LATE-NIGHT UPDATE: The University of Dayton -- though ultimately winning its game against Auburn, 60-59 -- went 0-for-24 on three-pointers. As a result, the following entry from the NCAA basketball record book must now be erased:

THREE-POINT FIELD-GOAL ATTEMPTS WITHOUT MAKING ONE
22—Canisius vs. St. Bonaventure, Jan. 21, 1995

[Update: I later learned of an 0-for-24 game by South Carolina State in 2004.]

Dayton entered tonight's game hitting from behind the arc at a .395 clip (for purposes of the calculations to come, the same figure can be expressed as a .605 failure rate, i.e., one minus the success rate).

To estimate the probability of a team with the Flyers' previous success rate going 0-for-24 on three-point attempts, we simply raise .605 to the 24th power, yielding .000006 or 6-in-1 million.

This analysis assumes independence of observations, that the outcome of one Dayton shot has no bearing on the next, like coin flips. Though reasons can be generated for why basketball shots should not be independent -- such as confidence, momentum, or fatigue -- sports performances have tended to be consistent with an independence model.

One reason a team might have such a disastrous night is that it fell way behind and jacked up a lot of desperation three attempts. This does not appear to be true of the Dayton situation, however, as the Auburn game appears to have been close throughout; the Flyers led 26-21 at the half and won in overtime.

Another line of inquiry is whether the lion's share of Dayton's trey attempts somehow were taken disproportionately by the team's weakest shooters from long distance, thus rendering the aforementioned .395 baseline inappropriate. Looking once again at the Flyers' pre-Auburn stats, Dayton's top three-point shooters coming in were Marcus Johnson, .500 (7-14); Mickey Perry, .455 (5-11); Chris Johnson, .417 (5-12); and Luke Fabrizius, .412 (7-17). According to the box score of the Dayton-Auburn contest, this quartet took 13 of the team's 24 shots, so at first glance, the Flyers' best long-distance shooters appear to have been reasonably well represented.

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Trailing 65-57 to Georgetown with 9:15 remaining in a battle of nationally ranked teams earlier today, Tennessee went on a 23-6 run to take an 80-71 lead right around the two-minute mark. Then, with the Hoyas starting to foul in desperation in the final minute, the Vols went 7-of-8 from the free-throw line to take a 90-78 victory. The second-half play-by-play sheet from ESPN.com can be viewed here.

Though Oklahoma and Texas Tech both came into their game last night with records of offensive explosiveness, only the Sooners kept the scoreboard operators busy, shellacking the visiting Red Raiders, 65-21. As the following brief excerpts from this morning's Lubbock Avalanche-Journal detail, Texas Tech was outplayed in all facets of the game:

Every element that the Raiders had deployed on the way to a 10-0 start – pass protection, the run game, Graham Harrell-to-Mike Crabtree and timely defense – fell flat on senior night at Owen Field/Memorial Stadium...

Tech had allowed only one 100-yard rusher all season, but OU had two. Tech had allowed only five sacks all season but, against OU, gave up four. The Raiders’ usually prolific offense was 1-for-13 on third down.

The latter bit of faltering, in particular, is highly amenable to statistical analysis; it will thus be the focus of the rest of this entry. Prior to last night, Texas Tech had a .64 (48/75) third-down conversion rate (i.e., success at getting first downs) in Big 12 conference play.

Using this online calculator for binomial probabilities (i.e., events that can have two outcomes, such as success and failure), one can ask what the probability is of a team with a prior .64 success rate achieving at a level of 1-for-13 (or worse) on third-down opportunities. Because any one specific occurrence, such as 1-for-13, is likely to be rare, statisticians add in the "or worse" element (or in other scenarios, "or better").

The answer is .00004, or 4-in-100,000. This fraction can be simplified further, allowing us to say that the Red Raiders' third-down performance last night would occur around once in 25,000 games!. Allowing for the fact that Oklahoma's defense (last night, at least) is better than that of Tech's other Big 12 opponents, the odds would be somewhat less astronomical. Still, the Raiders' dismal third-down conversion rate was pretty surprising.

This calculation can be broken down into different components. To estimate the probability of Texas Tech going 0-for-13 on third down, we simply raise .36 (the team's prior failure rate on third down) to the 13th power, yielding .000002.

For the probability of exactly 1 success and 12 failures in 13 opportunities, we take .36 to the 12th power, times .64 to the first power. This yields .000003. However, there are 13 different ways a team can go 1-for-13, namely getting its single first down on either its first, second, third,..., twelfth, or thirteenth opportunity. We thus multiple the previous .000003 by 13, yielding .00004. We would also add in the aforementioned probability of a 0-for-13 performance (.000002), but the solution would still round to .00004.

There would seem to be two major factors that determine success on third-down opportunities: whether a team finds itself with long distances to go to earn a first down; and how well the team moves the ball, even on short-yardage situations.

According to the OU-TTU play-by-play sheet, the distances to go on the Red Raiders' third downs were: 9, 10, 22, 3, 4, 2, 18, 10*, 11, 7, 21**, 6, and 1 (the single asterisk denotes the one successful conversion, which actually resulted in a touchdown, whereas the double asterisk indicates where an Oklahoma personal foul gave Texas Tech a first down, which apparently is not credited as an "earned" first down).

As can be seen, both of the above suggested factors appeared to be operative. The Red Raiders were left with several long third-down situations (7 with 9-or-more yards to go), but they also failed on several short opportunities.

This Saturday night, two of the most explosive offensive teams in college football -- Texas Tech and Oklahoma -- will meet in a game that has possible national championship implications. For starters, I thought I'd simply graph the two teams' offensive sequences (i.e., whether they resulted in touchdowns, field goals, or no score) against their five common Big 12 conference opponents (this information is available via ESPN.com's collection of college football team pages, by going to a given team's page, looking up particular games, and finding the Drive Charts). You can click on the following graph to enlarge it.


There are formal statistical tests one can do, such as the "runs test," which examines whether like events (such as touchdowns) are more commonly clustered together than would be expected by chance. Such statistical tests require large sample sizes, however, and the only way they could be obtained in the present situation is through the questionable practice of combining games into a long chain (i.e., have the final drive of one game be grafted onto the first drive of the next game).

Therefore, it's probably best to view the above chart only in a descriptive manner. As can be seen, both the Red Raiders and Sooners have put together several streaks of at least three consecutive touchdown-scoring drives. Though Oklahoma has recorded more such streaks than has Texas Tech, the Red Raiders seem to have more of a tendency to keep their streaks carrying over from one quarter to the next (and even over the halftime break).

In the games examined, Oklahoma has only one fourth-quarter touchdown, total. In many of games, however, the Sooners may have been trying not to run up the score.

One can also break down these streaks into smaller units than the scoring drive, such as pass completions. In Texas Tech's fast start against Kansas, for example, Red Raider quarterback Graham Harrell hit on 22 of his first 24 passing attempts. Oklahoma QB Sam Bradford once completed 18 straight passes in a game.

As a final note, amazing spurts are certainly not limited to Texas Tech and Oklahoma. Trailing Troy 31-3 in the third quarter last Saturday, LSU scored 37 unanswered points to win going away, 40-31.

Fittingly for Halloween night, the goaltenders for the Vancouver Canucks and Anaheim (Mighty) Ducks had to keep their masks on longer than usual.

Tied 6-6 after regulation, the teams played a five-minute overtime period, but there was no scoring. The game then went to a shootout, a sequence of one-on-one shooter-goalie encounters with the teams alternating roles. Vancouver won the shootout, 2 goals to 1, resulting in an official 7-6 final score (i.e., the shootout win counted as 1 goal in the final score). This was far from a normal shootout, however!

As per the rules, each team fields three shooters to go up against the other team's goalie, analogous to a three-inning baseball game. If the two teams are tied after the initial three rounds -- which was the case between Vancouver and Anaheim -- then an "extra-innings" system is used. As soon as one team scores in a round and the other team doesn't, the game is over.

After the Canucks and Ducks completed the main three-round shootout tied at a goal apiece, one extra round after another kept passing by with neither team able to score. Here is a line score I created from a narrative summary in the above-linked game article.


That's right, the shootout lasted for 13 rounds! Both goalies -- Vancouver's Roberto Luongo and Anaheim's Jonas Hiller -- sparkled in the shootout. Luongo was beaten only once by the Ducks in the shootout, whereas Hiller stopped 11 straight Canuck shots before giving up the game-winner.

(Unsuccessful attempts can be divided into saves, shots that would have gone in but for the presence of the goalie, and misses, shots that were off-target wide or high. I would argue that goalies still deserve some credit for misses, as good goaltending likely induces shooters to take risky shots, such as aiming for corners of the net.)

The question I decided to pursue was as follows: Given these goalies' prior success rates, what was the probability of each netminder doing as well as he did in last night's shootout?

In conducting this analysis, I was aided greatly by the amazing website NHLShootouts.com, which provides extensive, up-to-date data on shootouts.

Hiller did not have a lot of experience in shootouts; other than last night's, he participated in three shootouts last season, giving up 5 goals in 12 shots overall. The NHL Shootouts website gives Hiller a save percentage of .583 (evidently not distinguishing saves from misses). I next went to the Vassar College online binomial calculator and asked how likely it was that a goalie with a prior .583 success rate could stop 11 (or more) shots out of 13. The answer comes to a probability of approximately .05, a level social scientists would traditionally consider "statistically significant."

A similar analysis was conducted for the more experienced Luongo. Over the three seasons preceding the current one, Luongo had participated in 30 shootouts, compiling a cumulative success rate of .714. For a goalie with such a percentage to rebuff 12 (or more) shots out of 13 yields a probability of .08. Another way to look at this finding is that Luongo is a better shootout (if not overall) goalie than Hiller (albeit based on small sample sizes), so Luongo's stellar shootout performance would be less surprising.

For the record, last night's Canuck-Duck marathon was not the longest shootout since the NHL started using it as an ultimate tie-breaker in the 2005-06 season. The record is at least 15 rounds, from a November 2005 contest (the score was 4-3 within the shootout).

By finishing off the Tampa Bay Rays in Wednesday night's rain-delayed World Series finale, Philadelphia Phillies' closer Brad Lidge achieved relief-pitcher perfection on the "Save" statistic.

Combining regular-season (41 games) and post-season (7 games) play, every time Lidge had the opportunity to protect a Phillies' lead at the end, he succeeded. He thus ended up a perfect 48-for-48 on save opportunities.

A few years ago as a member of the Houston Astros, the 6-foot-5 right-hander was so powerful at the end of games that the team's middle/set-up relievers knew that their job was to provide a "Bridge to Lidge." As noted in this article:

With Houston in 2004, [Lidge] averaged 15 strikeouts per nine innings. In the seven-game National League Championship Series loss to St. Louis, he held the Cardinals to one hit in his eight innings...

The Astros and Cardinals met again in the next year's NLCS, and when Houston was one out from the World Series, Lidge gave up a game-deciding homer to Albert Pujols. Houston won the pennant in the next game, but somehow, Lidge seemed to become better known for that Pujols homer than for all his good work.

For now, at least, it looks like the "Heartbreak Lidge" moniker he obtained in Houston will likely be a thing of the past.

One week from tonight, two of college football's hottest teams (both undefeated) and most hot-handed quarterbacks will do battle when the University of Texas plays at Texas Tech.

The Red Raiders' Graham Harrell hit on 22 of his first 24 passes today, as Tech routed Kansas 63-21. The game was once tied 14-14 before the Red Raiders scored 49 straight points. The Jayhawks, fans will recall, won the Orange Bowl and finished as the No. 7-ranked team in the nation last season. Although clearly not as good this year, KU came into the game ranked 19th in the nation, thus serving as a quality opponent for Texas Tech.

The Longhorns' Colt McCoy has been similarly scintillating. As this ESPN.com article notes:

Through Texas' first six possessions Saturday [in a 28-24 win over previously unbeaten Oklahoma State], McCoy was ridiculously good. He led the Horns to four touchdowns and completed 30 of 33 passes including a school-record 18 straight at one point. Go back to last week's dissection of Missouri and McCoy was 59 of 65 (91 percent accuracy) across a span of 15 possessions, with a couple of drops and batted balls in there. On those drives Texas scored 77 points.

Inspired, presumably, by Boston's amazing comeback from a 7-0 deficit to win Game 5 of the American League Championship Series against Tampa Bay (although ultimately not the series), Tom Tango has just written a piece at Hardball Times probing the historical record of similar comebacks. Does evidence exist for comeback wins inspired by what the article calls "in-game momentum"?

Specifically, Tango identified games in which a team rallied from five or more runs down to tie a game, but not take the lead before the end of the inning. The idea is that the game would now have been back to square one -- dead even -- but one team would have had the momentum. Tango then investigated how often the latter team went on to win the game by scoring in a later inning, which might be seen as a sign of momentum on the part of the team that tied the game (or demoralization on the part of the team that squandered the lead).

Tango finds evidence that teams coming back from five or more runs have won games with a slightly greater frequency than that of teams coming back from smaller deficits (and thus with lesser momentum). However, he urges caution as follows:

...don't forget that we're talking extreme momentum here; in-game momentum in which the team scored five runs in an inning to tie the game. One must believe that the effect of momentum must be even less day-to-day.

Boston's postseason dominance over the California/Anaheim/Los Angeles Angels continued last night, with a 3-2 Red Sox victory to capture the first-round series, three games to one. According to a sidebar note with the above-linked article:

The Red Sox improved to 4-0 all-time against the Angels in postseason series, having beaten them in 1986, 2004, 2007, and 2008. Boston has also won 12 of the last 13 playoff games against Los Angeles.

The story begins, of course, with the 1986 American League Championship Series, in which the Red Sox, trailing three games to one, staged an unlikely comeback to take three straight and win the series, 4-3.

Boston then recorded 3-0 series sweeps against the Halos in 2004 and 2007, before winning 3-1 this year.

As I like to point out from time to time, streaks can arise from some combination of (a) sharp ability differences between the two competitors; (b) momentum and other psychological factors (though most statisticians are skeptical of this); and (c) random chance.

The idea that the Red Sox were substantially superior talentwise over the Angels -- for this year at least -- can be safely ruled out, as the Angels won eight out of the nine regular-season meetings between the teams.

Statistical research tends to show that athletes who have experienced consecutive successes (e.g., made baskets, hits in baseball) do not raise their probability of success on the next attempt, relative to their long-term baserates. For example, a long-term .50 basketball shooter will not be any more likely than .50 to make his or her next shot after making, say, three straight hoops. This runs contrary to the popular belief that the athlete is "hot" and therefore at an elevated rate of success. Another way of conveying the lack of a true "hot hand" is that athletes' instances of several successes in a row tend not to occur any more frequently than runs of several heads in a row (or tails in a row) from large numbers of coin tosses.

Despite most studies' lack of evidence for hot-handed performances beyond chance, however, I have never disputed that athletes may feel something special is going on during their runs of success. One type of perception, until recently (I thought) only in the realm of the anecdotal, is that relevant athletic stimuli (e.g., a basketball hoop, the baseball on the way from the pitcher) look larger or clearer than when the athlete is not in the midst of a streak.

An item from July on the "Nudge" blog, which I did not see until recently, cites evidence that successful athletes really do seem to see their targets as bigger.

The research in question is by Jessica Witt of Purdue University and colleagues, and is entitled "Putting to a bigger hole: Golf performance relates to perceived size" (published in Psychonomic Bulletin & Review, June 2008). The above-linked Nudge posting provides a concise description of the study's details.

In looking up Dr. Witt's faculty website, I noticed that she had published a similar study (with Dennis Proffitt) with recreational softball players ("See the ball, hit the ball: Apparent ball size is correlated with batting average," Psychological Science, December 2005).

The story does not end there, however. In their softball article, Witt and Proffitt cite a study by Wesp et al. (2004, Perception & Psychophysics) that "demonstrated that dart-throwing ability affects perceived size of the target. Participants who hit the target with fewer attempts selected larger circles as matching the size of the target than participants who were not as successful" (p. 938).

My apologies for not noticing these studies earlier, but better late than never!

With Boston's win over Cleveland last night, the New York Yankees have officially been eliminated from the postseason picture. The Yankees' streak of consecutive playoff appearances now ends at 13, one short of the Atlanta Braves' record of 14. The Bronx Bombers got more out of their postseason appearances, however, with four World Series championships to Atlanta's one.

The Houston Astros, whose hotness I wrote about in the posting immediately below, have now gone cold. In two games against the Cubs, the Astros were no-hit last night by Carlos Zambrano and then were held without a hit for six innings today by Ted Lilly. To provide some historical perspective, MLB.com reported several items of no-hitter trivia. The following appears most applicable to the Astros' latest futility:

According to the Elias Sports Bureau, the last time a team pitched or succumbed to a streak of 15 consecutive no-hit innings was June 2-3, 1995, when the Expos held the Padres without a hit for that span. On Sept. 25-27, 1981, the Astros held the Dodgers without a hit for 16 consecutive innings.

As they almost always seem to do, the Houston Astros are again pulling off a late-season hot streak. With a 6-0 win today against Pittsburgh, the Astros have now won 14 out of 15. As this ESPN.com/AP article notes:

Houston was 66-66 and 11 games back of Milwaukee, the NL wild-card leader, before the run began Aug. 27. The latest win closed the Astros within three games of the Brewers...

One method statisticians use to help determine if a given team's (or athlete's) streakiness is more than just a chance occurrence is to see if the team/athlete has shown any ability to repeat the special performance year after year. I recall the Astros' having had strong finishes in recent years, so I decided to look at the numbers for the last four completed seasons. Shown for each year are the Astros' records as of the last day of July, for the month of August, and for the month of September (plus any regular-season games played in October). You can click on the year to see the Astros' game-by-game log.

YEAR........THRU JULY..........AUGUST.........SEPT/OCT

2004........52-52 (.500)......17-11 (.607).......22-7 (.759)

2005........57-48 (.543)......13-14 (.481).......18-11 (.621)

2006........49-56 (.467)......17-12 (.586).......16-12 (.571)

2007........46-60 (.434)......15-14 (.517).......12-15 (.444)

Thus, with the exception of 2007, in recent years the Astros indeed have regularly put together strong finishes in August and/or September. Brewers, look out!

Leave it to those L.A. Dodgers to, once again, put together an amazing cold-to-hot pattern. In 2006, as I wrote about at the time:

The Dodgers started off the second half of the season, right after the All-Star Break, by losing 13 out of 14. They've now rebounded by winning 17 of 18...

Now in 2008, beginning August 22 with a loss at Philadelphia and culminating with today's 5-3 win at home against Arizona, the Dodgers have immediately followed up an eight-game losing streak with a winning streak of the same length (game-by-game log).

Given that a team has gone 8-8 during a 16-game stretch, how likely is it that such a record has been accomplished by losing eight straight and then winning eight straight (or vice-versa)?

Perhaps the easiest way to think about this problem is to imagine 16 boxes (representing the number of games) and eight cards, each of which has a "W" on it, for wins. Then we can ask: In how many ways can the eight wins be distributed into the 16 boxes? Obviously, there are lots of ways for this to happen. In addition to winning either the first eight or last eight games, a team might win games 1-2-4-5-7-10-11-12 or games 3-4-5-8-10-13-14-15, for example.

Fortunately, there's a relatively simple formula for determining how many ways eight wins can be distributed among 16 games. It's known as the "n choose k" formula, where in this case, n = 16 and k = 8. Using this online calculator, we find that there are 12,870 possible ways to distribute eight wins in 16 games.

So, indeed, the Dodgers' particular pattern is quite rare. Of course, it was the unusual nature of the sequence that drew me to analyze it in the first place, after the fact.

Perhaps it's a case of post-Olympic fatigue, but whatever the reason, the gold-medal beach volleyballers Misty May-Treanor and Kerri Walsh have had their 112-match winning streak snapped, in an Ohio tournament.

W.M. McEntire posted a message on the SABR members' e-mail discussion list, noting that the Cleveland Indians' current 10-game winning streak gives the team both 10-game losing and winning streaks this season (the losing streak occurred during late June and early July, whereas the more recent log of games is available here). According to McEntire, this is the 21st time since 1900 that a team has exhibited such a pattern.

A couple of major streakiness items from the Olympics:

The U.S. beach volleyball duo of Kerri Walsh and Misty May-Treanor won the gold medal for the second straight Olympiad. Overall, the pair has won 108 straight matches.

Another U.S. women's juggernaut, the softball squad, had its reign of three straight Olympic golds snapped by Japan. The gold-medalist pitcher, Yukiko Ueno, had quite a last two days. After pitching eight scoreless innings (seven innings is the regulation length) against the U.S. on Wednesday before losing in the ninth, Ueno came back later the same day to pitch all 12 innings of a 4-3 win over Australia. Then today, in a rematch with the Americans, she limited them to one run, the Japanese winning 3-1.

Given that Ueno gave up several runs during this stretch, I'm not sure we can say she always had the "Hot Arm," but I don't think anyone would dispute awarding her the title of the "Durable Arm!"

The Chicago White Sox hit four homers in a row Thursday afternoon, in a 9-2 victory over Kansas City. Such a feat has been accomlished six times in MLB history. Interestingly, half of these occurrences have taken place between 2006-2008 and the other half between 1961-1964 (see the linked article for a chart listing all the teams and players involved).

Baseball fans are still buzzing about Boston's 19-17 win over Texas last night, a game in which it looked like the Sox's 10-run first inning might go for naught. Obviously, several batters had big nights. The hottest hitter, though, appears to be the Rangers' Marlon Byrd who, according to this article, "went 5 for 6 and is 21 for 34 in his past eight games."

Trailing 10-11 early on against China, the U.S. women's Olympic basketball team closed out the first quarter on a 23-0 run (article, play-by-play sheet).

Deja vu, all over again. Similar to how he won the British Open last month, Padraig Harrington had the hot putter down the stretch to win the PGA Championship today.