skyking162

Hopefully we can all agree that the word “consistent” is used 90% of the time as a flowery synonym for “good”. I don’t like it, but it’s out there. Maybe part of the reason consistent is associated with good is because, in general, good teams want to be consistent and bad teams want to be inconsistent. I like to use the analogy of head-to-head fantasy football. Let’s take two teams — A and B — that compete against each other every week. Who will win? Well, we’d need to know which team averages more points. Let’s say Team A averages 60 points and Team B averages 40. Team A is the better team, obviously, but Team B would win sometimes, too. How often Team B pulls off the upset is dependent on the variability of each team’s score. Let’s say both teams’ scores have a standard deviation of 15 points. Here’s what the distributions look like:

In order for Team B to pull off the upset, they need to score in the upper range of their red curve, while Team A scores in the lower end of their blue curve. In general, the larger the overlapping area, the larger the chance Team B has of winning. One obvious way for Team B to win more games is to average more points. The following graph bumps up their average score to 50 points, versus Team A’s 60 points. You can see that the overlapping area increases in size:

But Team B can also increase their chances of winning without averaging more points. They can be a more inconsistent team. Increasing Team B’s standard deviation to 25 points results in a shorter, more spread out distribution, shown here.

If Team A’s standard deviation also increases to 25 points, then the overlapping area increases even more, helping out Team B. Team B is willing to take the risk of scoring very few points in order to sometimes scores a lot of points, and Team A is helping out Team B by more often playing down to their level. Inconsistency helps bad teams and hurts good teams.

And the opposite is true about consistency, as seen here. The distributions of both teams become much narrower and taller, meaning both teams will score closer to their average more often, resulting in more wins for the better team, Team A.

If we assume that teams can control their level of consistency (I’m not going there in this article), then Team B will choose to be less consistent and Team A will choose to be more consistent, resulting in a graph like this. It’s kind of like a game of tag, with Team B trying to touch Team A while Team A tries to run away.

Instead of just graphing one situation at a time, we could graph Team B’s winning percentage as a function of every possible level of scoring or every possible level of consistency. The following graph shows Team B’s winning percentage as its average point total ranges between 20 and 100 points, against Team A’s 60 points.

Notice that the steepest part of the curve is at a .500 winning percentage. Getting better counts for more near .500. An awful team or a world championship team that adds a superstar won’t benefit as much as a team stuck in mediocrity.

Now here’s a graph that shows Team B’s winning percentage as its standard deviation ranges from 0 to 50 points. Team B averages 40 points against Team A’s 60. Even with scoring 40 points every game (no variation), Team B will win occassionally when Team A happens to score fewer than 40 points. With less consistency, Team B will have a chance to beat Team A’s score by scoring more than 40 points, even though there’s also the (smaller) risk of losing to Team A when it scores fewer than 40 points. Theoretically, as the standard deviation of Team B’s point distribution gets extremely large, it’s winning percentage approaches 50%.

The following graph shows Team B’s winning percentage if the standard deviations of both teams’ points distributions increase together from 0 to 50 points.

It’s a similar curve to the one above it, except that Team B has zero chance of winning when the teams are perfectly consistent. And it’s a little more obvious that Team B will pull of victories half the time when both teams are basically throwing up random point totals — the difference between 40 and 60 points is arbitrarily small compared to the massive width of the curves, as seen in the example with standard deviations of 50 points.

Moral of the story? If you’re a good team, try to be consistent. If not, there’s nothing wrongn with inconsistency. Just remember that you want to be consistent because you’re already good, and it won’t help if you aren’t there yet.

Update: The html for one of the large graphs was messed up, preventing it and two paragraphs from showing in the post. All good now.

Popularity: 2% [?]

This entry was posted on Monday, May 8th, 2006 at 8:05 am and is filed under Baseball. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.