times density does not work for a march, because the crowd is strung out, its density changes along the route and people join and leave. Watson and Yip's method uses two counting points, A and B, well apart, with B near the end, and a short spot survey at B.

The estimate is N = N_A + (1 − q) × N_B, where N_A and N_B are the counts passing A and B and q is the share of people passing B who had already passed A, found by asking a sample at B whether they joined before A. Applied to Hong Kong's 2006 march, it gave a relative standard error of 2%.

A worked example: 30,000 people are counted past A and 42,000 past B. Of 1,000 people asked at B, 600 joined before A, so q = 0.6. The march is 30,000 + 0.4 × 42,000 = 46,800 people. The counts themselves are made at sampled intervals: count heads crossing a line for a fixed minute, several times an hour, and scale up. A people counter on a phone, one per counting point, keeps each interval's total.

When you can count people in, count them

If the crowd enters through gates, counting at the gates beats any estimate. Watson and Yip note that sports and concert crowds are "simple to count because entry to the event is controlled". With several entrances and no turnstiles, a counter on each entrance still narrows the estimate, and the guide to headcounts at events covers splitting the work between counters. A counted vote at a meeting is the same principle at its strictest: two tellers count the same hands separately and agree before the result is read, as in how to count votes at a meeting.

Entry counts and area estimates answer different questions. The gates give everyone who came over the day; area times density gives the people present at the moment of the photo. Both can be right and far apart at an event where people come and go. For a venue's legal limit, which is set before anyone arrives, use the floor space factors in the occupancy calculator rather than a density from a crowd.

Mistakes that inflate or shrink estimates

What this guide does not cover

This guide covers estimating the number of people in a crowd for reporting, research or event records. It is not crowd safety guidance. Still puts 5 people per square metre as the upper limit for standing and viewing, and a crowd approaching it needs managing, not counting; capacities for venues and events are set by the safety authority, licensing body or fire risk assessment that applies.

Two limits on the sources. Jacobs's 1967 article itself is not online, so his densities are given as Watson and Yip state them. Doig's figures are from his own 2009 article and from interviews quoted by others, and they changed over the years; the metric densities measured on the ground or from a photo of your own crowd always beat any of them. Simple Online Counter is a free, browser-based set of counters; its tally and multi counters count the heads in your sample squares, and the judgement of area and zones stays with you.