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NBA BASKETBALL March 2, 2002
Ray Allen tops Dirk and Peja as NBA's most efficient gunner

New stat, Points Per Scoring Opportunity, incorporates deuces, treys and
free throws. While Allen leads those who average at least 22 points,
Reggie Miller and Steve Nash would rank 1-2 if cutoff were 16 ppg.
Brent Barry dominates the 15 points and under crowd.

Among NBA players averaging at least 22 points, none scores as
efficiently as Ray Allen, according to a new statistic, Points Per
Scoring Opportunity (PPSO).

PPSO is not concerned with assists, rebounds and making your teammates
better. That's the job of Jason Kidd an Old Man Jordan. PPSO seeks to
answer one question: Who gets the most bang for his shooting bucks.
That's deuces, treys and free throws. Each is incorporated and given
its due weight. PPSO allows us to compare the inside banger who lives
at the line with the perimeter marksman with an instinct for

Back in the day, field goal percentage told a big part of the efficiency
story. But that was before someone painted an arc on the court. Today,
to compare Shaq and Peja from the floor, you have to count every Peja
trey as equal to 1.5 of Shaq's deuces.

That gets us closer to the truth, but we're still not accounting for the
fact that perimeter types tend to shoot far fewer free throws than paint
dwellers, and it's at the line where leading scorers not named Shaq make
hay. Tim Duncan, for example, averages 9 free-throw attempts (FTAs) per
game. Every time he shoots a pair, the .812 FT shooter delivers 1.624
points to the Spurs cause. That's far more than he contributes on a
typical un-fouled field goal attempt (FGA). Even with a solid .490 FG
mark, that works out to just .98 points per attempt. Clearly, Duncan's
FTAs are a huge factor in his overall scoring efficiency. Ditto for
Jerry Stackhouse, Karl Malone and Paul Pierce.

Points Per Scoring Opportunity (PPSO) assumes that a pair of FTAs equals
one FGA, for reasons explained below. Thus, a "scoring opportunity"
consists of a deuce attempt, a trey attempt, or a pair of FTAs. To
calculate a player's scoring opportunities, divide FTAs by two (which
will give you the number of "pairs" he has attempted) and add that
number to his FGAs. The PPSO is determined by dividing points by
scoring opportunities.

If Kobe Bryant goes 9 of 17 from the field, including 1 of 3 from
treyville, and 5 of 6 from the stripe, he has scored 24 points on 20
scoring opportunities (17 FGAs + 3 pairs of FTAs). Dividing 24 by 20,
we get an excellent one game PPSO of 1.2. If Kobe duplicated those
numbers every night, he'd lead all big-time scorers (22+ points per
game) in PPSO. Here are the NBA's top 16 scorers, ranked in order of
PPSO. (Don't forget to cut a little slack to the go-to guys who get
stuck with the ball when the play has broken down and/or the shot clock
is about to expire. This makes the 6-to-9 rankings of Bryant, Pierce,
Payton and McGrady all the more impressive.)

NBA's Top 16 Scorers, Ranked in Order of PPSO

Player Scoring Opps Total Pts PPSO
Allen, Mil 818 970 1.186
Nowitzki, Dal 1052 1231 1.170
Stojakovic, Sac 965 1125 1.166
O'Neal, LAL 867.5 967 1.115
Duncan, S.A. 1161 1282 1.104
Bryant, LAL 1207.5 1303 1.079
Pierce, Bos 1288 1389 1.078
Payton, Sea 1116 1172 1.050
McGrady, Orl 1168 1221 1.045
Malone, Utah 1202 1256 1.045
Webber, Sac 646.5 675 1.044
Stackhouse, Det 1017.5 1058 1.040
Carter, Tor 1232.5 1250 1.014
Walker, Bos 1236.5 1195 0.966
Iverson, Phil 1380 1302 0.943
Jordan, Wash 1314 1215 0.925

Why Two Free Throws Equals One Field Goal Attempt

I count a pair of free throws as one scoring opportunity because roughly
90 percent of NBA FTAs are shot in pairs. That pair constitutes a
team's scoring opportunity on that possession, just as a jumphook or
heave from treyville constitutes the scoring opportunity on other
possessions. Yes, there is the occasional "and one" attempt to complete
a 3-point play as well as single shots after a technical foul, illegal
defense or a second delay-of-game infraction. And once in a blue moon,
someone is fouled on an attempted trey, which results in three FTAs on a
missed trey and one FTA on a sunk trey. But the pair is the meat and
potatoes of free throws.

In 1996, when I first developed PPSO, I charted two NBA games
(Bulls-Magic, November 5; Magic-Cavs, November 19). Ninety-eight of 110
FTAs were shot in pairs, or 89.1%; toss out the four technicals and it
was 92.5%. Of the 110 FTAs, only eight were attempts to complete
3-point plays; no player was fouled firing a trey. I also charted two
WNBA games. Sixty of the 63 FTAs -- 95% -- were shot in pairs.

A Precise PPSO

Ideally, stats should be precise, not "pretty darn accurate." But
calculating a precise PPSO would require extensive research or the
computer wizardry of The Simpsons' Professor Frink. Should
Springfield's nutty professor be up to the programming challenge, I
would advise the following:

. Strip players of all FTAs and points stemming from technicals. The
guys who shoot technicals don't "earn" them in the sense that Iverson
earns a pair every time he's bludgeoned driving the lane.

. On "and one" three-point plays, the player is entitled to the third
point, but the FTA should not be included in calculating the number of
FTA pairs. The single "scoring opportunity" on that possession consists
of both the successful FGA and the one FTA, so don't saddle him with an
additional 0.5 scoring opportunity.

. Any time a player is awarded three shots, all three should count as a
single "pair," as those three attempts constitute the scoring
opportunity on that particular possession.

The sane person, facing such a monumental task, will say "Ninety percent
of FTAs are shot in pairs. Let's round that off to 100 percent and be
done with it." Yes, the pair rule imposes a tiny penalty on every
player. But the difference between the "most tiny" and "least tiny"
penalty is, well, tiny.

Fun With Numbers

The tiny imperfection of PPSO would slightly disadvantage Shaq if
compared to some perimeter dude who gets to shoot lots of technicals and
rarely gets fouled while sinking a field goal. But you can't go even a
wee bit wrong comparing compare like players. Shaq and Duncan aren't
getting fouled while firing treys. Sharpshooters Stojakovic and Allen
each get to shoot their team's T's. So too do exciting slashers Kobe,
Pierce, Stackhouse and Iverson.

A fun way to use PPSO is to imagine that each team gets 100 scoring
opportunities per game (which is fairly close to the mark). What would
happen if the Lakers faced Philly and Kobe and Iverson took all the
shots. If each matched his average, the Lakers would win 108 to 94. If
Nash squared off with Ray Allen, the Mavs would nip the Bucks 120 to
119. But if it were Allen vs. Nowitzki, the Bucks would win 119 to
117. Pray for a Dallas-Milwaukee championship series.

Points Per Scoring Opportunity, 2001-2002 (through Feb. 17, 2002)

Scoring Playmakers (Minimum 19 points per game average)

Player Scoring Opps Total Pts PPSO
Nash, Dal 861 1032 1.199
Cassell, Mil 848 918 1.082
Francis, Hou 634.5 682 1.075
Payton, Sea 1116 1172 1.050
Marbury, Phoe 985 1013 1.028
Davis, Char 1001.5 989 0.988

Old School Playmakers
Stockton, Utah 580.5 669 1.152
Kidd, N.J. 778 721 0.927

Backcourt Bullseyes
Barry, Brent, Sea 545 709 1.301
Miller, Ind 707.5 881 1.245
Allen, Mil 818 970 1.186
Smith, S.A. 526.5 612 1.162
Szczerbiak, Minn 869 991 1.140
Houston, N.Y. 985 1071 1.087
Bryant, LAL 1207.5 1303 1.079
Wells, Port 706.5 760 1.076

Frontcourt Efficiency Fiends (Minimum 16 points per game average
Marshall, Utah 471.5 558 1.183
Brand, LAC 842 962 1.143
Gasol, Mem 823 909 1.104

Shot selection? What's that?
Williams, Mem 691.5 630 0.911
Iverson, Phil 1380 1302 0.943
Sprewell, N.Y. 1017.5 981 0.964
Van Exel, Den 999 965 0.966
Walker, Bos 1236.5 1195 0.966

Battle of the Barry Brothers: Few "third bananas" as efficient as
Rick's kids
Barry, Brent, Sea 545 709 1.301
Barry, Jon, Det 319 400 1.254

Can play for my team despite putrid PPSO
Jordan, Wash 1314 1215 0.925
Kidd, N.J. 778 721 0.927
(Career wise, Jordan dominates high-scoring guards in PPSO. But this
season he's shooting .419 from the floor, with very few treys to boost
his True Percentage. Also, he visits the FT line less frequently these

Developing PPSO Back in the Day

Several years ago it occurred to me that field goal percentage had
become an imperfect measure of efficiency from the floor, given that
some players were firing 25 to 50 percent of their field goal attempts
(FGAs) from beyond the three-point arc. It didn't seem fair to compare
Shaq's astronomical FG percentage to Reggie Miller's or Steve Kerr's.
So I came up with Points Per FGA. If Shaq went 12 for 20 from inside
the arc, his 20 FGAs produced 24 points, or 1.2 points per FGA. If
Reggie went 9 for 20 from the field, including 4 of 9 from treyville,
his 20 FGAs produced 22 points (10 on the 5 deuces, 12 on 4 treys), or
1.1 points per FGA. While Shaq's efficiency was accurately rendered by
his 60% FG rate, Reggie was shortchanged by his 45% FG rate. In fact,
he produced the same number of points as an inside-the-arc 55% shooter.

I was quite pleased with myself for inventing an improvement on the
grand old FG%, but a trip to the bookstore revealed I had reinvented
Rick Barry's wheel. His annual book evaluating NBA players included
something called "True Percentage," which calculated exactly what I had

I looked at my calculations and discovered that a couple mediocre
players for the local Orlando Magic had an impressive PPFGA/True
Percentage. Might there be something misleading about a statistic that
made Dennis Scott and Nick Anderson look good? Indeed there was. Both
of these players had fallen into the bad habit of living behind the
line. Rare is the game when someone gets fouled firing a trey, and free
throws are critical to offensive efficiency. Yet these guys almost
never got to the line, and they were useless when it came to helping to
get the opposing team into the penalty situation.

The most efficient NBA possession is one that results in two free
throws. The league average hovers around 75 percent year after year, so
that trip to the stripe produces, on average, 1.5 points. That's the
equivalent of a 75 percent deuce attempt or a 50 percent trey attempt,
and no team has ever approached such percentages from the field.

While Scott and Anderson never got to the line, the likes of Karl Malone
and Kevin Johnson lived there. When I learned that about 90 percent of
FTAs are shot in pairs, I had my stat: A pair of FTAs was roughly
equivalent to an FGA; either constituted a team's "scoring opportunity"
on a given possession. Malone and Johnson had slightly higher PPFGAs
than the Orlando duo. But when I factored in the FTAs and the points
produced by those attempts, Malone and Johnson left Scott and Anderson
in the dust. Points Per Scoring Opportunity was born, and I learned
that no other NBA 20+ points per game scorer came close to Johnson's
1.229 PPSO. Not Shaq, not Hakeem the Dream, not even Jordan (though all
had excellent marks). Only WNBA superstar Cynthia Cooper was in KJ's

1996-97 PPSO, Gunners' Division (Minimum 20 ppg average)

Player Scoring Opps Total Pts PPSO
Johnson, Phoe. 1147.5 1410 1.229
Cooper, Comets 505.5 621 1.228
Rice, Char. 1780.5 2115 1.188
Malone, Utah 1916 2249 1.174
Miller, Ind. 1531.5 1751 1.143
Richmond, Sac. 1843.5 2095 1.136

Sprewell, G.S. 1736.5 1938 1.116
Jordan, Chi. 2180 2431 1.115
Olajuwon, Hou. 1649 1810 1.098
Pippen, Chi. 1511.5 1656 1.096
Webber, Wash. 1323.5 1445 1.092

Hardaway, Orlando 1113.5 1210 1.087
O'Neal, LAL 1230.5 1336 1.086
Hill, Det. 1575.5 1710 1.085
Smith, Atl. 1341.5 1445 1.077
Payton, Sea. 1659.5 1785 1.076

1996-97 PPSO. Minimum 850 Scoring Opportunities

Player Scoring Opps Total Pts PPSO
Stockton, Utah 921.5 1183 1.284
Mullin, G.S. 898.5 1143 1.272
Johnson, Phoe. 1147.5 1410 1.229
Hawkins, Sea. 942.5 1139 1.208
Rice, Char. 1780.5 2115 1.188

Malone, Utah 1916 2249 1.174
Hornacek, Utah 1019 1191 1.169
Dumars, Det. 1003 1158 1.155
Miller, Ind. 1531.5 1751 1.143
Mason, Char. 1039 1186 1.141

Bio: Dennis Hans is an occasional college professor, freelance writer,
unlicensed statistician, rec-league legend and self-proclaimed shooting
guru. His essays on pro basketball have appeared online at the Sporting
News, Slate, InsideHoops.com and The Black World Today (tbwt.com). His
writings on other topics have run in the New York Times, Washington Post
and Miami Herald. He can be reached at HANS_D@popmail.firn.edu.

©2002 by Dennis Hans, reprinted with permission


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