Did I mention that I was in the stands for Felix Hernandez’s perfect game?

Oh, I did? Cool.

On Wednesday, I asked a question that my friend Jon had asked me soon after that 2012 game: How many people alive today have seen a perfect Major League Baseball game in person?

In Wednesday’s post, I outlined an approach to solving this seemingly-difficult problem. To repeat:

1. We have the total attendance for each game. (It varies from the 6,298 people who saw Catfish Hunter’s perfect game in 1968 to the 64,519 who saw Don Larsen’s in the 1956 World Series.
2. Assume that the percentage of people at each age and sex at the game was the same as the percentage of people at each age and sex in the U.S. as a whole. Again, this is almost certainly not true, but I’m not sure how to do better.
3. All those people are older by the amount of years that have passed since the game. Someone who saw David Wells’s 1998 perfect game at age 48 would be 68 today.
4. Assume that anyone whose age today turns out to be greater than 76 (for men) or 81 (for women) has gone to the big game in the sky.
5. Add up the number of people still alive who saw each game to get the total alive across all games. That is the answer to Jon’s question.
6. Additional step! This extra step was suggested by my other friend Ed, who asked “What about the weirdos that have seen more than one perfect game?” Are there cases where adding up the attendance would double-count some people, and if so, how do we account for that?

I closed Wendesday’s post asking how to get the data we would need for this approach. We can get attendance figures from the Wikipedia article on MLB perfect games, and I suggested that we should be able to find demographic data (age and sex) from the U.S. Census. Did anyone think of how to find that data? No one commented, but that doesn’t mean you weren’t thinking about it.

But to jump to the reveal:

I got the U.S. population by age and sex from 1900 to 2000 from Demographic Trends in the 20th Century (table 5 on page A-7), and for 2010 from 2010 Census Briefs: Age and Sex Composition. Both are free to download, and use for any purpose, by following the links.

Of course, the U.S. census is only done every 10 years, and perfect games occurred in interim years like 1965 and 2004. So to get the age/sex populations in those interim years, I interpolated between the once-a-decade censuses by assuming a linear population growth, with slope (P₂ – P₁) / 10 for each of the categories. Note that this assumes that populations shifted within categories only every 10 years, which is not a good assumption but is good enough for this quick analysis. For games in 2011 and 2012, I carried the 2000-2010 growth rate forward another two years.

The census doesn’t publish population counts for every age; instead, they report a single count for all ages within five-year intervals (bins), starting with “0-5” and ending with “85+”. Within each age bin, they report the number of men and the number of women. They also give the total number of men and women within overall, and the U.S. bottom-line population total.

From these numbers, I calculated the percentage of people in each age/sex bin in years in which perfect games took place. I then multiplied this number by the attendance of each perfect game to find the number of people expected in each age/sex bin (as predicted by our simple model in which spectators at a baseball game is a representative sample of the U.S. population).

Then I subtracted the number of years between that perfect game and today, and added that number to each of the bins. That shows us how old that game’s crowd would be today. I added up the counts only the bins for people who are today 76 or younger (for men) or 81 or younger (for women).

So, calculate the percentage of people in each age/sex bin in each census. Multiply that percentage by the total number of people at each game to figure out how many members of each age/sex bin there were at the game. Increase the age of the population to today, and remove any bins that would be greater than 76/81 today. For example, Don Larsen’s perfect game took place 62 years ago, so I added up population counts for men who were then younger than (76 – 62 =) 14. That includes everyone from the age bins 0-4, 5-9, and 10-14. From a similar analysis of women, I added up everyone from age bins 0-4, 5-9, 10-14, 15-19. (In some cases the years overlapped bins – for example, Kenny Rogers’s perfect game in 1994 returns women 57 or younger, which overlaps the 55-59 age bin. So I added 2/5 of the count from that age bin.)

Adding up all remaining age bins leaves the number of people who attended each perfect game who are probably still alive in 2018. For example, of the 64,519 people who attended Don Larsen’s perfect game (it was in the World Series!), an estimated 21,422 are still alive today. We’ve made so many assumptions that I wouldn’t trust that number to be anywhere near exact, so let’s report it as “about 21,000”.

Doing this for all perfect games and then adding up for the total gives 363,000. But then, there’s Ed’s question: were there some people who saw multiple perfect games, meaning that we’ve double-counted some people? Do we need to adjust our estimate down to make up for it?

If this were 2011, I’d say no. Before then, perfect games had happened in different cities in different years. But then in 2012, Philip Humber’s and Felix Hernandez’s perfect games took place at Safeco Field in Seattle. Thus, a Mariners 2012 season ticket holder would have seen metaphorical lightning strike twice – two perfect games in one stadium in one season.

So to account for this effect, we need to estimate the percentage of the crowd at a Major League Baseball game that are season ticket holders who attend every game (or at least attended two games, and were incredibly lucky in choosing the two games they attended). I have no idea how to estimate that percentage. But consider that we know for sure two people that attended Hernandez’s perfect game but not Humber’s – me and my lovely spouse. And I know several Baltimore Orioles season ticket holders, none of them have attended every single game this season. So, let’s pick an estimate that I think is on the high side of reasonable: 10 percent.

That means that to account for the double-counting, we need to subtract about 10% of the crowd for Hernandez’s perfect game because we suspect they had already seen Humber’s perfect game four months before. Ten percent of 21,889 is (rounding off to remind ourselves this is just an estimate) about 2,000. So subtract out 2,000 people from our preliminary estimate of 363,000 to leave 361,000 people who have seen one or more perfect games.

Again, there are a lot of potential sources of systematic error in this analysis, so I don’t think we can be confident enough in our estimates to go down to the level of 1,000 people either way. So again, let’s round to 360,000.

Thus, our estimate shows that about 360,000 people alive today have seen a Major League Baseball perfect game. And I am one of them.

This has been a quick analysis of a small, self-contained question, but it showcases many features of the thought process that data scientists go through each day. I hope it’s been fun to follow along. The most important part is to always keep in the back of your mind: How might I be wrong?

With that in mind, then: how might this analysis be wrong? What assumptions did we make that we should not have made? What can we do to improve our estimates?

I’d love to hear your thoughts on this, no matter what your experience with math and science have been. The entire point of this blog is to bring the excitement of science to everyone.

Don’t make Eeyore sad, comment below with your thoughts!