I looked at the average height of NBA players, weighted by minutes played. For most of basketball history the players got bigger and bigger, but they peaked in the late 1980's. Since then the league has actually gotten smaller.

Starting with 1952, the average NBA player was 6'4. Four years later he was 6'5. In 1965, with Russell and Chamberlain dominating the game, the average player was 6'5.7. One more decade and the average topped 6'6 for the first time in 1976, and 6'7 in 1986. The average height stayed above 6'7 for just 3 years (1986-88), which is the time I was in high school. So you kids out there today, you didn't have to faced the kind of trees on your way to the rim that I did.

Since then the giants have become more rare. Average height fell to 6'6.5 in 2000, rebounded a bit, and then back to 6'6.5 in 2010. While there is a bit of year to year fluctuation, the average height has been within a half inch (6'6.5 and 6'7) for over 30 years now, 1981-2011.

## Sunday, March 18, 2012

## Saturday, March 17, 2012

### They Play The Best When It Matters Most - Part 2

And now the top dozen postseason performers. To get this list I'm looking at postseason win shares, above and beyond what you would expect given their regular season win shares per minute rates. An adjustment has been made for the fact that with improved competition, WS/48 is on average 0.018 lower in the postseason.

12. Reggie Miller (+3.0 wins) If you need 8 points and only have 9 seconds to get them, then you want Reggie on your team.

11. Scottie Pippen (+3.2 wins) Scottie was a forward who had the skills of a guard because he was a 6 foot 1 or so guard when he went to college, and a late growth spurt made him a 6 foot 7 lottery pick 4 years later. What some people don't know is that he kept growing the longer the postseason went on (to be reset after the season to 6'7). Scottie was HUGE in the playoffs. By the time he got to the finals he was 7 foot 4, and still could handle the ball like a guard.

10. Ben Wallace (+3.4 wins) - here's a speech from the 2004 NBA finals, never before reported:

Piston: The Lakers are too many.

BW: Sons of Detroit,

I am Benjamin Wallace...

Piston: Benjamin Wallace

is seven feet tall (including his fro).

WW: Yes, I've heard.

Blocks shots by the hundreds,

and if he were here,

he'd consume the Lakers with fireballs from his eyes

and bolts of lightning from his arse < laughter >

I AM Benjamin Wallace,

and I see a whole army of my countrymen

here in defiance of tyranny.

You've come to fight as free men,

and free men you are.

What will you do with that freedom?

Will you fight?

Piston: Against that? No, we will run, and we will live.

BW: Aye, fight and you may die.

Run, and you'll live... at least a while.

And dying in your beds many years from now,

would you be willing to trade all the days,

from this day to that,

for one chance,

just one chance,

to come back here and tell our enemies

that they may take our lives,

but they'll never take our Championship Trophy!!!

9. Derek Fisher (+3.6 wins)

Fisher has gotten a lot of criticism lately because he's too old andslow to guard speedy point guards. But he had a fine career and was an important part of 5 Laker championship

teams. Shot 40% (37 from 3) for his career, but in the playoffs he upped that to 43% and 40%.

8. Walt Frazier (+4.1 wins) Improved both his volume (PPG) and efficiency (FG%) in leading the Knicks to two championships.

7. John Havlicek (+4.6)

Won 8 championships, including two after the Bill Russell era was over. He took on more of a scoring role in the playoffs, and from what I understand had a pretty important steal in one game.

6. Elvin Hayes (+4.7) This one is a bit of a surprise. He did improve his FG% while shooting more and rebounding more. Won one championship with the Bullets, towards the end of his career.

5. Robert Horry (+4.8)

I would not trust this list if the numbers didn't give me Big Shot Bob. In the playoffs his 3 pointers made jumped from 0.7 per game to 1.1, with an increased percentage, but these numbers can't even tell you how big some of those were, at the last second to win or tie games. Won 7 championships, 2 with the Rockets, 3 with the Lakers, and 2 with the Spurs.

3T. Jojo White and Dave Cowens (+5.0). These were the big stars of the 1970's Celtic championship teams, in between the Russell and Bird eras. White was the 1976 Finals MVP, leading the NBA playoffs in field goals, free throws, and assists that year. For Cowens my numbers show him getting better in the playoffs, Basketball-reference's numbers do not. He did increase his rebounds and scoring, but his FG% is slightly lower.

2. Isiah Thomas (+6.3)

To be frank about it, by regular season numbers Thomas looks like a very good but hardly dominant point guard. 19 points, 9 assists, 45% shooting. You can find a ton of players who could do that. What makes Thomas the all-time great is his playoffs. His WS48 is .146 for the regular season (very good, .100 is average) but .200 for the playoffs (among top 25 all time for all positions). His biggest statistical jump in the playoffs is his rebounding (4.7 vs. 3.5), he scored more while playing on a team that shut down opponents and made every point more valuable. He just got better as the games got bigger, and even a severely twisted ankle could not stop him when he got on a roll.

1. Michael Jordan (+7.3)

It's really not fair - he's already the best player in the game and then when everyone else's stats drop in the playoffs he gets even better? Jordan's WS48 is .286 for the regular season (actually slightly behind David Robinson) but .315 for the playoffs (100 points ahead of the Admiral).

12. Reggie Miller (+3.0 wins) If you need 8 points and only have 9 seconds to get them, then you want Reggie on your team.

11. Scottie Pippen (+3.2 wins) Scottie was a forward who had the skills of a guard because he was a 6 foot 1 or so guard when he went to college, and a late growth spurt made him a 6 foot 7 lottery pick 4 years later. What some people don't know is that he kept growing the longer the postseason went on (to be reset after the season to 6'7). Scottie was HUGE in the playoffs. By the time he got to the finals he was 7 foot 4, and still could handle the ball like a guard.

10. Ben Wallace (+3.4 wins) - here's a speech from the 2004 NBA finals, never before reported:

Piston: The Lakers are too many.

BW: Sons of Detroit,

I am Benjamin Wallace...

Piston: Benjamin Wallace

is seven feet tall (including his fro).

WW: Yes, I've heard.

Blocks shots by the hundreds,

and if he were here,

he'd consume the Lakers with fireballs from his eyes

and bolts of lightning from his arse < laughter >

I AM Benjamin Wallace,

and I see a whole army of my countrymen

here in defiance of tyranny.

You've come to fight as free men,

and free men you are.

What will you do with that freedom?

Will you fight?

Piston: Against that? No, we will run, and we will live.

BW: Aye, fight and you may die.

Run, and you'll live... at least a while.

And dying in your beds many years from now,

would you be willing to trade all the days,

from this day to that,

for one chance,

just one chance,

to come back here and tell our enemies

that they may take our lives,

but they'll never take our Championship Trophy!!!

9. Derek Fisher (+3.6 wins)

Fisher has gotten a lot of criticism lately because he's too old andslow to guard speedy point guards. But he had a fine career and was an important part of 5 Laker championship

teams. Shot 40% (37 from 3) for his career, but in the playoffs he upped that to 43% and 40%.

8. Walt Frazier (+4.1 wins) Improved both his volume (PPG) and efficiency (FG%) in leading the Knicks to two championships.

7. John Havlicek (+4.6)

Won 8 championships, including two after the Bill Russell era was over. He took on more of a scoring role in the playoffs, and from what I understand had a pretty important steal in one game.

6. Elvin Hayes (+4.7) This one is a bit of a surprise. He did improve his FG% while shooting more and rebounding more. Won one championship with the Bullets, towards the end of his career.

5. Robert Horry (+4.8)

I would not trust this list if the numbers didn't give me Big Shot Bob. In the playoffs his 3 pointers made jumped from 0.7 per game to 1.1, with an increased percentage, but these numbers can't even tell you how big some of those were, at the last second to win or tie games. Won 7 championships, 2 with the Rockets, 3 with the Lakers, and 2 with the Spurs.

3T. Jojo White and Dave Cowens (+5.0). These were the big stars of the 1970's Celtic championship teams, in between the Russell and Bird eras. White was the 1976 Finals MVP, leading the NBA playoffs in field goals, free throws, and assists that year. For Cowens my numbers show him getting better in the playoffs, Basketball-reference's numbers do not. He did increase his rebounds and scoring, but his FG% is slightly lower.

2. Isiah Thomas (+6.3)

To be frank about it, by regular season numbers Thomas looks like a very good but hardly dominant point guard. 19 points, 9 assists, 45% shooting. You can find a ton of players who could do that. What makes Thomas the all-time great is his playoffs. His WS48 is .146 for the regular season (very good, .100 is average) but .200 for the playoffs (among top 25 all time for all positions). His biggest statistical jump in the playoffs is his rebounding (4.7 vs. 3.5), he scored more while playing on a team that shut down opponents and made every point more valuable. He just got better as the games got bigger, and even a severely twisted ankle could not stop him when he got on a roll.

1. Michael Jordan (+7.3)

It's really not fair - he's already the best player in the game and then when everyone else's stats drop in the playoffs he gets even better? Jordan's WS48 is .286 for the regular season (actually slightly behind David Robinson) but .315 for the playoffs (100 points ahead of the Admiral).

### They Play The Best When It Matters Most

As an analyst, I was first exposed to baseball and only got into basketball later. I've always tried to make an effort to recognize the areas where approaches in one sport do not apply to the other. One of these is the treatment of the playoffs.

In baseball you can safely ignore the playoffs when discussing where players rank on the greatest of all time lists. There are fewer teams in the playoffs, fewer rounds and games, and the regular season is twice as long. I estimate that an average baseball player will play about 1% of his games in the playoffs. The maximum in a season is (as of 2012) is 20, or 12% of the regular season.

In basketball, an average player plays about 7 percent of his games in the playoffs, and a player who maxes out the postseason (going to the finals and playing to 7 games in each series) will play 34% as many postseason games as regular season games. It's even more extreme this year with the abbreviated schedule.

In baseball the argument for mostly ignoring postseason stats in player evaluation comes down to small sample size. There aren't enough games for the stats to be meaningful (look at Billy Hatcher some time) and usually it's such a small amount of games that it's not going to change an evaluation anyway, with Mariano Rivera being a notable exception.

In basketball I think a better argument could be made for looking only at postseason stats than for looking only at regular season stats. In the regular season you have three types of teams:

First tier teams are your championship contenders. These teams know they are going to the playoffs, the only question is which seed they get. Younger teams of this tier might play at max effort and put up 60+ win seasons. As they get older and have taken a few trips to the playoffs, the regular season means less to them, they will approach it more as a tune-up for the real season. Towards the end of Shaq's Laker run, he would use the start of the regular season to have necessary surgery, miss the beginning of the season to rehab, use the rest of the season to get into shape, and come out at full force for the playoffs. You didn't think he was going to do surgery and rehab in the summer, did you? That's his personal time.

Second tier teams actually have something to play for. Some of them are going to make the playoffs as lower seeds, some will stay home. They are unlikely to go far if they get in though, they are mostly fighting to see who gets to be knocked off by the top contenders in the first round.

Third tier teams are the hopeless dregs of the NBA. They can play for pride, they can play because they don't want to keep getting embarrassed, or they can play for pingpong balls.

In contrast, the playoffs are the time when you can be sure everyone is giving max effort and trying their best to win. There have been many years when I barely bother to watch the regular season, and tune in when the playoffs, the real season, starts.

Next post I'll present the 12 players who turned their games up the most in the playoffs.

In baseball you can safely ignore the playoffs when discussing where players rank on the greatest of all time lists. There are fewer teams in the playoffs, fewer rounds and games, and the regular season is twice as long. I estimate that an average baseball player will play about 1% of his games in the playoffs. The maximum in a season is (as of 2012) is 20, or 12% of the regular season.

In basketball, an average player plays about 7 percent of his games in the playoffs, and a player who maxes out the postseason (going to the finals and playing to 7 games in each series) will play 34% as many postseason games as regular season games. It's even more extreme this year with the abbreviated schedule.

In baseball the argument for mostly ignoring postseason stats in player evaluation comes down to small sample size. There aren't enough games for the stats to be meaningful (look at Billy Hatcher some time) and usually it's such a small amount of games that it's not going to change an evaluation anyway, with Mariano Rivera being a notable exception.

In basketball I think a better argument could be made for looking only at postseason stats than for looking only at regular season stats. In the regular season you have three types of teams:

First tier teams are your championship contenders. These teams know they are going to the playoffs, the only question is which seed they get. Younger teams of this tier might play at max effort and put up 60+ win seasons. As they get older and have taken a few trips to the playoffs, the regular season means less to them, they will approach it more as a tune-up for the real season. Towards the end of Shaq's Laker run, he would use the start of the regular season to have necessary surgery, miss the beginning of the season to rehab, use the rest of the season to get into shape, and come out at full force for the playoffs. You didn't think he was going to do surgery and rehab in the summer, did you? That's his personal time.

Second tier teams actually have something to play for. Some of them are going to make the playoffs as lower seeds, some will stay home. They are unlikely to go far if they get in though, they are mostly fighting to see who gets to be knocked off by the top contenders in the first round.

Third tier teams are the hopeless dregs of the NBA. They can play for pride, they can play because they don't want to keep getting embarrassed, or they can play for pingpong balls.

In contrast, the playoffs are the time when you can be sure everyone is giving max effort and trying their best to win. There have been many years when I barely bother to watch the regular season, and tune in when the playoffs, the real season, starts.

Next post I'll present the 12 players who turned their games up the most in the playoffs.

## Sunday, March 4, 2012

### Credit for scorers in Win Shares

Last post was all about the 3 pointer. For this one, lets travel back in time to 1978 and look at only 2 pointers. In addition, free throws are excluded from this discussion. The base credit for scoring in win shares is (points - missed shots). This tells us that a player will earn positive points so long as he shoots better than 33%. Some would tell us that this is terribly wrong and will overrate high volume, low percentage scorers.

This confuses absolute production with marginal production. My comparison is to the Wins produced metric from the Wages of Wins blog, although I don't mean to pick on them, there may be others who do it that way. The metric is (points - shot attempts), so a player will only add positive value if he shoots more than 50%.

Let's build a model:

Our player can shoot 60% if he only takes about 6 shots a game, 500 on the season. He mostly takes layups and dunks in transition, or short jumpers when a double team leaves him wide open. He can opt to take more shots, but if he does the opportunities are not as easy. For the next 100 shots, he only makes 57%. If he chooses to further expand his range, and take shots despite more defensive contest, he only makes 54% of the next 100. And so on. By the time he's up to 2400 shots, he's only able to make 3% of the last, most difficult 100 shots taken.

To maximize his win shares rating, he would take 1300 shots total. He only made 36 of the most difficult 100, and to take another 100 he'd only make 33, and drop his rating slightly. But at 1300 shots, his cumulative shooting percentage is a quite healthy .517. What if we observe this player shooting 1800 shots for a .448 percentage? Win shares will still show a positive total for him, but not as good as if he had stopped at 1300 shots and a .517 percentage.

Wages of Wins will give this player maximum credit up to the point where his last, marginal shots are made at 50%. This happens when he takes only 800 shots, at a cumulative rate of .578. Win shares will say that the 800 shot, .578 player has as much value as a 1900 shot, .434 player. Both are inferior to the 1300 shot, .517 player.

(Let's put some faces to the stats: This is saying Larry Bird > Dominique Wilkins, and Larry Bird > Robert Parish)

The average team last year took 6660 shots, which is 1332 per position. Shots MUST be taken, because there is a 24 second clock in the NBA. If you take a below average number of shots at a high efficiency, then one or more teammates have to make up for this by taking more shots.

I would fully endorse the Wages of Wins model if a team had the option of sitting for 5 minutes in the halfcourt, passing the ball around waiting for the perfect shot. Take away the shot clock and we might see a 19-18 final score again. Since we have the shot clock, taking a shot does not have the opportunity cost that Wins Produced assumes. They give credit for the absolute number of steals, rebounds, assists, etc. It is inconsistent to do this and then only give credit for points when they are scored at an above average rate.

This confuses absolute production with marginal production. My comparison is to the Wins produced metric from the Wages of Wins blog, although I don't mean to pick on them, there may be others who do it that way. The metric is (points - shot attempts), so a player will only add positive value if he shoots more than 50%.

Let's build a model:

Our player can shoot 60% if he only takes about 6 shots a game, 500 on the season. He mostly takes layups and dunks in transition, or short jumpers when a double team leaves him wide open. He can opt to take more shots, but if he does the opportunities are not as easy. For the next 100 shots, he only makes 57%. If he chooses to further expand his range, and take shots despite more defensive contest, he only makes 54% of the next 100. And so on. By the time he's up to 2400 shots, he's only able to make 3% of the last, most difficult 100 shots taken.

To maximize his win shares rating, he would take 1300 shots total. He only made 36 of the most difficult 100, and to take another 100 he'd only make 33, and drop his rating slightly. But at 1300 shots, his cumulative shooting percentage is a quite healthy .517. What if we observe this player shooting 1800 shots for a .448 percentage? Win shares will still show a positive total for him, but not as good as if he had stopped at 1300 shots and a .517 percentage.

Wages of Wins will give this player maximum credit up to the point where his last, marginal shots are made at 50%. This happens when he takes only 800 shots, at a cumulative rate of .578. Win shares will say that the 800 shot, .578 player has as much value as a 1900 shot, .434 player. Both are inferior to the 1300 shot, .517 player.

(Let's put some faces to the stats: This is saying Larry Bird > Dominique Wilkins, and Larry Bird > Robert Parish)

The average team last year took 6660 shots, which is 1332 per position. Shots MUST be taken, because there is a 24 second clock in the NBA. If you take a below average number of shots at a high efficiency, then one or more teammates have to make up for this by taking more shots.

I would fully endorse the Wages of Wins model if a team had the option of sitting for 5 minutes in the halfcourt, passing the ball around waiting for the perfect shot. Take away the shot clock and we might see a 19-18 final score again. Since we have the shot clock, taking a shot does not have the opportunity cost that Wins Produced assumes. They give credit for the absolute number of steals, rebounds, assists, etc. It is inconsistent to do this and then only give credit for points when they are scored at an above average rate.

## Saturday, March 3, 2012

### Value of the 3 pointer

In a basketball overall rating system, how much should the 3 point shot be valued? The easy answer is "3 points", and that's the way I've done it for a long time, but it actually doesn't quite work.

Ratings that incorporate all the box score stats will give you credit for scoring points, and then remove credit for missing shots. In the Tendex system, developed by Dave Heeren back in the 1980's (or maybe before, I read his books in the 80's) a player taking 10 shots and making 6 of them will get a credit of 8 - 12 points scored minus 4 missed shots. A player taking 10 3 pointers and making 4 of them scores an identical 12 points on the same number of shots, but he misses 6 so his credit is only 6. Further compounding the imbalance is that if these two players are playing each other, 1 on 1, and all missed shots are rebounded by the defender, the guy taking two pointers will get 6 rebounds, putting him up to 14 credits, while the 3 point shooter only gets 4 boards (and 10 credits).

Not all of the statistical systems work that way. The Wages of Wins system gives .032 for a 2 pointer and double that, .064 for the 3 pointer. So these players are getting even credit for the points and missed shots, but even still they are out of balance when you consider the rebounds.

Let's try an experiment. One team shoots 50%, taking only 2 pointers. Assume no free throws are taken, no turnovers from either team, and rebounds are split at a league average rate (27/73) between the offensive and defensive teams. What shooting percentage would a team have to shoot from 3 pointers to tie the score?

If you said 33.3%, then you are wrong. That only works if every rebound is by the defensive team. The actual answer is about 30.5%. When the teams score, then they are giving the ball to the other team 100% of the time. When they miss, they get the ball back 27% of the time. So if you can score the same number of points per shot, and do so while missing more shots, then you are going to get more shot attempts than the other team, and score more points overall.

In light of this, as the starting point for my win shares formula I will now add 0.9 for every three pointer to balance out the formula. For offense it is (PTS + 3pt made *.9 - (fga-fg) - (fta-ft)*.5)

Ratings that incorporate all the box score stats will give you credit for scoring points, and then remove credit for missing shots. In the Tendex system, developed by Dave Heeren back in the 1980's (or maybe before, I read his books in the 80's) a player taking 10 shots and making 6 of them will get a credit of 8 - 12 points scored minus 4 missed shots. A player taking 10 3 pointers and making 4 of them scores an identical 12 points on the same number of shots, but he misses 6 so his credit is only 6. Further compounding the imbalance is that if these two players are playing each other, 1 on 1, and all missed shots are rebounded by the defender, the guy taking two pointers will get 6 rebounds, putting him up to 14 credits, while the 3 point shooter only gets 4 boards (and 10 credits).

Not all of the statistical systems work that way. The Wages of Wins system gives .032 for a 2 pointer and double that, .064 for the 3 pointer. So these players are getting even credit for the points and missed shots, but even still they are out of balance when you consider the rebounds.

Let's try an experiment. One team shoots 50%, taking only 2 pointers. Assume no free throws are taken, no turnovers from either team, and rebounds are split at a league average rate (27/73) between the offensive and defensive teams. What shooting percentage would a team have to shoot from 3 pointers to tie the score?

If you said 33.3%, then you are wrong. That only works if every rebound is by the defensive team. The actual answer is about 30.5%. When the teams score, then they are giving the ball to the other team 100% of the time. When they miss, they get the ball back 27% of the time. So if you can score the same number of points per shot, and do so while missing more shots, then you are going to get more shot attempts than the other team, and score more points overall.

In light of this, as the starting point for my win shares formula I will now add 0.9 for every three pointer to balance out the formula. For offense it is (PTS + 3pt made *.9 - (fga-fg) - (fta-ft)*.5)

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