Last season, North Carolina’s John Henson made 47.9% of his 142 free throw attempts. What is a reasonable expectation for Henson’s free throw percentage this season?

One way to make a prediction is to look at players with a similar free throw percentage in a single season and see how they performed in the following season. Since 2005, there have been 24 players that have shot at least 120 free throws in a season, were within four percentage points of Henson’s accuracy and took at least 100 free throws the following season.

This is that group sorted by the player’s free throw percentage in the following season:

    Player            Yr1   Yr2 
08  Nick Murphy       51.3% 64.9%
07  Thomas Sanders    50.0  63.4
06  Will Thomas       51.6  62.2
10  Olu Ashaolu       47.2  60.3
05  Darius Glover     47.1  59.3
09  Dinma Odiakosa    46.2  57.7
05  Corey Rouse       50.0  57.3
07  Shaun Pruitt      51.3  56.4
07  John Fields       45.5  54.3
07  Trent Plaisted    49.2  54.2
06  Juwann James      45.9  54.1
05  David Berghoefer  47.7  53.7
06  Ryan Bright       44.7  53.6
05  Joe Martin        51.3  53.1
10  Damian Saunders   48.4  52.3
10  Hamidu Rahman     51.5  51.8
08  Marcus Lewis      45.5  50.3
07  Jermaine Griffin  50.8  50.0
05  Bruce Brown       45.6  46.0
07  Lamar Sanders     51.3  44.7
05  Dillion Sneed     48.0  38.7
07  Joey Dorsey       46.7  37.8
06  Durrell Vinson    48.1  32.5
08  Arinze Onuaku     44.5  29.8
    AVERAGE           48.3  51.6

As with many basketball skills, it is rare to show massive improvement from one season to the next in free throw shooting. Even though the average doesn’t change much, there is a range of possibilities, from Jacksonville State’s Nick Murphy, whose 51% mark as a freshman rose to 65% as a sophomore and eventually 75% as a senior, to noted masonry aficionado Arinze Onuaku who stretched the bounds of human performance by missing over 70% of his attempts as a junior.

Based on this analysis, the future doesn’t look all that bright. But as Carolina fans would be quick to point out were the investigation to end here, Henson might be a special case because he had a split personality at the free throw line, producing one of the most noteworthy mid-season improvements in college hoops last season*. And I suppose it’s possible he had an epiphany sometime between February 16th and February 22nd.

            FTM-FTA  Pct
Before 2/22  39-101 .389
After  2/22  31-47  .660

The issue at hand is whether this is a statistical artifact or whether Henson did, in fact, consume some magic beans (or learn some kind of “secret”) sometime during the third week in February. In order to get a handle on this, we need to know how well Henson’s metamorphosis can be explained by chance. If Henson’s true ability last season was to make 47.9% of his free throws, what are the chances he would luck into a stretch of making 31-of-47?

In order to do this, I simulated John Henson’s free throw shooting from last season a million times using my shooting simulation program, RandoShoot 3000TM. It’s the same kind of cutting-edge technology that allowed us to guess at Derrick Williams’ chance of breaking the all-time three-point accuracy record late last season. I assumed that Henson had a 47.9% chance of making a free throw, and that he would shoot exactly 142 free throws in a season. For each of one million simulated seasons, RandoShoot 3000TM recorded Henson’s best performance over any 47 consecutive free throws. The results follow.

Most free throws made in any 47 consecutive attempts (with number of cases each was observed)

14:      1
15:      0
16:      3
17:     23
18:    133
19:    621
20:   2398
21:   7402
22:  19380
23:  41438
24:  73907
25: 110703
26: 141340
27: 153992
28: 143412
29: 116909
30:  83329
31:  51890
32:  29173
33:  14189
34:   6217
35:   2397
36:    797
37:    265
38:     68
39:     10
40:      1
41:      2

Even a 47.9% shooter would be expected to have a stretch as good as 27-of-47 (57.4%) at some point during the season. But he would only have about a 10% chance to have a run as good or better than Henson’s closing 31-of-47. From a statistician’s perspective, that’s not exactly convincing evidence that Henson’s finish was anything but random noise. Of course, the case in question is a bit more unusual since Henson’s hot streak came at the very end of the season.

One other way to attack this is to examine history, looking for shooters that finished the season shooting much better than they started and then checking how they shot in the following season. So I went back to the 2010 season for which I have nearly every play-by-play on file. I looked for players with similar splits to Henson. The rules I established were the following…

– Player must have shot at least 100 free throws in the 2010 and 2011 seasons.
– A point in the season can be identified where the player shot at least 20% better after that point than before.
– The sample size after the identified point must cover at least 35 free throws but no more than half of a player’s season total.

The results are pretty amazing. The player with the biggest transformation in 2010 was Quinnipiac’s Justin Rutty. He made exactly half of his first 88 attempts that season before closing the season on an astounding 51-of-63 (81.0%) run.  Clearly, any tests for significance would say “that’s significant”. You don’t flip a coin 63 times and get heads 51 of those times without something being messed up with the coin.

However, somehow Rutty fooled his own ability. In two prior seasons, he had posted season totals of 36.9% and 41.9%. The hot streak in his junior season lifted his FT% to 62.9. Given the closing performance in 2010, it would have been reasonable to assume that he would be closer to, or even better than, his season-long junior year performance as a senior. But in 114 attempts last season, he made just 44.7% of them. (Rutty did have bone chips in his non-shooting elbow which required surgery in the middle of his senior season, but he shot no better post-surgery.)

Rutty’s case is fascinating and actually, it’s somewhat unusual. But the collective performance of the 15 players identified by my criteria indicates that a late and sudden improvement in free throw accuracy isn’t as meaningful as you might think. Here’s a list of all of the players that had a 20% jump in their FT% at some point late in 2010 and how their season-long FT% compared between 2010 and 2011.

Player           Imp  2010  2011   Chg 
Justin Rutty    .310  .629  .447  -.182
Herb Pope       .271  .496  .548  +.052
Orlando Johnson .257  .703  .804  +.101
Kendall Cutler  .247  .699  .686  -.013
Jake Cohen      .244  .709  .773  +.064
Brian Conklin   .236  .643  .622  -.021
Kwadzo Ahelegbe .221  .739  .786  +.047
Hillary Haley   .220  .703  .729  +.026
Devin Brown     .220  .773  .806  +.033
Jay Couisnard   .208  .718  .637  -.081
Casper Ware     .207  .784  .810  +.026
Verdell Jones   .205  .758  .672  -.086
Reggie Jackson  .204  .733  .796  +.063
Noah Dahlman    .203  .702  .758  +.056
Shawn Lewis     .203  .727  .714  -.013
                         AVERAGE  +.005

Imp = Improvement in FT% after epiphany point
Chg = Difference in FT% between 2011 and 2010

Some guys on the list improve and some guys don’t, but in reality there’s no evidence that a late hot streak has much more predictive value than a player’s season-long FT%. Herb Pope is the most comparable player to Henson. The two have astonishingly similar splits. Pope closed on a 32-of-49 (65.3%) streak after starting 26-of-68 (38.2%) but he was just slightly better than his overall 2010 percentage last season.

Henson will probably do better than last season, because as shown in the first table, guys who shoot as poorly as he did for a whole season tend to improve a little. It’s simple regression to the mean. Or in regular-person’s terms, there were a lot of things to fix in Henson’s stroke and he’s bound to fix some of that. The relative hot-streak to close last season may have partially been a result of that. (Still though, it could have just been noise.)

A reasonable expectation for Henson’s FT% is somewhere in the low 50s this season. You can’t completely rule out that he’ll sustain his late-season run in the mid-60s for an entire season, because there are cases of players improving like that. But that outcome is just as likely to occur as Henson channeling his inner Justin Rutty.

*Henson’s turnaround was the third-highest in D-I last season. East Tennessee State’s Adam Sollazzo started 83-of-138 (60.1%) before finishing a sizzling 37-of-39 (94.9%).

(Update: Henson weighed in on twitter. Hilarious.)