I must confess to all my readers that I've fallen into a bad trap. I too have been caught quoting the meaningless statistic. Well, that's not true. It's not as bad as the old, "Jones is hitting .400 in night games on the same day as a family member's birthday," that's for sure. But statistics like one which made the rounds earlier this year about the Sonics always winning when they scored 100 and this current one, the Sonics' 10-3 record without Vin Baker, aren't really meaningful on their own. They have to be given some context. The 100 points stat might indicate that the Sonics play better in faster games, as I demonstrated in a previous column (more later). In that sense, it's important. Similarly, the record without Baker is only meaningful if it meets two conditions -- statistical significance (more later) and the opponents aren't significantly worse than the Sonics' other opponents.

So I started out this afternoon to take a look at those two conditions and whether they were met or not. While I was working, the Sonics threw a wrench into things by upping the record to 10-3 with a pair of convincing wins over Sacramento and Phoenix, so I've included those as well now.

Is the 10-3 record without Baker statistically significant?
Let's first get out of the way the definition of statistically significant. The idea here is how likely something that we see (ie the 10-3 record) could be essentially the same as something else (in other words, the Sonics are really equally good without Baker as they are with him) by pure chance. Most fields require for something to have only a 5% or lower chance of being pure chance before they are willing to attach the term significant to it.

My Method:
Feel free to read on to the conclusion if you haven't taken a statistics class or just don't care; there will be no quiz after the column.

I had to start out by finding the standard deviations of the winning percentages with and without Vin. The standard deviation is a measure of how far from average a typical value is and generally of how centered the data is. For the no-Baker games, this came to .1775, with .2475 with Vin.

I then plugged these into the formula for the 'standard error' of the difference, which is:

SQRT ((.1775/13)+(.2475/40)) = .141 = standard error.

The next step was to divide the observed difference (.769-.45 = .3) by .141, which gave a t-value of 2.27. There are 16 degrees of freedom; I don't show the calculations here because I doubt they interest anyone. If you want to know what I did, e-mail me. I then used Excel to find the probability of finding a t-value this large or larger if the Sonics were really as good with Baker as they are without him, which is 1.88%.

Conclusion:
Well, even with the statistics, this isn't so cut and dried. Whether the difference is currently significant or not depends on how we 'frame' the question -- "Are the Sonics a different team with Baker out of the lineup?" or "Are the Sonics a better team with Baker out of the lineup?"

The two questions provide different answers. Quoting from my statistics book, "In order to use a one-sided test, you must be sure that no matter how the data had come out, you would still have used a one-sided test on the same side ("larger than" or "smaller than") as you will use. . . . If in doubt, use a two-sided test." (Siegel, Andrew F. Practical Business Statistics, p.360)

Based on that, I think it would probably be unreasonable to use a one-sided test ("better than?"), because had the Sonics played worse without Baker, I'd be studying that.

Using the other question ("different than?") means having to double the value I got before for the odds of the 10-3 record being simply random chance, giving a 3.77% degree of randomness.

We can thus say with statistical confidence that the Sonics are indeed a better team without Vin Baker, despite the fact that it may feel like a very small sample.

Editor's Note: I have not updated this section to reflect the Phoenix game.

More on those pesky games where Vin was injured:
Oddly enough, the games where Baker hasn't played have been the exact definition of NBA average. The opposition's average winning percentage has been .500, with half of the 12 games at home and half on the road. Of course, there's an inherent problem with this kind of analysis, because it assumes that all games a team plays are equal (for example, the Sonics were a bit more likely to win against Toronto with Vince Carter out, don't you think?), but it's all we've got.

You might be thinking then that the games where Vin played were against the same average .500 team, right? Well, that's wrong, because the West is stronger than the East. In fact, the average opponent when Baker has played has had a winning percentage of .531. If you're savvy enough statistically, you might wonder what the chances of the Sonics beating a .500 team and a .531 team are. Because the Sonics are close to .500 themselves, the three percent difference holds. The technical way to do this, from Dean Oliver's Journal of Basketball Statistics, is to use the following formula:

((win%A)*(1-win%B))/(((win%A)*(1-win%B))+((1-win%A)*(win%B)))

This gives the Sonics a 48.8% chance of beating the .531 team and a 51.9% chance of beating the .500 team; right about a three percentage difference. Of course, what's been seen is ten times this. . . .

. . . But why, exactly, have the Sonics done better? Improved offense? Better defense? Faster pace? The main idea postulated by fans is that the offense is more effective without Baker because of better ball movement and more fast breaking. How does this hold up?

In the games without Baker, the Sonics have had an offensive rating of 113.0, meaning that for every 100 possessions (defined as the time they first get the ball to the time they lose it; an offensive rebound is not considered a new possession) they'd score 113 points. When Baker has played, the offense has been rated a 103.7. Somehow, the Sonics have gained nearly 10 points in an extremely fast game!

Is this a significant difference? I tested this, including the statistics from the Phoenix game, and found a p of about 3.5% or so, meaning it too is statistically significant. Again, we can be fairly confident that the Sonics play better offense without Vin Baker.

On defense, the difference is not so dramatic. With Baker in the lineup, the Sonics have a defensive rating of 105.1. Without him, it's 102.6. This surprises me; I figured the Sonics would have a bigger improvement, given many of the lost minutes are/were going to Art Long, Jerome James, and Calvin Booth, all considered better defenders than Baker. I guess the effect of Peja Drobnjak playing more and smallball makes up for that.

So, in theory, if the Sonics played a 100 possession game against an average opponent with Baker, they'd expect to lose by a point. Without him, they'd expect to win by 10; quite a difference indeed.

As for that playing faster idea, it's a myth. The Sonics' average game is slightly faster when Baker plays (3.81 possessions per minute) than when he sits (3.78 possessions per minute); this one can definitely be chalked up to chance.

But where is this improvement coming from? What players are picking up the slack? Is there any evidence that Baker's ability to draw double teams helps any teammates?

In order to answer that question, I summed up the stats of the nine players who have played in at least half of the games Baker has missed and compared their minutes per game, field goal percentage, and points, rebounds, and assists per game with their season averages. Here is the result (positive values indicate that the player has improved in Baker's absence):

player	mpg	fg%	ppg	rpg	apg
Payton	0.1	3.4%	1.6	-0.7	-0.2
Barry	0.2	8.3%	2.6	0.1	-0.5
Lewis	-1.8	-0.9%	-0.2	0.4	-0.5
Long	4.5	5.6%	1.3	0.8	0.2
James	4.0	11.1%	2.6	2.2	0.2
Watson	1.5	12.2%	1.3	-0.1	0.7
Mason	3.5	2.4%	2.3	-0.6	1.0
Radmanovic	-2.7	-1.8%	-0.9	-0.5	-0.3
Drobnjak	8.3	12.2%	5.0	1.7	0.4
Team	-	4.1%	1.4	-3.2	0.0

I'm legally required to run this table with a warning, that being that all of this data might not be worth the computer screen it's written on. It could just be that a player happened to be going through a hot streak at the same time as Baker was injured, completely independent events. Fortunately, he's spread his recent absences, making this slightly more random.

That said, there is clear evidence there of the Sonics' other interior players stepping up their games in Baker's absence. Together, Long, James, and Drobnjak play 17 more minutes, shoot about 9% better from the field, score nine more points, grab five more boards, and dish out an extra assist than they do when Baker does play. Obviously, part of this is due to the increased playing time, but these players aren't collectively averaging anything near nine points or five rebounds per 17 minutes during the regular season, so the quality is improving along with the quantity.

The assists trend is rather odd. The theory that the Sonics move the ball better with Baker out doesn't really hold. The average total number of assists is exactly the same, and the assist ratio (assists/field goals made) is actually lower (.583 vs. 601) because the Sonics are making more shots on average with Vin out. What's strange is that the assists are distributed differently. The starting guards have less while the other bigs have more. My best guess is that they are getting some of the touches Baker normally would, while the guards' touches aren't really changing.

The three less rebounds per game is probably misleading. Frank Hughes reports in yesterday's News-Tribune that the Sonics are outrebounded by two boards a night with Baker, but are even without him. So the opposition is getting less boards; I'm assuming this is because the Sonics are shooting so much better. Lewis has stepped up his rebounding when Baker has been out. I noted last week that his rebounding was good in November but only mediocre the last two months, skewing his season mark. Perhaps this is explained by Baker's absence; more emphasis for Rashard on rebounding and more minutes at the four position.

And why exactly is that? How is it you take a guy shooting nearly 50% out of the lineup and everybody else (and the team, then) shoots better? Regular minutes seem to be doing the trick for the other big men. Apparently, getting five minutes of run just isn't enough for them to get warm. As for why Barry has improved his shooting so much, I'd attribute that to more freedom to drive the lane without Baker clogging the interior. The last two games, the Sonics have had significantly more points in the paint than their opponents despite lacking a post presence. Driving the lane more explains this.

The Big Picture:
As untenable a conclusion as it may be, all available evidence points to the Sonics playing better without Vin Baker. Start up the trade talks, please!

Some Other Statistical Notes, since I Already had the Data There . . .:
Well, let's go back to the issue of pace. Perhaps things have changed in the last month in a half, or perhaps my method was wrong, because it now appears the Sonics are not better in a faster paced system.

Part of it might be that I've changed how I've defined pace. Before, I based it on what Oliver calls 'minor possessions', essentially every time a team gets a shot or turns the ball over (with free throws also counting, of course). Now, I'm using 'major possessions', where an offensive rebound does not start a new possession but is considered part of the old one. That might have changed things up a little, but what I've found now is that both the Sonics' offensive and defensive ratings improve as pace goes down.

However, for whatever reason, this doesn't translate very conclusively to results. Looking just at wins and losses and ignoring the ratings, the Sonics seem to be about even, though slightly better in a slower pace.

I'm going to need to spend more time thinking about this one and hopefully I'll come up with a revised theory to explain the data.

. . .

Editor's Note: I have not updated this section to reflect the Phoenix game.

Have the Sonics been unlucky? I noted yesterday that a poster around here (theSonicsMan?) noted that the Sonics had a better point differential than the Boston Celtics, despite having a worse record by four games (now two and a half). So I decided to see what the Sonics' record ought to be based on point differential.

Taking more methods from Oliver, there are two accepted ways of seeing how much a team should win. The first is called the "Pythagorean Winning Percentage" and adopted from baseball. It is rather simple: points^16.5/(points^16.5 + points against^16.5). This formula projects the Sonics to have a .580 winning percentage; or, over 52 games, 30 wins. The Sonics have really underachieved by this measure, as three games is quite a bit of difference.

Of course, one caveat is that this also can measure performance in close games. The Sonics have, quite obviously, struggled of late in such games. As to whether that's luck or lack of 'clutch' ability, well, that's a question for a different day.

The other method is much more complicated, using the aforementioned offensive and defensive ratings. I won't go into the method (you can find it at the website linked above if, by some strange chance, you're actually interested in this sort of thing). It gives the Sonics' expected winning percentage as .531, which works out to 28 wins (or one extra) so far. It is, on average, a better predictor.