How Much of an Advantage is Hosting Middle Three Games in NBA Finals 2-3-2 Format?

Since 1985, the NBA has used a 2-3-2 format for the final series. In other words, for a theoretical seven-game series, the team with the better regular-season record hosts the first two games, then the next three games are played in the other team's arena, with the final two games returning to the home of the team with the better regular-season record.

A question that sometimes comes up is whether having three games in a row at home (as opposed to a 2-2-1-1-1 format) gives the team with the poorer regular-season ledger a special momentum-based advantage. This question is again relevant, in light of how this year's final series has gone. The Miami Heat, after splitting the first two games of this year's finals in Oklahoma City, has, of course, won Games 3 and 4 at home and will be going for the championship tonight while again playing in Miami.

Based on this list of NBA championship match-ups (with further details available by clicking on the year in the chart in the linked document), I counted eight years (prior to the current season) in which a team had won Games 3 and 4, giving it the opportunity to "ride the wave" to win also in Game 5. In three of these eight final series, it was actually the road team that caught fire in Games 3-5. The 1990 Detroit Pistons, 1991 Chicago Bulls, and 2001 L.A. Lakers all split the first two games of their respective series at home, then won three straight on the road to win titles.

More relevant for tonight's Thunder-Heat tilt are the five cases from 1985-2011 in which the home team won Games 3 and 4 at home. They are listed in the following table.

Year	Home Team Games 3-5	Chance to Close Out Series in Game 5?	Home-Team Result in Game 5
1997	Utah	No	Lose
1998	Chicago	Yes	Lose
2004	Detroit	Yes	Win
2005	Detroit	No	Lose
2006	Miami	No	Win

HTML Tables

Given the small sample size, these findings should be interpreted very cautiously. For what it's worth, however, three of the five teams that won Games 3 and 4 at home went on to lose in Game 5. The two series that best resemble this year's in terms of the win-loss patterns for the home and away teams are those from 1998 and 2004 (highlighted above in red). These two series offer a mixed verdict for 3-4-5 momentum.

In 1998, Michael Jordan's Chicago Bulls split the first two games of the finals in Utah, then returned to the Windy City to win Games 3 and 4. With a chance to close out the series at home in Game 5 (akin to the Heat tonight), the Bulls faltered. They ultimately did win the series, though, back in Utah with a Game-6 victory over the Jazz.

In 2004, the Detroit Pistons split the first two games against the Lakers in L.A., then won Games 3 and 4 in Michigan to take a 3-1 lead. There was no stopping the host Pistons in Game 5, as they dispatched the Lakers 100-87 to win the championship.

Again, there are not many data points on the specific situation facing Miami and OKC tonight. However, the little historical evidence we have suggests that the Heat is not an absolute shoo-in for coronation in Game 5.

Dominant Pitching by Mets' R.A. Dickey

ESPN.com has a couple of interesting articles on the hot streak currently being enjoyed by New York Mets pitcher R.A. Dickey. Monday night, the 37-year-old knuckleballer threw his second consecutive one-hitter, but his pitching dominance goes back further than these last two starts.

One article, entitled "R.A. Dickey on One of Great Rolls of All Time," lays out the basic facts: "Over his past six starts, Dickey is 6-0 with a 0.18 ERA (one earned run in 48 2/3 innings), 63 strikeouts, five walks and a .131 average allowed." The author of this article, David Schoenfield, then offers his opinions on what are some of the other great streaks of pitching mastery. Here's his list, with some illustrative quotes:

• Orel Hershiser: "spun six consecutive shutouts in September 1988."
• Greg Maddux: "In 1994... posted a 1.56 ERA; he was mostly a model of consistency that season, allowing two earned runs or fewer in 21 of his 25 starts."
• Pedro Martinez: "Over his final seven starts [in 1999], he allowed seven runs (five earned) with eight walks and 96 strikeouts in 55 innings. In consecutive starts, he struck out 15, 11, 15, 17, 14, 12 and 12."
• Fernando Valenzuela: "at the start of the 1981 season ...he threw five shutouts in seven starts and allowed two runs in 63 innings."  (Note: I analyzed the "Fernandomania" streak here.)
• Bob Gibson: "in 1968, had an 11-start stretch in which he threw 11 complete games and allowed three runs." 

In my book Hot Hand, which came out before the 2012 season, I discussed all-time streaks of pitching greatness (pp. 91-93). I also mentioned Gibson. Another pitcher I discussed is Don Drysdale, who for whatever reason was omitted from Schoenfield's article. In 1968, a great year for pitchers due in part to their getting to stand on a 15-inch-high pitchers' mound (lowered the next season to 10 inches), Drysdale pitched 58 2/3 consecutive scoreless innings. This record held until Hershiser topped it 20 years later.

For a short-term, high-intensity streak, there's the record set in 2009 by then-White Sox pitcher Mark Buehrle of retiring 45 consecutive batters, with the stretch encompassing a perfect game.

The other ESPN.com article I alluded to asks in its headline, "Will Dickey's Dominance Continue?" Before addressing that question, the writer, Tristan Cockcroft, adds another name to the list of great pitching reigns: "Johan Santana's 13-0, 1.21 ERA and 11.13 K's-per-nine ratio during the second half of 2004..."

What I see as the main contribution of Cockcroft's article is his division of Dickey's statistics -- apparently based on extensive watching of Dickey's performances on video -- into those when he throws his knuckleball 78 mph or faster vs. 77 mph or slower. The differences are pretty dramatic, such as a .111 opponents' batting average (or batting average against, BAA) on the faster knucklers, compared to .283 on the slower ones.

Cockcroft notes that extensive pitching-velocity data are a relatively new development, thus limiting historical comparisons. For what it's worth, Tim Wakefield (who pitched from 1992-2011) "never topped 74.7 mph since 2009."

The message from Cockcroft's article seems to be that, if Dickey can continue to throw his "fast knucklers," he can indeed maintain his streak of dominance. However, due to ordinary uncertainty and the relative lack of historical precedents for knuckleball pitchers, we really don't know what will happen. Cockcroft chooses to be optimistic, asking "who is to say [Dickey] can't continue at or near the pace he's currently on?"

Heat on Free-Throw Hot Stretch

Whereas Miami forward Shane Battier has exhibited some hot three-point shooting this postseason (which he extended by going 2-of-2 behind the arc in Game 3 of the finals), the team as a whole has stepped up its free-throw success rate. The following chart shows the Heat's free-throw shooting percentages -- and those of its offensive stalwart LeBron James -- in each of Miami's 2012 postseason games (you may click on the graphic to enlarge it).
  

In the two most recent final-round games against the Thunder (labeled as OKC 2 and OKC 3 on the horizontal axis), Miami has registered free-throw shooting percentages of .880 (22-of-25) and .886 (31-of-35). Not since its series-clinching Game-6 win over Indiana in the second round had the Heat hit from the stripe at a .800 clip (indicated by the white line in the graph). In none of the seven games of the Eastern Conference finals vs. Boston did Miami make 80% of its free throws, nor did it happen in Game 1 against Oklahoma City. The Heat did accomplish the feat twice in the first round against New York (Game 3 and Game 5).

James has been a major part of Miami's free-throw renaissance, going a perfect 12-for-12 in Game 2 vs. OKC, in addition to nights of 7-of-9 in Game 1 and 6-of-8 in Game 3. Overall for the championship round, James is 25-for-29 (.862). Perhaps the combination of the importance of the finals and the tightness of the games has prompted James to concentrate better on free throws. In three of the Heat's earlier playoff games, he made fewer than half of his shots from the stripe (3-of-7 in Game 2 against the Knicks; 1-of-3 in Game 3 vs. the Pacers; and 1-of-5 in the third game of the Boston series).

During the regular season, Miami's team free-throw percentage was .775. For the playoffs as a whole thus far (21 games), it's even worse, .741. Can the Heat sustain its recent 88% free-throw success in the remaining games of the finals?

Battier Steps Up His Three-Point Shooting Percentage

I would say the dominant story of this year's NBA finals, two games in, is the drama of Miami's LeBron James and Oklahoma City's Kevin Durant trying to will their respective teams to victory. Not far behind, however, is the way Heat forward Shane Battier, the 11-year veteran from Duke, has lifted his three-point shooting accuracy. Typically thought of as a defensive specialist, Battier wasn't exactly scorching from behind the arc in the regular season, registering a .339 percentage (62-for-183). His game-by-game statistics are available here.

His overall postseason shooting percentage on threes at this point is .371 (36-of-97), up slightly from the regular season. However, in the latter part of the Eastern Conference finals against Boston and now thus far against Oklahoma City, Battier is on a run that has him making more than half of his three-point attempts. The following graph shows his game-by-game three-point shooting in the playoffs, with his recent hot streak highlighted in a box on the right-hand side.

As depicted in the graph, Battier had a lot of ups and downs in his three-point shooting in the first round vs. the New York Knicks and the second round vs. the Indiana Pacers; however, in many of these games, he had few attempts. Now he's getting a lot more attempts, including 9 in the seventh game against the Celtics (of which he made 4), 6 in the opener against the Thunder (of which he made 4), and 7 in Game 2 against OKC (of which he made 5).

There will be at least three more games in the final series. Whether the factors that have allowed Battier to shoot so well thus far, such as Oklahoma City's defensive scheme and a little old-fashioned luck (as on this shot that he banked in), continue in his favor, remains to be seen.

MLB.com Article on Hitting Streaks

MLB.com, the official website of Major League Baseball, has a new article on hitting streaks, with a focus on the recent offensive outburst by Josh Hamilton of the Texas Rangers. The article quotes a variety of sources, including yours truly.

College Softball's Turang on Torrid Hitting Stretch

The NCAA women's softball super-regionals are taking place this weekend, with teams trying to advance to next week's Women's College World Series. One of the hottest hitters in the postseason, if not the hottest, is Oklahoma junior outfielder Brianna Turang, whose team has already qualified for the World Series. As shown in the following chart, Turang has hit .750 (9-for-12) in five postseason games (three in the initial regional round and two against Arizona in the super-regionals).You may click on the graphic to enlarge it.

The left-hand part of the display tracks Turang's batting average during the regular season. It first lists her batting statistics at the end of preconference play (i.e., as of the close of play on March 16) and then updates the statistics on a roughly weekly basis, coinciding largely with Big 12 conference play (even after conference play begins, however, there are still some nonconference games sprinkled through the schedule and these games are included in the stats). The data come from OU's softball news releases throughout the season.

As can be seen, Turang's batting average varied within a fairly narrow range -- either a little bit above or below .330 -- over the final two months of the regular season, ending up at .328. The right-hand portion of the display shows her game-by-game postseason statistics, documenting her cumulative 9-for-12 performance.

The relevant statistical question is the following: What is the probability of someone with a .328 regular-season batting average (i.e., long-term success rate) getting a hit in 9 or more of her 12 postseason at-bats? The answer, according to this online binomial-probability calculator, is .0034 or roughly 3-in-1,000.

Raising her game in the postseason is nothing new for Turang, however. Her regular-season batting average a year ago was .318. According to OU's news release for this year's super-regionals: “In 2011, Turang led all Sooners in postseason play with an average of .533 off a team-leading 16 hits…”

Spurs' on 18-Game Winning Streak (Combining Regular Season and Playoffs)

With their win last night to sweep the L.A. Clippers 4-0 in a second-round NBA playoff series, the San Antonio Spurs now have a combined regular-season/playoff 18-game winning streak. The Spurs won their final 10 games of the regular season and then swept Utah in four in the first round of the playoffs.

San Antonio's hot streak goes back even further. Sitting at 26-13 after play on March 9, the Spurs closed the regular season on a 24-3 run to finish the abbreviated 66-game NBA season with a 50-16 record. Adding in the eight playoff games, San Antonio is now 32-3 in its last 35 games.

The Spurs' roster has an aging core, led by 36-year-old Tim Duncan and 34-year-old Manu Ginobili. Perhaps the shortening of the NBA season from the usual 82 games to 66, due to the owners' lockout, has helped conserve the energy of the Spurs' players. A year ago, the silver and black appeared to hit the wall after compiling a 57-13 record through 70 games. The Spurs went 4-8 in the remainder of the regular season (including a six-game losing streak) to finish at 61-21, and then exited in the first round of the playoffs at the hands of the Memphis Grizzlies.

Anecdotes Aside, Trouncing Opponent in Game 1 of NBA Playoff Series Usually Presages a Series Win

Watching the closing minutes of last night's 119-90 Oklahoma City Thunder thumping of the L.A. Lakers in the teams' opening game of their NBA playoff series, I was reminded of two other Game-1 blowouts in NBA playoff history.

In what is known as the "Mother's Day Massacre" of 1982, the Boston Celtics routed the Philadelphia 76ers 121-81 to open that year's Eastern Conference finals. Yet, Philly took the series in seven games.

Three years later, the Celtics opened up the 1985 NBA Finals with a 148-114 thrashing of the Lakers, the "Memorial Day Massacre." Again, though, it was the team on the losing end of the massacre that ultimately prevailed, as L.A. hoisted the championship trophy.

Judging by these two precedents, therefore, this year's Laker squad didn't seem to be in such bad shape, after all. Rather than go purely by anecdote, however, I decided to investigate all opening games of NBA playoff series since 1982 (i.e., roughly the past 30 years) in which a team won by 25 or more points, to see how often that team ultimately lost the series. (I only examined 4-out-of-7 series; the NBA used to have 2-of-3 and 3-of-5 opening rounds.) The Wikipedia's year-by-year NBA playoff summaries supplied the necessary data. Here is what I found.

Well, what do you know? From 1982 forward, there have been 27 playoff opening-games in which one team triumphed by 25 or more points (excluding this year's Thunder-Lakers game, as the series is incomplete). And in 25 of those 27 series (93%), the team that won easily in Game 1 went on to win the series. In other words, the two counter-examples I initially came up with (the 1982 Sixers-Celtics and 1985 Lakers-Celtics series) were the only two exceptions to the larger trend.

Apparently, the outlook for this year's Laker squad is a lot worse than I had thought!

In thinking that a team that gets blown out in Game 1 has a good chance to win the series, I appear to have succumbed to what is known as the "availability heuristic." Because the 1982 Sixers' and 1985 Lakers' comebacks against the Celtics were so dramatic, they stuck in my mind (i.e., remained available) and evidently led me to overestimate the frequency of overcoming an opening-game disaster.

The late Amos Tversky, a co-developer of hot hand research, also launched the research on availability with his longtime collaborator Daniel Kahneman.

Josh Hamilton's Four-Homer Game

With Josh Hamilton of the Texas Rangers on Tuesday night joining the club of sluggers who have belted four home runs in a single game, I decided to examine Hamilton's and all previous four-homer games in detail to see what we might learn about the continuity and timing of such outbursts. For example, when a player has homered four times in a game, have the four blasts tended to come in four consecutive at-bats? Against the same pitcher each time? If not the same pitcher, has the throwing arm at least been consistent across pitchers faced? You get the idea.

Hamilton, of course, is best known for overcoming drug-abuse problems to bring his beautiful bat swing -- reminiscent, in some observers' eyes, of the main character in "The Natural"  -- to the major leagues. In the 2008 Home Run Derby during the All-Star Break, Hamilton socked 28 homers in a single round. Thus, as rare a feat as hitting four home runs in a game is, one had the feeling that if anyone would do so in the near future, Hamilton could well be the person.

My starting point was this list, from Wikipedia, of the 16 major-league players to hit four homers in a game. For each instance, I then consulted Retrosheet, an online archive of MLB box scores and play-by-play sheets. Box scores routinely list all home runs hit in a game by hitter, pitcher, and inning. The accompanying play-by-play sheets supplement the box scores by listing in narrative fashion what happened on all at-bats in the game, thus documenting what the hitter with four home runs did in all his other at-bats (if any) during the game.

For the 11 batters who had four-homer games from 1950 to the present, the chart below thus documents what they did in each at-bat of their big game. For three batters whose four-homer game occurred in the 1930s or '40s, Retrosheet had box scores only, so the non-homer at-bats could not be charted. Finally, for Bobby Lowe (1894) and Ed Delahunty (1896), neither box scores nor play-by-play sheets were available, so they do not appear in the following tables.

Here are the charts. You may click on them to enlarge them.

Here are some findings from the charts:

• Among the 11 players for whom the full scope of data was available, four (Carlos Delgado, Mike Cameron, Mike Schmidt, and Rocky Colavito) homered in four consecutive at-bats, with no other kind of outcome intervening.
• Of the same 11 players (excluding Delgado, who had only four plate appearances), three (Hamilton, Shawn Green, and Joe Adcock) had additional hits (singles, doubles) beyond their home runs, without ever making an out.
• Among the documented outs (pink squares), five were via fly balls or pop-ups, only one was a ground-out, and none were strike-outs. Thus, these hitters were getting the ball in the air with great regularity.
• Hamilton (3 of 4), Delgado (3 of 4), Green (3 of 4), Cameron (3 of 4), Mark Whiten (all 4), Chuck Klein (all 4), and Lou Gehrig (all 4) hit their homers predominantly or exclusively against pitchers who threw with the opposite arm relative to the handedness of the batters' swings. To really know if there's any significance to this statistic, we would need to know the percentages of opposite-hand match-ups in all at-bats and all home-run at-bats in MLB generally.
• Among the 14 players whose home-run innings can be documented, all except Schmidt, Colavito, and Pat Seerey got off to a quick start by homering in the first or second inning.

L.A. Kings Dominating Their NHL Playoff Series

The big surprise of the National Hockey League playoffs thus far is the largely dominant play of the Los Angeles Kings. The Kings qualified for the playoffs as the eighth (and final) seed in the Western Conference, but easily dispatched No. 1 seed Vancouver in the first round, four games to one, and are currently leading No. 2 seed St. Louis two games to none in the second round. Other than during a possible letdown after winning the first three games against the Canucks, the Kings have hardly been challenged.

I created the chart below to illustrate how many minutes, cumulatively, the Kings (in purple) and their opponents (in light blue) have held the lead in the two series (you may click on the graphic to enlarge it). Links to box scores of all the Kings' games are available here. To walk through an example, in Game 1 vs. Vancouver, the Canucks scored first 4 minutes and 17 seconds into the contest, thus coloring the chart blue. The Kings then scored 13:31 into the game, tying things up and returning the tracker to white. L.A. scored the next goal, 16:33 into the second period, bringing purple coloration. And so forth.

As circled in red, Vancouver led for only 9 minutes and 14 seconds total in the first three games (180 minutes) of the Kings-Canucks series. Now, two games into the Kings-Blues series -- both played at the home of the Gateway Arch   -- St. Louis has led during only 7:42 of the 120 minutes played.

The Kings and Blues resume their series tomorrow night in Los Angeles. Fortunes can change in a hurry in hockey, especially with the seemingly random bounces of the puck. For now, though, the Kings seem to have the right formula.

Pitcher Colon Throws 38 Straight Strikes

Last Wednesday night, the Oakland Athletics' 38-year-old Bartolo Colon threw 38 consecutive strikes in shutting out the Angels 6-0. (My thanks to Richard Zitrin, whose message to the SABR discussion group brought Colon's streak to my attention.)

This article provides links to statistical breakdowns of Colon's consecutive-strikes streak and to a video of the sequence (less than four minutes long, edited to include only the pitches and related game action). The streak is said to be the longest of its kind since at least 1988. There's an aspect of the streak that makes me wonder if it can really be considered official, however.

Embedded within the 38 pitches are 10 balls put in play. For example, on one of the pitches in the video (the one resulting in the Angels' first out of the seventh inning), the broadcaster says, "That's a strike... and it's grounded to short."

A called strike by the umpire, a swing-and-miss, and a foul ball are all officially designated as strikes (putting aside whether a foul with two strikes is officially deemed a strike). On a ball put in play, I wondered, what authority is there for someone seemingly to pronounce a pitch a strike because it looked to be in the strike zone.

I sent my own message to the SABR group, as did other members, and a bit of a discussion got going. Dvd Avins wrote in that, "...all balls in play are always considered strikes for this purpose... So I very much doubt that the streak, as reported, depends on the announcer's or anyone else's perception of what pitches would have been called had a batter not swung."

My view is that it would be more defensible to say either that: (a) on pitches not put in play, Colon threw 28 consecutive strikes; or (b) Colon threw 38 consecutive pitches on which the umpire never called a "ball."

U of Houston's Inning Hit Streak

This past weekend, the University of Houston baseball team played at Texas Tech. The Cougars lost all three games -- and if not for a rally Sunday to force extra innings, all three losses would have been blowouts. Still, however, UH recorded an unusual offensive accomplishment, of which Tech radio announcers Robert Giovanetti and Mike Gustafson kept listeners informed. Namely, from the sixth inning of Friday's opener to the fourth inning of Sunday's finale, the Cougars recorded at least one hit for 17 straight innings. Below, I've illustrated the streak with red shading based on play-by-play sheets for Friday's, Saturday's, and Sunday's games.

I have no idea whether Houston's streak is a record of any kind. Still, getting hits in 17 straight innings and scoring only four runs during these innings seems pretty rare!

Tyshawn Taylor's (Kansas) March Madness Cold Hand

Those who closely followed the recent NCAA men's college basketball tournament are probably aware of the three-point shooting difficulties experienced by Kansas senior guard Tyshawn Taylor. He shot 0-for-21 from beyond the arc in this year's NCAA tourney, getting off the schneid only by making a three in last Monday night's title game vs. Kentucky (on what turned out to be his sole attempt from deep in that game). For the "drama" he created both on and off the court during his four years at KU, this Sporting News article referred to Taylor as a "Shakespearean character."

One of the key points in analyzing hot and cold hands is that streaks can be better understood by looking at the athlete's long-term performance record prior to the streak. For example, a stretch of 50 consecutive made free throws will be more surprising coming from a long-term 70% shooter than from someone who hits 90% from the stripe. For this reason, I examined Taylor's three-point shooting statistics for every game of his collegiate career (this link takes you to Taylor's game-by-game statistical logs, with the ability to select any of his four seasons).

What I found was what I consider a surprisingly consistent pattern in Taylor's career, as shown in the chart below. If you look at the final 10 pre-NCAA-tournament games of each season (when players should be aiming to reach peak sharpness for the important competition ahead), Taylor had three-point shooting percentages in the mid-40s (at least), an impressive level. However, each year, once NCAA March Madness began, he could hardly hit a shot from behind the arc. In four years of NCAA-tournament play combined, he made only 4-of-41 attempts (.098).

Whether Taylor can make the NBA, I don't know. He was a third-team All-America selection this season, so I suppose he has a chance. However, I think we can be confident that, even if he makes it, his role as a professional probably won't be as a three-point specialist!

50th Anniversary of Wilt Chamberlain's 100-Point Game

Fifty years ago today, March 2, 1962, Wilt Chamberlain of the then-Philadelphia Warriors scored an NBA record 100 points in a single game. The game was played in Hershey, Pennsylvania, as the Warriors defeated the New York Knicks, 169-147. Chamberlain, who lived from 1936-1999, stood 7-foot-1 and was associated with many amazing statistical feats (e.g., a 33-game winning streak with the 1971-72 Los Angeles Lakers; playing over 1,000 games without ever fouling out). However, it is the 100-point game that is his signature accomplishment.

Despite the magnitude of Chamberlain's performance, information on his 100-point game is relatively scarce. No video or film of the game seems to exist and, because it was played 95 miles outside of Philly in a much smaller city, attendance was low (4,125) and there was less media coverage than usual. One really has to do some work to learn the details of the game, and luckily writer Gary M. Pomerantz did so for readers of his 2005 book Wilt, 1962: The Night of 100 Points and the Dawn of a New Era (there's an excellent website accompanying the book, which includes audio of the radio broadcast of the game).

Here at the Hot Hand website, our focus is statistical analysis, so that is what we shall pursue. As can be seen from the box score of the game, Chamberlain hit 36-of-63 from the floor for 72 of his points and went 28-of-32 from the free-throw line.It is the latter performance, a nearly 90% success rate (.875), that is really amazing. Wilt's free throw percentage during the 1961-62 season was .613, which actually was the best single-season free-throw accuracy rate of his career. Thus, on this night, he really overcame his free-throwing difficulties to make the 100-point game possible.

Noting that Chamberlain had made 21-of-22 free-throws through the first three quarters of play, Pomerantz asks in his book, "How does a sixty percent free-throw shooter throughout the season convert ninety-five percent on a night in Hershey?" (p. 116). Updating the percentages to reflect the full game and rephrasing the question a bit, we can ask how likely it is that a .613 shooter would hit 28 (or more) free throws out of 32, assuming independence of outcomes (i.e., what happens on one shot having no impact on the next shot, like coin tossing).

This binomial-probability calculator allows one to type in the underlying probability of an outcome (p = .613), the number of opportunities (n = 32), and what the website calls the number of "stipulated" successes (k, in this case 28 made free-throws). Because the probability of exactly 28 (or any other specific number) made free-throws would tend to be very small, statisticians typically inquire into the probability of a particular result or one more extreme. Hence, we have the reference above to "28 (or more)" successful free-throws out of 32.

In the following chart, I've graphed the probability of Chamberlain making anywhere from 9 to 32 free-throws out of 32, given his season-long percentage of .613 (below 9, the likelihood is extremely low). You may click on the graphic to enlarge it.

As can be seen, the probabilities of Chamberlain making 28, 29, 30, 31, or 32 free-throws were all tiny, so that the probability of 28 or more made shots was around 1 or 2 in 1,000, according to this annotated screen caputre from the statistical calculator website:

With a roughly 61% free-throw percentage, Chamberlain would have been most likely to make, you guessed it, 61% of his attempts, which translates into 19.5 out of 32 (note how the bars for 19 and 20 are the highest in the graph above). Thus, he greatly exceeded statistical expectations.

Pomerantz tried to answer his original question about how Chamberlain could have been so successful on free-throws (which we now know to be a roughly 1-in-1,000 phenomenon) by claiming that the baskets in Hershey featured "soft," rickety rims. When a ball hits a soft rim, the reasoning goes, it will bounce gently around and possibly fall through the hoop, rather than banging hard against it and flying off. Whether that's a valid explanation or not, I can't say. Still, Wilt must have been shooting the ball close enough to the basket on his free-throw attempts to (allegedly) benefit from the soft rims.

A famous quote from the writings of the late Harvard paleontologist Stephen Jay Gould is that, "Long streaks always are, and must be, a matter of extraordinary luck imposed upon great skill." Indeed.

Mississippi Valley State's Unusual Win/Loss Distribution

My friend Gregg from Los Angeles recently drew my attention to Mississippi Valley State's unusual distribution of wins and losses in men's basketball this season. As seen in this game-by-game log, the Delta Devils completed non-conference play with a 1-11 record (the only victory coming by a single point in double overtime). However, once league play in the Southwestern Athletic Conference (SWAC) began on January 3, MVSU really took off, winning all 17 games from then on! After the 15th straight win, the national media started taking notice.

Quite a turnaround, right, from nearly winless to unbeatable? Well, even a quick glance at MVSU's schedule reveals that the non-conference segment (played nearly all on the road) was much harder than the conference games. I created the following table to document this fact, using Ken Pomeroy's computer rankings (as of Monday night). Non-conference games are shown in white text, with conference games in yellow. For comparison purposes, Mississippi Valley State's ranking is 246.

Opponent	MVSU	Opponent Rank
@Notre Dame	L	32
@DePaul	L	169
@N. Carolina	L	7
@S. Carolina	L	160
Tennessee St.	W	182
Cal Poly	L	184
@Arkansas	L	111
@Northwestern	L	56
@Ole Miss	L	103
@Florida	L	17
@Wisconsin	L	5
@Iowa State	L	31
Arkansas-Pine Bluff	W	333
@Alabama State	W	321
@Alabama A&M	W	341
Alcorn State	W	342
Southern U.	W	327
@Prairie View A&M	W	329
@Texas Southern	W	283
Jackson State	W	335
Grambling State	W	345
Alabama State	W	321
Alabama A&M	W	341
@Alcorn State	W	342
@Southern U.	W	327
Prairie View	W	329
Texas Southern	W	283
@Grambling State	W	345
@Jackson State	W	335

HTML Tables

Looking at the table, MVSU's turnaround may seem less astounding and more explicable than when first hearing the team's records during its losing (1-11) and winning (17-0) stretches. In the non-conference portion of the schedule, all opponents were good enough to be ranked 184 or better as of yesterday, and five were ranked in the top 35. In the SWAC, with the exception of Texas Southern (283), all other opponents were ranked from 321-345 (Grambling State, at 345, is dead last in the nation).

One could argue that MVSU's season simply consists of losing to superior teams and beating inferior ones. I don't think this characterization does justice to what the Delta Devils have accomplished, however. First, winning 17 straight games under any circumstances is not easy. Second, the team clearly did not feel its season was lost after starting out 1-11 and kept plugging away as conference play started (one might say the team had a "Devil may care" attitude).

At worst (if it loses in the SWAC tournament), MVSU will be playing in the post-season National Invitation Tournament (NIT), which guarantees a berth to all regular-season conference champions. At best, if it wins the SWAC tourney, the Delta Devils will be playing in the NCAA "Big Dance."

As I suspected, MVSU's road schedule had to do with "The Delta Devils crisscross[ing] the country, playing some of the nation’s top teams on the road in exchange for money to supplement the athletic budget," according to the above-linked Washington Post article. That MVSU lost nearly all of these non-conference games is not that surprising. That the Devils have won all their conference games (even in a relatively easy conference) is impressive.

Accessing My BBC Radio Interview on Jeremy Lin

I had a nice discussion Sunday night with the host of BBC Radio's "Up All Night" show, about the recent success of New York Knicks' guard Jeremy Lin. If you want to listen to the archived audiotape of the interview, click here first to get to the appropriate BBC webpage. Then, unless you want to listen to 3 and a half hours of programming before my interview, advance the time marker as shown below:

Daily Beast Article on Jeremy Lin

There's a nice article on Jeremy Lin and hot hand research in the Daily Beast. I am interviewed in the piece, as are several other researchers who study statistics and streakiness.

Putting Jeremy Lin's Hot Hand in Perspective

The athlete creating the greatest amount of "buzz" in U.S. sports is almost certainly Jeremy Lin, a 6-foot-3 second-year New York Knicks point guard from Harvard. In addition to his ability to hit game-winning shots, Lin has gotten attention for his scoring. In fact, he holds the modern-era (i.e., post NBA-ABA merger) record for most points in a player's first five starts (136).

According to David Berri and Martin Schmidt's book Stumbling on Wins, points scored appear to carry great weight in determining basketball players' salaries and receipt of awards (e.g., being named to the All-Rookie Team). However, shooting percentage (or efficiency) appears more important for actually winning games. Thus, I decided to focus on Lin's game-by-game shooting percentages.

Lin recently had a four-game streak of shooting .500 or higher from the floor in each contest: February 4 vs New Jersey (.526, 10-of-19); Feb. 6 vs. Utah (.588, 10-17); Feb. 8 at Washington (.643, 9-14); and Feb. 10 vs. the L.A. Lakers (.565, 13-23). How does this stretch compare to those of other leading guards?

It's been over a year since Kobe Bryant had such a four-game streak. Between January 17-25, 2011, Bryant had consecutive games of shooting .583, .556, .533, and .636.

It's been almost two years since Derrick Rose had one. Between Feb. 22-27, 2010, Rose had consecutive games in which he shot .500, .526, .600, and .643.

Of course, Lin is not the only NBA guard who has shot well recently. Miami's Dwyane Wade is currently on an eight-game streak of shooting .500 and above (.500, .571, .529, .625, .611, .500, .571, and .500).

Miami and Duke Trade Scoring Runs in College Hoops

Yesterday's Miami (FL) at Duke men's basketball game, won by the Hurricanes 78-74 in overtime, featured many examples of hot and cold streakiness. Miami darted off to a 9-2 lead, only to have the Blue Devils outscore the Canes 20-10 to take a 22-19 lead. Miami then responded by going on a 23-6 run to close out the first half and lead 42-28 at the break. The Canes upped their lead to 48-32 early in the second half, before a Duke 16-2 run put the Blue Devils back in the game. Things were tight the rest of the way and, of course, it took overtime to decide the outcome.

Ultimately, however, the lasting impression will likely be how Duke went 0-for-6 on free throws in OT (see overtime play-by-play sheet). As shown below, three players -- two of them above 80% free-throw shooters coming into the game -- each missed a pair from the stripe.

Duke Players' Free-Throw Percentages Within Different Time Frames

Player	For the Season  (Before  Miami)	Regulation  vs. Miami	Overtime  vs. Miami
Seth Curry	.887 (63-71)	1.000 (4-4)	.000 (0-2)
Austin Rivers	.684 (65-95)	.714 (5-7)	.000 (0-2)
Quinn Cook	.812 (26-32)	1.000 (2-2)	.000 (0-2)

HTML Tables

To estimate the probability of the six consecutive overtime misses, we simply multiply the individual miss probabilities on each shot (two for each player). Each player's miss probability is 1 minus his long-term free-throw success rate (in this case, I'm using the players' season-to-date FT percentages prior to the Miami game). The calculation, using the miss probabilities for Curry (1 - .887), Rivers (1 - .684), and Cook (1 - .812), is:

.113 X .113 X .316 X .316 X .188 X .188 = .000045 or roughly 1 in 22,000.

The usual cautions apply to analyses such as the present one. It is post hoc, selected after the fact only because of the unusual nature of Duke's free-throw misses. To paraphrase the words of one statistician, presumably no one asked before the overtime started, "How likely is it that Duke will miss all of its free throws in the extra period?"  Finally, although 1 in 22,000 seems pretty improbable, if one considers all the basketball games that have been played -- in the U.S. and internationally; high school, college, and pro; men's and women's -- perhaps a slump like Duke's was bound to happen at some point.

Ups and Downs in Matt Carlino's Three-Point Shooting

One of the definitions used by statisticians for a hot hand is that a player's successes cluster together sequentially, as do his or her failures. A college basketball player whose three-point shooting has been fitting this pattern lately is Brigham Young University frosh guard Matt Carlino, a transfer from UCLA. (One of our Texas Tech grad students, Tim Oblad, who did his undergraduate work at BYU, brought Carlino's stats to my attention.)

As seen in his game-by-game log, Carlino missed BYU's first 10 games this season before playing in the team's last 14 contests. He started off nicely from behind the arc, going a combined 17-of-33 (.515) in his first six games. After a 1-for-7 (.143) game January 5 at Loyola Marymount, Carlino resumed his hot three-point shooting over his next three games, hitting a combined 7-of-13 (.538).

Since then, in his last four games, Carlino appears to have lost his shooting touch, making only 2 of 21 attempted threes. It's also possible that the most recent opponents have figured out how to defend him better than did his earlier opponents. I've plotted Carlino's game-by-game shooting percentages from long distance (with opponents displayed along the horizontal axis), as shown in the following figure. (You may click on the graphic to enlarge it.)

Admittedly, we're dealing with short sequences of games. However, Carlino seems to get "stuck" either in sequences of hot shooting or of cold shooting. As a point of comparison, Creighton's Doug McDermott (whose three-point shooting I analyzed here) seems able immediately to snap back after a poor shooting night. Hence, at this stage of the season, Carlino appears to be more of a streaky shooter -- with both hot and cold spells -- than McDermott.