{"id":1615,"date":"2011-12-22T12:04:18","date_gmt":"2011-12-22T02:04:18","guid":{"rendered":"http:\/\/www.somethinkodd.com\/oddthinking\/?p=1615"},"modified":"2011-12-22T12:04:18","modified_gmt":"2011-12-22T02:04:18","slug":"could-fewer-people-attend-regular-events-than-attendees-think","status":"publish","type":"post","link":"https:\/\/www.somethinkodd.com\/oddthinking\/2011\/12\/22\/could-fewer-people-attend-regular-events-than-attendees-think\/","title":{"rendered":"Could fewer people attend regular events than attendees think?"},"content":{"rendered":"

Introduction<\/h3>\n

I run a juggling club. The fact that it is dedicated to juggling isn’t important here. What is important is that the format is very casual. If you turn up, you pay a small fee to cover the cost of the hall rental and other incidentals. If you don’t turn up, you pay nothing. Each week you have an individual decision about whether to turn up.<\/p>\n

As a result, the attendance varies. On a great night, we might have two dozen jugglers turn up. On a poor night, we might have seven. The fee is the same each time, so I am wearing a risk that on average we will get the approximately 15 paying jugglers required to break even. I am not aiming for a profit; in fact, I have and will carry over minor losses to keep the thing going – because *I* enjoy it.<\/p>\n

I’ve noticed a phenomenon where the occasional jugglers think that it could be run cheaper, because they have rosy impressions of how many people turn up on average.<\/p>\n

I have attributed part of this misunderstanding to a sampling error. The nights that are very busy are witnessed by a large number of jugglers. The nights that are quiet are witnessed by only a small number of jugglers. So the average perception of the jugglers is higher than the real figure.<\/p>\n

I’ve never convinced myself how big that effect is, or even that it is true. (Each individual seems to take a genuine random sample, so how could there be a bias?) I decided to do a Monte Carlo simulation. This is the result.<\/p>\n

Method<\/h3>\n

I made a few dangerously large assumptions.<\/p>\n

First, I assumed each juggler made the decision to attend independently. In fact, that’s not the case, with couples and cliques tending to turn up as groups rather than individuals. This also ignores the effects of weather, major sporting matches, juggling events and the like which tend to group attendance behaviour. Both factors which would further accentuate the effect.<\/p>\n

Second, I assumed that each juggler made the decision to attend each week independently of their previous record and the previous attendances. It was based purely on an internal probability checked each week. This discounts people going through phases of attending each week, or going away, etc. It also discounts people deciding whether to come the next week based on the number of people who turned up the previous week. (Massive over-attendance turns off some of the jugglers because the hall gets too busy for their comfort. Under-attendance turns off many more who come for social interaction.)<\/p>\n

Finally, I had to assign the number of jugglers and their likelihood of attendance, based on a finger-in-the-air guess. This modelling decision affected the magnitude of the result greatly, and should be treated with suspicion!<\/p>\n

I decided there was one person who attended 100% of the time (me). Two attended 90% of the time, four attended 70%, six 50%, eight 30%, ten 10% and fifty 1% of the time.<\/p>\n

That was arbitrary, and gave an expected average attendance of 12.5 people. Slightly less than real life, but close enough to be going with.<\/p>\n

Using this model, I did a Monte Carlo simulation of 10,000 weeks.<\/p>\n

I averaged the results at each rank.<\/p>\n

Results<\/h3>\n