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Tampa Bay Rays' Attendance: Why Not to Worry, Part 2

For a quick update, Friday night's game against the White Sox ended up drawing 28,927 people and Saturday night's game drew 31,916, which are quite impressive numbers.  So if anyone was questioning Thursday night's attendance, I hope at least this helped assure you that it was simply a product of it being a lousy night and a low-drawing team.  Really, nothing to get worked up about.

However, I do still want to address the statement made by a commenter on my piece a couple weeks ago about hoping the Rays' attendance never fell before 20K for any game.  While made in good faith, such high expectations set us up to be disappointed.  Even if the Rays were to draw an average of 30,000 fans a game, there would still be individual games that dipped under the 20,000 mark.  That's the nature of averages - you're going to have some games much higher than your average and some much lower. 

What I want to know, though, is two things: how low is too low and how many low attendance games can we tolerate before we should get worried?  Obviously, if we only draw about 5,000 fans to one game, that's a huge cause for concern.  And also obviously, if we consistently start drawing only 13K fans per game, that's also a problem.  So where do we draw the line between what's low attendance based on random fluctuations and what's low attendance based on...well, low attendance?

Okay, so this explanation is going to get a bit mathematical/statistical and I'm going to do the best I can to explain everything in terms hopefully everyone can understand.  If anything is not clear to you, though, please leave a comment and I'll be happy to elaborate and/or fix the section that you were confused by.

In order to answer these questions, I'm going to use the statistical concept of the normal curve.  The normal curve goes by a bunch of different names - bell curve, normal distribution, Gaussian function , stuff like that - and it's a pretty common concept (at least I feel like it is, though I may be biased) so hopefully you already have a pretty good idea of what it is. If you don't off the top of your head, all you really need to know for now is that the normal curve basically is a way of approximating any distribution of data that groups around an average.

Whether you're familiar with the normal curve or not, though, I'm going to walk us through this step by step.  Take a look at this image - that's the normal curve.  That u-like symbol (the μ thingy) in the middle of the x-axis represents the mean (AKA "average") of whatever sample it is you're dealing with.  In our case, we're dealing with attendance figures over the course of a season.  The percentage figures represent how much of your sample should theoretically end up within each distance from the mean.  For example, look at the section directly to the left of the mean.  In that section, which is slightly less than the average, you should theoretically find about 34.1% of your total sample.  In our case, that would mean that we should find about 34.1% of Rays' home games have attendance figures falling in this range of being slightly less than the average for the year.  On the flip side, about 34.1% of Rays' games should have attendance figures slightly higher than the average.  And then, also looking at the graph, about 13.6% of Rays' games should have attendance figures slightly higher than that section.  Make sense?

Now the trick is, how do we determine where these cut-offs between sections are?  That's where the term "standard deviation" comes in (represented by the symbol σ in the picture), which is basically just a statistical way of calculating how spread apart your sample is and therefore, where these different intervals are cut off.  The standard deviation for the Rays' attendance per game last year was 8,340, meaning that about 68% of the Rays' games should have had attendance figures that were within ±8,340 of the average (22,370 per game), which ends up being a range between 14,030 and 30,710 fans per game.  In actuality, the Rays had 44 games that fell into that range last year out of 78 home games (I discounted the ones at Disney because the smaller capacity of that stadium screwed with things), which is about 56% of our home games - damn close to the theoretical estimate of 68%.

If you'll notice, though, that huge range still only accounted for 56% of our games, meaning that there are still 34 games to be accounted for.  If you look at the next two sections of the normal curve (within two standard deviations of the mean, or anything between the 2σ cutoffs), you'll see that each section should hold about 13.6% of our sample.  In 2008, there were 16 games that fell between -1σ and -2σ (between 14,030 fans per game and 5,690) and 18 games that fell between 1σ and 2σ (between 30,710 fans per game and 39,050).  That's around 20% (16 out of 78) of the Rays' home games that fell within each of these intervals.

We've only looked at 2008 so far, but the distribution of games is very similar each of the last few years.  Here's a summary of the last 3 years: Normal CurveVery similar, with even a couple games in '06 and '07 reaching into the category between 2 and 3 standard deviations above the mean.  This was possible those years because we sold out days like our home opener, but our overall attendance average for the year was very low.

So this is all well and good, but what does all this mean going forward?  Basically, we now know that most (if not all) of the games the Rays will play at home this year will fall within ±2σ (plus/minus two standard deviations) from the mean.  I'm going to estimate our standard deviation this year as 8000, which is about where our standard deviations have been for the last three years (8,340; 7,960; and 7,500).  Using this standard deviation, let's look at where our cut-offs for this year would be depending what our average attendance per game was:

(2σ) (1σ) Mean
9000 17000 25000 33000 41000
11000 19000 27000 35000 43000
13000 21000 29000 37000 45000
15000 23000 31000 39000 47000
17000 25000 33000 41000 49000

Based on this, even if we were to average 33,000 fans per game, there would still be a good chance that we'd have some games bottoming out as low as 17,000 fans.  I personally think it is more realistic to expect our average attendance for the year to end up around 25,000 - 28,000 per game (see this piece on the potential effects of the recession), meaning that we could theoretically see games as low as 9,000 per game.  Considering the lowest we got last year was 10,500, I really doubt we'll get that low, but it's not unreasonable to expect a couple more games down around 11,000 - 13,000 fans in attendance.

How many games, though?  Based on the distribution of games over the past couple of years, here's my estimate of how many games to expect fitting into each section of the normal curve:

(2σ - 1σ) (1σ - Mean) Mean - 1σ 1σ - 2σ
13 28 18 19

If that looks really similar to what we did last year...well, that's because it is.  I expect us to have fewer games on the very low end (since last year it took about half the season before Tampa-St. Pete recognized how good we were and started attending games) and  about the same number of games in each other category.

Anyway, to summarize, I expect the Rays draw an average of about 25K-28K per game, yet to bottom out around 11,000 - 12,000 fans in a few games and to have about 13 games this year with under 20,000 fans per game.  As the season progresses, I'll revisit these estimates and see how we're fairing but for now, we're looking pretty darn fine.  Again, if you have any questions, please don't hesitate to ask.