As has been discussed a great deal here on South Side Sox, the White Sox offseason feels incomplete. Many have understandably suggested that the Sox are "one move short" and we're all waiting hopefully for more reinforcements.

I wanted to gain a greater understanding of what acquiring one more piece would mean for the White Sox' playoff odds, so I took a quick journey to my statistical roots. Let's work with the current Steamer projections as a starting point. Right now, here are the projected win totals for the AL Central:

Indians : 84

Tigers : 81

White Sox: 81

Royals : 79

Twins : 78

We can debate the merits of these projections until we're blue in the face, but the two most important takeaways for this analysis are that the White Sox are probably close to a true-talent .500 team and that the AL Central teams are really close to one another in quality.

Before continuing, we need to understand what an 81-win projection means. If we were to play the 2016 season a very large number of times, we'd expect the Sox to win 81 games on average. However, even though 81 wins is the estimated total, the odds that the Sox will win exactly 81 games are well less than ten percent. Sometimes they'd win significantly more than that, sometimes significantly less. How then, do we model the possible outcomes?

I'm going to do this using a simple normal distribution*, which looks something like this:

For this purpose, the horizontal axis represents the number of wins and the vertical axis is the probability of the Sox winning a specific amount of games. Imagine the point on the horizontal axis that touches the vertical line in the middle represents the average of 81 wins for the White Sox. The higher the curve is above a given win total on the horizontal axis, the higher likelihood of the White Sox winning that amount of games. Here's the important bullet points about using this distribution to model wins for the Sox:

The average outcome is 81 wins

The distribution is symmetrical, meaning the odds the White Sox win 83 games (two more than the average) are the same as the odds the White Sox win 79 games (two fewer than the average), and so forth

Outcomes closer to the average are more likely than outcomes further away from the average

The other key variable here is the standard deviation, which is a measure of how far win totals tend to stray from the average. I'm going to use six wins as the standard deviation for this exercise, as there seems to be some consensus in statistical circles that variation around true talent is close to that amount. To get a feel for what that means for an 81-win team, we'd expect the Sox to win between 75 and 87 games about 68 percent of the time. They'd win between 69 and 93 games about 95 percent of the time, with the remaining five percent representing extreme outcomes.

Of course, the other four teams in the division also have their own distributions using their projected win totals as a central point. To simulate a random trial, I generated a random win total from each team's distribution and determined who won the division. If a team won at least 86 games, but did not win the division, I also gave them some chance** of winning one of two wild card slots, which I gave half the value of winning the division to account for the play-in game.

To come up with playoff odds, I ran 300,000 random simulations of win totals for the AL Central. Here's the percentage chance I came up with for each AL Central team making it to the divisional round of the playoffs (so, these odds exclude outcomes in which the team in question makes it to the Wild Card game and loses that game).

Indians: 41.9%

White Sox: 22.4%

Tigers: 22.4%

Royals: 13.7%

Twins: 10.5%

Now, let's say the White Sox bring in a 2-win upgrade, such as Dexter Fowler , moving their average to 83 wins***. Here's how those odds shake out:

Indians: 38.1%

White Sox: 33.0%

Tigers: 20.1%

Royals: 12.1%

Twins: 9.1%

And finally, here's what a 4-win upgrade would look like, such as a healthy Yasiel Puig ****, or bringing in both Ian Desmond and Dexter Fowler.

White Sox: 45.0%

Indians: 33.8%

Tigers: 17.3%

Royals: 10.4%

Twins: 7.7%

Playoff odds calculators at sites like Fangraphs and Baseball Prospectus are more sophisticated than this, but this process is good enough for a reasonable approximation.





The White Sox may have missed out on the big-ticket outfielders, but as the above shows, adding a mid-tier guy like Fowler could still make a significant difference, perhaps as much as a 50 percent (multiplicative) increase in their playoff odds. Based on Steamer's projections, bringing in two league-average regulars would about double their odds and make the Sox division favorites. Plus, as MelidoPerez pointed out in the comments awhile back, the improvement would be even more significant if you go by the ZiPS projections , which rate the White Sox a little better and the Indians a little worse.





The conclusion is nothing different than what we've thought all along, that ceasing to add talent to the 2016 roster at this point would be a mistake and that the value of putting an additional projected win or two on the roster has never been higher. The goal of this is not to precisely estimate the White Sox' playoff odds (or those of other teams in the division), but to point out just how much things could change, even this late into the offseason. If there were ever a perfect moment for Hawk Harrelson to yell, "Don't stop now, boys!", this is it.





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*There are some issues with doing this. For one, teams voluntarily choose to improve or reduce the quality of their teams in-season based on how the first half of the games go, so there's a greater chance for very high or very low win totals than implied by this analysis. I chose this for simplicity and don't feel that I'm losing much in accuracy by doing so.

** 86 wins = 25%, 87 wins = 50%, 88 wins = 75%, 89+ wins = 100%. This is far from perfect and changing this scale a little doesn't affect the analysis much, particularly because the AL Central has relatively low Wild Card odds using the Steamer-projected win totals.

***The White Sox play just shy of 1/8 of their games against each divisional foe. For each projected win added to the White Sox, I subtracted 1/8 of a projected win from each of the other teams. This isn't precise, but it's closer to reality than doing this without adjustment.

****Just dream with me for a second, OK?