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What’s the Probability That James/Wade’s Declines Are Due to Chance?

Posted by Neil Paine on November 30, 2010

This post is a follow-up to this morning's piece about LeBron James & Dwyane Wade's current slumps, so should probably read that one first, if you haven't yet.

In response to the hand-wringing about Wade & James' sub-standard production thus far, some have suggested it's merely a pair of slumps that just happened to coincide with the duo joining forces in South Beach. How legitimate is this theory? Well, thanks to the magic of Monte Carlo simulation, I can test exactly how likely that explanation actually is.

Specifically, I'm going to simulate 10,000 18-game samples based on the career distribution of James & Wade's Hollinger Game Scores. (Yes, there are countless other, better metrics, but hey, this is a quick-n-dirty study.)

How do you do this? First, you start out with Hollinger's Game Score formula:

GmSc = PTS + 0.4 * FG - 0.7 * FGA - 0.4*(FTA - FT) + 0.7 * ORB + 0.3 * DRB + STL + 0.7 * AST + 0.7 * BLK - 0.4 * PF - TOV

Calculate that for every game of James/Wade's careers. Then examine a histogram of the game-by-game distributions of Game Score, making sure the data is approximately normal (it was, for both players). Find the per-game average and standard deviation of each player's career Game Scores through the 2010 season:

Player > James Wade
Avg 22.06 19.61
Stdev 8.49 9.15

Also, find the players' probabilities of playing in any given team game through 2010:

Player > James Wade
Games Missed: 26 103
Total Tm Games: 574 574
prob(Play): 0.955 0.821

Finally, find the per-team game average of James & Wade's game scores so far in 2011:

Player > James Wade
2011 GmSc/TmG: 18.29 14.38

Now we have all the tools necessary to run the Monte Carlo simulation. For every game of an 18-game sample, I used a random number generator to determine whether James/Wade played (using the probabilities above), and if so, what their game score was (taking a random number from a normal distribution with the mean and standard deviation listed above). Average those numbers over 18 simulated games, checking if both players were at or below their actual 2011 averages. Then repeat this process ten thousand times, and count how many simulations contained 18-game averages at or below the real-life 2011 marks (this will be the probability that James/Wade's declines have been due to random fluctuations in performance).

The result? Out of 10,000 simulations, only 276 contained 18-game stretches where both James and Wade simultaneously put up average game scores as bad as they've posted so far in real life. This means there's just a 2.76% chance that Wade & James currently have the same inherent "Game Score skills" as their 2004-2010 averages suggest, but have merely gone through two simultaneous slumps.

Instead, it seems far more likely that either one or both players' inherent Game Score skills have declined from the 2004-10 period. This doesn't explain why those skills have declined (certainly, contextual effects like the new scheme and new teammates seem much more probable than any physical drop-off), but it does mean it's extremely unlikely that the 18-game sample we've seen out of Wade and James so far in 2011 represents two players with the same inherent game score ability as in 2004-10 simply having a bad stretch of games.

56 Responses to “What’s the Probability That James/Wade’s Declines Are Due to Chance?”

  1. huevonkiller Says:


    I'm calmly explaining to you, that you didn't say anything insightful in this thread. At every step neil disproved your criticisms, now and in the past.

    Neil has posts that account for external factors, like two conflicting playing styles that might take time to mesh.

  2. huevonkiller Says:

    #48 I meant.


  3. EvanZ Says:


    Point taken, but the take-home message was not intended to be "this is when you use the eye test", rather simply explaining the difference between slump and SLUMP. I suppose one could argue the eye test is never necessary (maybe sufficient in certain cases?). We can have that discussion another day.

  4. lorrance Says:

    @ 51 don't see where i really criticized(to strong a word for just disagreeing) Neil and where those criticisms were proven wrong, really don't get the issue. My Point is simple, they both were slumping and you could tell this both statistically and by using your eyes. Think you are confusing me with someone else. This is the first time i've ever remember disagreeing with Neil on something so I don't get the tone of you're comments. I know this is a stat site but trying to figure out the PROBABILITY of something we see ACTUALLY happening doesn't seem to be helpful to me, but maybe I'm missing something.

    @EvanZ- Wasn't looking for any cheers or any of the sort. Point is actual observation of actual game play is useful without made up probability models. Plain statistical comparison is enough to see they are slumping at the same time and by watching footage of HOW they play now and in the Past is enough to figure out Why. And Yay to you to EvanZ!LOL

  5. EvanZ Says:

    Ok, Lorrance. Nice to see you still don't get it.

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