Earlier this year, Brian Burke of the fantastic Advanced NFL Stats site wrote a post in which he detailed and graphed a series of scatter plots that featured the power law distribution. If you have absolutely no idea what that means, click that link; his explanation is superior to anything I can offer. Anyways, he went on to address the idea of whether coaching job spans were a normal distribution or a data set that followed the power law. He found the latter, and I’ve finally gotten around to seeing how baseball managers compare.

First, let me detail my data set. I went team-by-team and collected each manager holding a position that was hired between 1995 and 2005. This means if a manager was hired in 1995, fired in 1996, and a new manager was hired then both count. If a team went through a manager per season they are all included. However I did not include interim managers who managed less than a season. So Bruce Kimm is out of luck, as are managers hired prior to 1995 or after 2005 – sorry Bobby Cox and Joe Maddon.

I ended up with 76 managerial cases. I then took down their organization, length of tenure, and win/loss record with that organization. With such I found the amount of seasons survived with that team by each manager, which was then used to create this graph:

For those who prefer charts:

Years Managers
15 1
12 2
10 1
9 1
8 2
7 4
6 5
5 8
4 15
3 21
2 14
1 2

The data set follows the power rule much like Burke’s examples. There are other questions to be answered though, which I’ll attempt to do now:

How long does the average managerial job last?

The mean of the tenures is 4.3 years, the median is 4, and the range is 14 years. So the mean is skewed to the right, but only barely. This matches up pretty well with what we see above since nearly 66% of the population falls into the 2-4 years group.

How does winning affect the lifespan?

It doesn’t:

That steep red line represents the .500 mark. Some were well below and lasted five or more years, some were well above and received the axe far earlier. That means to last as a baseball manager, you have to juggle player attitudes and egos, get along with management and ownership, and show competence at winning a fair share too. Most people would note that anyways and the numbers back it up.

To summarize, I think all of this is rather intuitive. Most managerial contracts seem to last 2-4 years, which is the average lifespan, and we can’t evaluate mangers well enough to say there’s a huge difference between any two skippers, which means firing a guy is more of a “gut” feeling. If the manager is friendly enough to the media he can probably buy time even if he makes questionable in-game decisions.

I’m not claiming this is a perfect measure and most people will probably read this and think “Duh“, but hopefully it did something for someone.

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Matthew Nolan
6 years 10 months ago

You’ve drawn an incorrect conclusion about the managers tenure, although it is not that far off of being correct.

You are predicting it is a power distribution when it is obviously not, as the mode is not the lowest value, the mode is 3 and you are ignoring this, as well as the obviously low value at 1 and 2.

This follows a gamma distribution with the argument to the gamma function (k) being 3 and theta being 1.5. This would predict a distribution of the correct shape with mean 4.5 and mode 3

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Joe R
6 years 10 months ago

Would that really affect the linear relationship (or lack thereof) in the 2nd chart, though?

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Ryan M
6 years 10 months ago

I think you’re being kinda pedantic with what constitutes a power law. The classic example of a power (pareto) distribution is income distribution, and its mode is only the lowest value when you’re using and interval of something like \$0-\$25,000/year. This data could easily follow that as well (and probably does, but I’m not going to recreate that data set by eyeing the graph) using an interval of 0-3 years. The sample is too small to see this either way.

The author’s more general point (whether he knows it or not) is the application of a long tailed distribution to the data, which is definitely right.

I actually think such distributions are greatly underutilized in sabermetrics. I’d go as far as saying that a majority of distributions are really a long tailed/craziness extreme values rather than normalish/well ordered patterns. Like player development- situations like a Ben Zobrist this year or Carlos Pena three years ago or a Cliff Lee last year or Mark Loretta several years ago happen waaaaayy too often to be accurately captured otherwise.

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Matthew Nolan
6 years 10 months ago

Just to clarify why it is close, a gamma distribution with k = 1 is just an exponential (power) distribution with mean theta

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David A
6 years 10 months ago

I was going to say it looks like a lognormal distribution, but gamma probably fits even better. We use these in insurance to model the expected payout on claims. Much like high-value claims, this data suggests that managers with long tenures are rare. Though in both cases, they tend to skew our expectations.

Guest
6 years 10 months ago

Thanks for the kind words. If you through out the 1-2 yr guys, which may be mostly interim managers or other special-circumstance managers, the distribution from 3+ yrs on looks like it’s very power-law. It’s bit of a stretch, I know.

I’d bet All-Star game appearances are power-law distributions. Team playoff or World Series appearances/victories are probably power-law too. The reason is that MLB baseball team strength is a rich-get-richer system like college football. The revenue and prestige that comes from success begets more success. See “Yankees, New York.”

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TanGeng
6 years 10 months ago

Looks like a poisson distribution. Ahh some one already mentioned gamma distribution.

Also applying context to winning percentage might help. Comparing managers against historical winning percentage or winning percentages around the time of tenure might give a more meaningful conclusion, or it might not at all.

Guest
6 years 10 months ago

“threw” not “through”

Ug–It’s a Monday after a late Sunday night game.