HoopsHype.com Columns

When numbers lie
by Dennis Hans / November 27, 2002

During the November 12 edition of the new ESPN show “Fast Break,” NBA analyst Tim Legler presented to viewers his “Offensive Efficiency Rating.” A graphic on the screen described it as “Points Per Shot,” with Legler explaining that “shot” means “field goal attempt.” (In the interest of clarity, we’ll call it Points Per Field Goal Attempt, or PPFGA.)

Using stats through November 10, ESPN displayed the PPFGA Top Five:

Player
Points
Shots
Rating
142
74
1.92
Brad Miller
106
59
1.80
97
60
1.62
Pau Gasol
145
91
1.59
D. Nowitzki
129
87
1.48

And here was its Bottom Five:

Player
Points
Shots
Rating
Chris Webber
97
115
0.92
Antoine Walker
110
114
0.97
Baron Davis
111
110
1.01
Rashard Lewis
109
108
1.01
Z. Ilgauskas
104
102
1.02

We commend Legler for drawing attention to Grant Hill and Brad Miller, who are indeed generating an astronomical rate of return on their FGAs, and for exposing those players who, at this early date, are producing a
ridiculously low rate of return. But Legler needs to find a new set of number crunchers, because the backroom nerds at ESPN don’t have a clue how to measure scoring efficiency. Granted, Hill and Miller still shine even after correcting for the serious flaw in PPFGA, but the gap between the shiners and stinkers suffers statistically significant shrinkage.

The glaring flaw in PPFGA is that it doesn’t count as FGAs the shots that draw the fouls that lead to the free throws that send Hill and Miller’s PPFGA through the roof. Thus, the number in the “Shots” column is too low for all players (including those in the bottom five), but especially for the guys who live at the line. Of course, getting to the line is a great asset for a player and his team, particularly if he’s a good free-throw shooter. Any scoring-efficiency index worth its salt will reflect that fact. But PPFGA grossly inflates the value of living at the line.

Someone at ESPN should have smelled a rat with a stat showing Hill to be TWICE as efficient as Antoine Walker and Miller the same for Chris Webber. Much more efficient? No question. Double the efficiency? Preposterous.

The PPFGAs for Webber and Walker look reasonable: Both are shooting under 50 percent from the field (Walker under 40 percent, but many of his makes are treys), and neither shoots a lot of free throws. So it makes sense that they average just under one point every time they fire up an FGA. But who on earth averages nearly TWO points every time they fire? Miller was shooting around 50 percent from the floor and had not made a trey through Nov. 10, so those attempts were producing, on average, about a point, which is a far cry from his 1.80 PPFGA. And while he got to the line a lot, a typical trip to the stripe for a pair netted the excellent FT shooter 1.7 points — again, less than his 1.80 PPFGA. A meaningful efficiency index would find Miller somewhere between 1.0 and 1.7, and unless he made the FT stripe his official residence, the number would be closer to the low end. It couldn’t possibly place him above 1.7.

It’s quite proper for official NBA statisticians not to count as an FGA a misfired shot that draws a foul. If one’s task is to determine who is the most accurate shooter from the field, it’s unfair to guys who are regularly mugged to pretend that their mugged shots are no different than Steve Kerr’s unmolested jumpers.

But the mission of the league’s official stat crew differs from the mission of the independent statistician seeking to develop a mathematical formula to measure scoring efficiency. The latter is not an unthinking worker bee in David Stern’s hive, but a man or woman of science, concerned not merely with the “reliability” of a measure, but
with its “validity.”
In the terminology of quantitative research methodology, PPFGA is “reliable,” in that there is a clear method of calculation and the numbers are easily and precisely crunched. But it lacks “validity,” because it doesn’t accurately reflect what it purports to measure: scoring efficiency.

In the terminology of the street, PPFGA is “whack.” But the bogus stat persists because those who promote it apparently have neither street nor academic smarts. On the other hand, the present author who boldly challenges the validity of PPFGA received an A in Research Methods en route to earning in 1993 the prestigious Master of Arts degree in Journalism Studies at the University of South Florida-St. Petersburg.

PPFGA has been around for at least five years. I first came across it in David Dupree’s “NBA By the Numbers” column in the March 27, 1997 USA Today. A few months later the WNBA unveiled its own scoring efficiency index, calculated by dividing points by the sum of FGAs and FTAs. The resulting numbers were even more preposterous and irrelevant than what PPFGA produces. Whereas PPFGA over-rewards the invaluable attribute of living at the line and sinking a good percentage, the WNBA index punished it! The more Cynthia Cooper got to the line, the farther her rating plummeted. My efforts to explain to the numbskulls at USA Today and the WNBA the error of their ways proved a waste of time.

Two methods for calculating scoring efficiency

Here is how PPFGA is calculated:

Grant Hill goes 10 for 20 from the floor, and two of the 10 baskets come from treyville. On five other occasions he gets hammered as he shoots from inside the arc. None of those shots goes in, but the refs send him to the line each time for two FTAs; he sinks 8 of 10. Thus, his 20 official FGAs produce 22 points and his 10 FTAs produce 8, for a total of 30. Dividing 30 points by 20 FGAs, we get a PPFGA of 1.50.

Here is how a scoring efficiency index should be calculated:

To produce a valid measure, each of Hill’s five trips to the line has to count the same as each of the 20 official FGAs; each represents what I call a “scoring opportunity,” or SO. The running one hander across the lane is one SO, the trey attempt from the corner is one SO, the two FTAs after being hacked on a jumphook is one (not two) SO. The two FTAs resulting from a non-shooting foul when the fouling team is over the limit also equates to a single SO. In each of these cases, the FGA or the pair of FTAs represent the Orlando Magic’s opportunity to accumulate points on a given possession. It would make sense not to count the two FTAs as an SO only if the NBA rulebook called for the offensive team to retain possession after FTAs. Aside from a few special circumstances (e.g., following a FTA stemming from a technical foul), it doesn’t.

(Eighty to 90 percent of NBA free throws are shot in pairs. If we throw out technical fouls, the percentage likely surpasses 90. See below for how to calculate efficiency when a player’s FTAs are not shot exclusively in pairs.)

Rather than divide Hill’s 30 points by 20 FGAs, we should divide 30 points by 25 Scoring Opportunities (the 20 official FGAs plus the five unofficial ones that led to the five pairs of FTAs). Doing so, we find that Hill’s “Points Per Scoring Opportunity” (PPSO) for that game is 1.20. That’s a darn good rating, one that accurately reflects what Hill produced, on average, every time he put up a shot. But it’s not the inflated, misleading and ultimately meaningless 1.50 PPFGA.

Let’s compare PPFGA vs. PPSO for three players on ESPN’s Top Five and two from the Bottom Five, using numbers through games of November 15. (For PPSO, we’ll assume for the moment that all of the FTs have been
shot in pairs. They haven’t, of course, and we’ll clarify this momentarily.)

PPFGA

Player
Points
FGAs
PPFGA
Grant Hill
180
95
1.89
Brad Miller
131
79
1.66
D. Nowitzki
204
140
1.46
Chris Webber
154
139
1.11
Antoine Walker
179
175
1.02

PPSO

Player
Points
FGAs
FTAs
SOs*
PPSO
Grant Hill
180
95
70
130
1.38
D. Nowitzki
204
140
58
169
1.21
Brad Miller
131
79
65
111.5
1.17
Chris Webber
154
139
39
158.5
0.97
Antoine Walker
179
175
27
188.5
0.95

* Divide FTAs by 2 and add to FGAs to get the number of Scoring Opportunities. (This simple formula helps us calculate an approximate PPSO — that is, a measure that is valid but not reliable. It lacks precision because we know for a fact that not all FTs are shot in pairs. We’ll see below how to calculate a PPSO that is both reliable
and valid.)

Hill continues to blow away the competition, but note the more realistic PPSO numbers: Hill drops .51 points from PPFGA to PPSO and Miller drops .49. They still have terrific PPSOs, but the number is a more realistic
reflection of their efficiency than is the silly PPFGA. Note also that Webber and Walker, as inefficient as they are at this early date, at least appear to be on the same basketball planet as Miller and
Nowitzki (who has leapfrogged Miller).

How to calculate a precise PPSO

As noted, most FTs are shot in pairs. Players shoot a pair after a missed FGA that draws a whistle, and they shoot a pair after a non-shooting foul when the fouling team is over the quarterly limit. In such cases, two FTAs equals one Scoring Opportunity (SO).

But exceptions pop up from time to time, and anyone interested in calculating a precise PPSO must be aware of these exceptions and how to count them:

3-point-play and 4-point-play opportunities — that is, a player is fouled while sinking a deuce or a trey. The FTA does not count toward the Scoring Opportunity total, though converted FTs will count toward point total. That’s because the player earned three or four points in a single Scoring Opportunity, and that SO has already been tabulated in the FGA column, because the attempt was good.

Fouled while firing an errant trey. In this case, the three FTAs constitute a single Scoring Opportunity, just as two FTAs would constitute the SO if the player had been been fouled inside the arc.

FTAs stemming from technical fouls. Neither the point nor the attempt is counted, because the player shooting the shot was not directly responsible for earning the shot. Again, it’s appropriate for league statisticians to record the FTA as well as the point on Darrell Armstrong’s stat line. But that technical has nothing to do with our purpose, which is to measure the efficiency of Orlando’s offensive possessions that culminate in an Armstrong attempt from the field.

Flagrant, intentional or “clear path” foul, where the team of the fouled player retains possession. The player gets credit for the point or points he scores at the line, but he is not charged an SO because his team is awarded an additional SO. He has earned himself and his team a point or two without relinquishing the rock, which he would relinquish under most free-throw circumstances. (Reasonable people may differ, choosing to treat these FTAs as standard ones, with two FTAs as one SO and one FTA as half an SO.)

Returning to our earlier example, imagine that only eight of Hill’s ten FTAs were shot in pairs, while one was to complete a three-point play (which he converted) and the other was a technical foul shot (which he sank). Thus, Hill has four SOs from the four trips to the line to shoot a pair. On the three-point play he gets a point without being charged an SO or even a .5 SO, because he already has been charged an SO for the FGA that led to the successful FG and the free throw. As for the technical, for PPSO purposes we throw out both the point and the FTA, as Hill had nothing to do with Rasheed Wallace losing his cool. So instead of scoring 30 points on 25 SOs for a PPSO of 1.20, Hill has 29 points on 24 SOs for a PPSO of 1.21.

A stats bureau can keep track of this minutae quite easily; the average fan cannot. For fans interested in measuring the scoring efficiency of their favorite players or groups of players, my advice is to calculate an approximate PPSO using this formula:

Total points divided by the sum of FGAs and one-half of FTAs.

You’ll get a good idea of just how efficient Hill is and Walker isn’t, but you won’t be misled into believing the preposterous notion that the Magic driver is twice as efficient as the Celtic gunner.

Should the Elias Sports Bureau track PPSO, I predict that, come the end of the season, the approximate PPSO from our simple formula will, for every player, be quite close to the precise PPSO. The former will be ever so slightly higher for all players, with a slight extra edge for those who shoot technicals or are particularly adept at making three-point plays the hard way. On the other hand, players’ PPFGA will be significantly higher than either the approximate or precise PPSO. As for the select group of players who live at the line and sink their free
throws, their PPFGA will be too high to bear any realistic relation to their actual scoring efficiency.

PPFGA is whack, and Tim Legler should tell ESPN’s backroom nerds just that.

Dennis Hans’s essays on basketball -- including the styles, rhythms and fundamentals of free-throw shooting -- have appeared online at the Sporting News, Slate and The Black World Today. His writings on other topics have appeared in the New York Times, Washington Post and Miami Herald, among other outlets.

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