Great Olympic Streaks: Volleyball

Starting today and running each day until next Friday's (July 27) opening ceremony of the Summer Olympics in London, I will be reviewing great winning streaks and dynasties in Summer Olympic history.

I would like to start today with volleyball, both of the beach and indoor varieties. Two of America's best known and most successful Olympic athletes in recent times are the beach volleyball duo of Misty May-Treanor and Kerri Walsh.

According to a USA Volleyball profile: “…the pair won the 2008 Olympic beach volleyball tournament to become the first team to win two beach volleyball Olympic gold medals. The pair finished the tournament with a 7-0 record and did not drop a set... The team finished the [2008] Olympic Games with a 108-match winning streak… [It also] went 14-0 to win the [2004] Olympic Games in Athens.”

May and Walsh will be going for a third consecutive gold medal on the sand in London. The two players have faced some challenges over the past four years, including a torn Achilles tendon by May in late 2008 while practicing for the television series Dancing with the Stars.

In women's indoor volleyball, the US will be seeking to end an 0-for-11 slump of never winning the gold medal. The most noteworthy string of success in recent decades belongs to the Cuban women, who won the indoor volleyball gold in 1992, 1996, and 2000.

One name is most synonymous with success in modern Olympic men's volleyball history: Karch Kiraly. The former UCLA setter helped lead the USA to gold in the 1984 and '88 indoor competitions. Kiraly then returned to Olympic play in 1996, in the then-new sport of beach volleyball, and took home a third gold.

Cubs' Ryan Dempster's Streak of Scoreless Innings Pitched

It's been a rough season for the Chicago Cubs, but the team has played better of late. One major reason is the pitching of Ryan Dempster, who has not allowed a run in his last 33 innings pitched. According to this article, "The 33 scoreless innings is the club’s longest such streak since Ken Holtzman in 1969 and is the longest in the major leagues this season."

Based on Dempster's 2012 game-by-game log, I created the following chart of how many runs he has allowed in each inning he has thrown this season (you can click on the chart to enlarge it).

By looking at the games on the left-hand side, we can obtain Dempster's pre-streak proportion of innings started in which he retired the side without allowing a run. As detailed in the summary statement on the chart, that proportion is .803. We then apply this baseline percentage to the streak by raising it to the 33rd power (to represent the 33 straight innings). That calculation yields .0007, or roughly 1-in-1,428, for the probability of Dempster's 33 scoreless-inning streak, given his prior pitching proficiency this season.

Because Dempster has pitched so well all season, allowing relatively few runs during April and May, the probability of his throwing 33 consecutive shutout innings, while not large, is not astronomically small, either. Note that, in his first nine starts of the season, in only two games did he give up a run in more than one inning.

In fact, the statistics charted above don't do justice to how well Dempster has pitched, in my view. In his season debut vs. the Washington Nationals, for example, he was officially credited with allowing 1 run, in the 8th inning. As can be seen in the play-by-play sheet, Dempster pitched to three Washington batters in the top of the 8th (Chad Tracy, Ian Desmond, and Danny Espinosa), who respectively struck out, singled, and struck out. However, because reliever Kerry Wood allowed the runner to score, that run was attributed to Dempster.

Dempster's next start is scheduled to be this Friday night at St. Louis, although he is the subject of rampant trade rumors as the Cubs seek to rebuild with younger players.

Stricker's Streak of Winning John Deere Classics Ends

Steve Stricker has fallen short in his attempt to make it four straight wins in the John Deere Classic golf tournament.

According to this Associated Press article, "Stricker [was] attempting to join Tom Morris Jr., Walter Hagen, Gene Sarazen and Tiger Woods as a winner four straight times in the same tournament. Woods has accomplished the feat twice."

Woods won the Bay Hill Invitational (now known as the Arnold Palmer Invitational) in Orlando, Florida each year from 2000-2003, and the Buick Invitational (now known as the Farmers Insurance Open) in San Diego, California, each year from 2005-2008.

Sarazen won the Miami Open four times between 1926-1930 (there officially was no 1927 version; the tournament was moved from December to January so what would have been the December 1927 playing of it was moved to January 1928).

Hagen won the PGA Championship four times from 1924-1927.

Morris was an old-time golfer, who won the British Open four times between 1868-1872 (the tournament was not held in 1871).

Let's come back to Stricker and estimate the probability of his winning three straight John Deere Classics. There were roughly 150 golfers in this year's event, so a very simple statistical model might assign Stricker a probability of 1/150 of winning the John Deere in a given year. However, my brother Steve, a very knowledgeable golf observer and also a Stricker fan, acknowledges that the John Deere tends not to have as deep a field of top professionals as other tourneys, apparently because many elite golfers prefer instead to use this time of year to prepare for the upcoming British Open. The weaker the John Deere field, the greater Stricker's likelihood of winning.

In fact, I checked this year's list of the Top 25 PGA money winners, and only five of them played in this year's John Deere Classic: Zach Johnson (who ended up winning the event), Carl Pettersson, Kyle Stanley, Mark Wilson, and Stricker.

Of course, someone outside the Top 25 tour money winners can win a tournament. If we assume there are 10 golfers each year in the John Deere field who are capable of winning it, Stricker among them, his single-year win probability would be 1/10. Raising this value to the third power to estimate Stricker's probability of three straight John Deere wins would yield 1-in-1,000.

If we assume there are 20 golfers in the John Deere field capable of winning, Stricker's probability of winning three straight is  (1/20) to the third power, yielding .000125 or 1-in-8,000. Finally, if we assume 50 entrants conceivably could win the John Deere Classic, Stricker's probability of three straight wins is .000008 or 1-in-125,000.

Regardless of which assumption one makes, Stricker's three straight John Deere victories before this year are noteworthy. Before one considers the probabilities above as being astronomical, however, one should realize that there are many fine golfers and many tournaments out there, creating a large number of opportunities for someone to win three straight.

There have been 21 instances of a player winning the same tournament exactly three straight times. If one adds the five occurrences (listed above) of a player winning the same event four consecutive times, then we have 26 instances in which a player has won the same tournament three or more times in a row.

Senator Collins Nears 5,000th Roll-Call Vote Without an Absence

It's not in the realm of sports, but it's definitely a long streak! U.S. Senator Susan Collins (R-Maine) is on the verge of casting her 5,000th consecutive vote on Capitol Hill without ever missing a vote due to absence. Heading into this week's legislative sessions, her streak stood at 4,997. This article (located via Political Wire) describes some close calls Collins has had since her election to the Senate in 1996. The article also presents some other statistics on consecutive votes cast:

• "Among sitting senators, only Republican Chuck Grassley of Iowa [first elected in 1980] has a longer voting streak, having cast 6,444 consecutive votes dating back to 1993, when he missed votes to join President Bill Clinton in touring flood damage in Iowa."
• "Grassley and Collins pale in comparison with the Senate's record holder, Wisconsin Democrat William Proxmire, who had 10,252 consecutive votes from April 20, 1966, to Oct. 18, 1988."

Not everyone is a fan of consecutive-votes streaks in government, however. Former Senator Eugene McCarthy (D-Minnesota) has said that voting on the Senate floor "is in many ways the least important thing that senators do." He went on to add that, "Shortly after I left the Senate, a roll-call vote was taken on what to do with the remains of Smokey the Bear."

D-Backs' Hill Hits for the Cycle -- Twice Within 10 Games!

Three nights ago, Friday, June 29, Aaron Hill of the Arizona Diamondbacks hit for the cycle, the second time he had done so in 10 games. The previous occasion was on June 18 (click here for the D-Backs' game-by-game log).

According to a Wikipedia page on the subject, "Cycles are uncommon in Major League Baseball (MLB), and have occurred 294 times... The cycle is roughly as common as a no-hitter (272 occurrences in MLB history); it has been called 'one of the rarest' and 'most difficult feats' in baseball."

With the rarity of hitting for the cycle, one might think it extremely rare for any player to have done it twice (or more) in a career, never mind twice in one season or twice in the same month. Remarkably enough, multiple-cycle seasons for the same player have occurred three times previously, although you have to go back 81 years for the last instance before 2012 (Babe Herman in 1931). Even more remarkably, the first two players to hit for the cycle twice in the same season (Tip O'Neill in 1887 and John Reilly in 1883) apparently accomplished the feat in quicker succession than did Hill (O'Neill and Reilly are each listed as having hit for the cycle 7 days apart; unless either of them faced a heavy supply of doubleheaders, it is almost certain each player's two cycles occurred within fewer than 10 games).

So how likely (or should I say, unlikely) was Hill's recent pair of cycles within 10 games? An excellent starting point is Jeff Sackmann's 2010 article, "The Odds of a Cycle." The foundation of Sackmann's tutorial is the relatively simple "multiplication rule" for joint probability (e.g., obtaining the probability of double-sixes on dice by multiplying together the probability of a six on each die, 1/6 x 1/6 = 1/36).

Also helpful was a brief analysis specifically focused on Hill's recent pair of cycles, by Justin Hunter, which provided some of the necessary statistical inputs for Hill. I used Hill's empirical frequencies of recording different types of hits, provided by Hunter, rather than what Sackmann used (something known as CHONE projections). A key insight of Hunter's was to identity Hill's low rate of producing triples historically (which was actually around 0.4% instead of the 0.3% figure Hunter gave).

Sackmann starts out with the simple case of a "natural cycle," that is, hitting a single, double, triple, and home run in that specific order, in four total plate appearances (PA). Filling in Hill's probabilities of the different types of hit, shown in the following table, we then multiply the numbers in the first four columns by each other, yielding the answer in the fifth column.

p (single)	p (double)	p (triple)	p (homer)	p (Nat Cycle | 4 PA)
.162	.053	.004	.027	.000000927

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As Sackmann further notes, having the benefit of 5 or 6 plate appearances in a game instead of 4 improves a player's odds of hitting for a natural (or any) cycle.

With 5 plate appearances, there are 5 ways a natural cycle can happen, as listed below (1, 2, 3, and H represent, respectively, a single, double, triple, and home run; x = non-fitting outcome, such as an out, whose probability we apparently don't care about):

x-1-2-3-H
1-x-2-3-H
1-2-x-3-H
1-2-3-x-H
1-2-3-H-x

For 5 PA games, we thus multiply our original probability (.000000927) by 5, yielding .00000464.

With 6 PA, there are 15 ways a natural cycle can happen  (those with semi-advanced mathematical training may be familiar with the n-choose-k principle, in this case, 6 choose 4). These are the 15 ways:

x-x-1-2-3-H.....x-1-x-2-3-H.....x-1-2-x-3-H.....x-1-2-3-x-H.....x-1-2-3-H-x
1-x-x-2-3-H.....1-x-2-x-3-H.....1-x-2-3-x-H.....1-x-2-3-H-x.....1-2-x-x-3-H
1-2-x-3-x-H.....1-2-x-3-H-x.....1-2-3-x-x-H.....1-2-3-x-H-x.....1-2-3-H-x-x

For 6 PA, we would multiply our original probability (.000000927) by 15, yielding .0000139.

To get one overall probability of a natural cycle for Hill, we next average the probabilities given 4, 5, and 6 plate appearances, weighted by the frequency of occurrence of the different numbers of PA. Sackmann provided relative frequencies of how often players (in general) get different numbers of PA in a game -- not figures specific to any particular player, but overall averages. Because it would be too time-consuming to look at box scores for Hill's nearly 1,000 games played, we'll use the overall relative frequencies, too.

PA	Proportion of Time (for avg. player)	p (Nat Cyc) with those # PA	Column 2 x Column 3
3	.101	---	---
4	.591	.000000927	.000000548
5	.274	.00000464	.00000127
6	.034	.0000139	.000000473

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Summing the yellow numbers (and dividing by 1.00 for the aggregated PA proportions) yields .00000229, the weighted average for the probability of Aaron Hill attaining a natural cycle (based on his batting statistics, weighted by PA patterns for an average player).

In reality, however, we're interested not just in natural cycles, but all possible kinds of cycles. As Sackmann says, “There are 24 permutations of the sequence 'single, double, triple, home run.' Thus, a garden-variety cycle is 24 times more likely than a natural cycle.” For scenarios beginning with a single, there are 6 possible sequences (only one of which, shown in red, is a natural cycle):

1-2-3-H
1-2-H-3
1-3-2-H
1-3-H-2
1-H-2-3
1-H-3-2

There are also 6 sequences beginning with a double, 6 beginning with a triple, and 6 beginning with a homer, thus yielding the 24 permutations. Multiplying our previous probability by 24 yields the following probability of Hill hitting for the cycle in a game (regardless of how the single, double, triple, and homer are sequenced).

24 x .00000229 = .000055 (roughly once every 18,182 games)

Before we get to the final steps, let's reflect on what we've discovered thus far. We have a player, Aaron Hill, who would be expected to hit for the cycle once every 18,182 games. Yet, he has done it twice in only 10 games!

We now bring in an online binomial probability calculator, which can help us answer questions of the form: How likely is it that, with an underlying probability of .000055 of an event (the cycle) occurring, 2 or more occurrences will be observed in 10 trials. This probability is .000000136, or approximately once in every 7 million (7,352,941) 10-game sets.

Ten-game sets are pretty numerous, however. Each time a player appears in his first career game, he is establishing a 10-game set (i.e., Games 1-10), then the next day, he establishes another (Games 2-11), etc. Sackmann notes that, “When 30 teams play a 162-game season, that's 43,740 player-games …”

In earlier times, there were fewer teams than 30 and fewer games on a team's schedule than 162. For simplicity, however, let's use Sackmann's figure of 43,740 player-games per year, each of which inaugurates a new 10-game stretch, with rare exception (e.g., when a player has 9 or fewer games left in his career).

Dividing 7,352,941 (the expected number of 10-game stretches for Hill or a player of similar abilities to hit for the cycle 2 or more times) by 43,740 (a rough estimate of the number of player-specific 10-game stretches launched in a single season) yields 168.1. We would thus expect a double-cycle by the same player within 10 games about once every 168 years.

Major League Baseball (and its organizational forerunners) have been around for 143 years, so it is not all that surprising to see two cycles by the same player occur within 10 days, at least once. In fact, it apparently has happened 3 times, but only once after 1887.

If you've stuck with me this far, you're probably really interested in the subject matter! I would recommend the following article to gain additional perspective:

Stefanski, Leonard A. (2008). The North Carolina lottery coincidence. The American Statistician. 62, 130-134.

Stefanski reflects upon the different interests of statisticians and laypersons (including the media) in understanding rare occurrences involving lotteries, sports, and perhaps other phenomena. Whereas laypersons seem to be interested in what Stefanski calls a "narrow" perspective (e.g., Aaron Hill's achievement), statisticians embrace a "wide" perspective, seeking to contextualize an occurrence in the larger set of opportunities for an event to occur. Writes Stefanski:

Statisticians should point out when seemingly rare events are not really that rare. But in doing so we should not lose sight of the fact that for some human interest stories, a probability calculation from the “narrow perspective” is appropriate. My hunch is that we sometimes do lose sight of the human-interest angle because we are geared toward the “wide” perspective (p. 131).

If you notice any faulty assumptions or calculations, please let me know in the Comments section or by e-mailing me via my faculty homepage (see links section to the right).

Dodgers' Offense Cools Off

The L.A. Dodgers, who started out the season with a 30-13 record (.698), averaging 4.58 runs per game during these 43 contests, have hit the skids.

With yesterday's series-concluding 3-0 loss to San Francisco by Los Angeles -- coming on the heels of 2-0 and 8-0 Dodger losses to the Giants -- the two teams are now tied for first place in the National League West. The Dodgers had led their northern California rival by eight games earlier this season. The Bay Area, on the whole, has not been kind to the Dodgers lately, as they were swept by the A's in Oakland June 19-21 by scores of 3-0, 4-1, and 4-1.

Going back to the Dodger-Giant series, however, the L.A. Times article on yesterday's game noted the following:

... the Dodgers have been held scoreless the last 30 consecutive innings. It was the first time the Los Angeles Dodgers had been shut out by one team in three consecutive regular-season games. The Brooklyn Dodgers suffered the same fate in 1937.

The Dodgers moved west in 1958, so the specific type of scoreless streak alluded to in the quote is unprecedented for 54 years. Note the important qualifier in the quote, however:

... the first time the Los Angeles Dodgers had been shut out by one team in three consecutive regular-season games... 

This qualifier suggested to me that the Los Angeles-era Dodgers must have been blanked three straight at some point before this week, but involving more than one opponent (i.e., being shut out once or twice by one team to close out a series and then being shut out once or twice in a series against a new opponent). I thus perused Dodger game logs going back in time and found the following sequence in 2007:

Sun, Aug 5 vs Arizona L 3-0

Tue, Aug 7 @ Cincinnati L 4-0

Wed, Aug 8 @ Cincinnati L 1-0

There could well be additional three-game Dodger scoreless streaks against multiple opponents prior to 2007, of course. But even though a Dodger three-game runless streak may not be quite as rare as it originally seemed, there's no question the L.A. bats are quiet.

The most obvious reason would have to be the absence of Matt Kemp, who has played in only two games since May 13 due to hamstring problems. He had been going wild offensively before his injury. For those who prefer traditional baseball statistics, Kemp hit 12 home runs and recorded a .355 batting average before his extended absence; for those into sabermetrics, Kemp had compiled a stratospheric 1.163 OPS (On-base [.444] Plus Slugging [.719] percentages).

The absence of one player alone, presumably, would not be enough to quell the entire offense of a team that had been doing so well. I thus examined the monthly offensive output (again using OPS) for the five Dodgers with the most at-bats so far this season (which would not include Kemp). From ESPN.com's Dodger statistics page, you can click on a player's name and then click on "Splits" to see the monthly breakdowns. The following graph presents what I found.

Two players in particular, right-fielder Andre Ethier (who himself was injured in yesterday's series finale against the Giants) and catcher A.J. Ellis, were hitting at exceptionally torrid paces in April and (especially) May. An OPS around 1.000, which Ethier and Ellis reached in May, is very rare. Currently only four National Leaguers are above 1.000 for the season. As shown in the graph, Ethier and Ellis have now come down to the .600s thus far in June, well below each player's respective career OPS (Ethier, .843; Ellis, .766).Other Dodger players in the graph have either remained level or dipped slightly in their OPS from May to June.

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

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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.

10th Anniversary of Hot Hand Website

Today marks the 10th anniversary of the launching of the Hot Hand in Sports website. The page has gone through a few different looks, as shown below. From 2002 to 2006, I used just a very basic FrontPage HTML format for the site, which was hosted on my college's server at Texas Tech University.

In 2006, I transferred the site to Blogger/Blogspot, where it has remained ever since. However, I've changed the template somewhat during the past six years. Here's how the page looked in its early years on Blogger.

Finally, 2011 and 2012 have brought major new developments. The book Hot Hand has now been published (see upper-right column). Also, I have started issuing brief updates on streaky performances right as they occur, via Twitter (http:/www.twitter.com/alanreifman), reserving the blog for more elaborate analyses.

As long as athletes keep going on streaks and I enjoy writing about them, this blog and associated activities will keep on going!

New York Giants Bring Unusual Stat into Super Bowl

The New York Giants are the first team with a four-game losing streak during the season to make it to the Super Bowl, since the 2002 Oakland Raiders. No other team in American football's Super Bowl era (beginning with the 1966 season) has lost four straight and made it all the way to the title game in the same season.

I find it very surprising that a team with four straight losses during a 16-game season could make the Super Bowl, for two reasons. First, because the season is so short, it doesn't give teams much time to recover and compile a playoff-worthy record (although it is possible to make the playoffs with a weak record). Second, if a team loses four straight, one has to wonder how good that team really is. A lot of Super Bowl teams didn't even lose four games the entire season!

In fairness to New York, the four straight losses were to three eventual playoff teams and a fourth team (Philadelphia), which was good in spurts. Plus, the losses were mostly close. The Giants lost on November 13 to San Francisco (27-20), Nov. 20 to Philadelphia (17-10), Nov. 28 to New Orleans (49-24), and December 4 to Green Bay (38-35). In the playoffs, however, the Giants avenged their losses to the Packers and 49ers. (Click here for the Giants' 2011-12 game-by-game log.)

Interestingly, the Giants played their upcoming Super Bowl opponent, the New England Patriots, during the season. New York won that game, 24-20 on Nov. 6, and immediately went into its four-game skid!

Looking at this all-time chart of Super Bowl teams (with won/loss records during the season and links to game-by-game logs), we see that a few teams during the past decade nearly lost four straight the season they made the Super Bowl. The 2005 Pittsburgh Steelers and 2003 Carolina Panthers each lost three straight, whereas the 2008 Arizona Cardinals lost four out of five.

In 1978, the NFL adopted a 16-game schedule to replace the previous 14-game docket. Also, prior to 1978 there was only one wild-card playoff team per conference, making it extremely unlikely that a team with four consecutive losses could even make the playoffs.