The Hot Hand in Sports

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.

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.

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.

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:

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.

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).

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.



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.

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.

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!

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.

It's only late afternoon, and we've already seen instances of pronounced hot and cold shooting in men's college basketball.

Florida State's Deividas Dulkys hit 8-of-10 on three-pointers in Seminoles' 90-57 shocker over No. 3 North Carolina. Dulkys, a senior guard, has made almost exactly one-third of his shots from behind the arc during his junior (.333) and senior (.321) years, making this afternoon's 8-of-10 performance extremely unusual. How unusual?

Using what is known as a binomial probability calculator, we can answer the question of how likely a long-term .333 three-point shooter is to make 8 (or more) out of 10 attempts from downtown. The answer is .003 or roughly 3-in-1,000.

Jaron Nash is a Texas Tech sophomore forward who plays about 10 minutes per game (Nash stats). He doesn't get to the free-throw line much, but when he has, he hasn't shot well. In fact, before making a pair from the stripe late in the Red Raiders' 67-54 loss to Texas A&M, Nash had missed 11 straight free throws.

It's not like Nash got flustered and missed several free throws in one game while in a funk. Rather, he compiled the streak gradually over five games.

• Against Oral Roberts, he missed his last free-throw of the game, after two previous makes (box score, play-by-play).
• He then went 0-for-4 against Cal State Bakersfield (box score).
• And 0-for-3 vs. Southeast Louisiana (box score).
• He had no free-throw attempts in Texas Tech's next two outings, against Oklahoma State and Baylor.
• He missed his only attempt against Kansas (box score).
• Finally, this afternoon vs. Texas A&M, he missed his first two before making a pair (box score, play-by-play).

Science News has just published an article on research by German and Austrian investigators purporting to document a hot hand in volleyball spiking, and the reporter was nice enough to contact me for comment. A hot hand in this context would mean that a player who has successfully put away a few spikes in a row (known as "kills") would have a higher likelihood of a kill on his or her next spike than the player's long-term kill percentage would suggest. A cold hand would represent the opposite, that a player whose last few spike attempts have resulted in errors (e.g., ball hit out of bounds) would have higher than usual odds of an error on the next attempt than his/her long-term percentages would suggest.

Within the constraints of the data set to which the authors had access (partial game-sequence data from top players in a German men's professional league), the analyses were conducted with full rigor and in a manner consistent with previous hot hand research. However, as I elaborate below, I feel there was at least one major limitation in the available data.

One type of analysis done by the authors used the runs test. This statistical technique requires the researcher first to list the sequence of events, in this case, a given player's order of kills (K) and errors (E). A "run" is an uninterrupted sequence of the same outcome, either all K's or all E's. The following hypothetical sequence, with few runs, would indicate streaky performance (i.e., clustering of K's and of E's):

KKKKEEEKKKKK (3 runs)

Another hypothetical sequence (with the same number of total attempts), this time with many runs, would indicate less (or absent) streakiness:

KKEKEEKKKKEK (7 runs)

According to the Science News piece:

An analysis of playoff data from the 1999/2000 season for 26 top scorers in Germany’s first-division volleyball league identified 12 players as having had scoring runs that could not be chalked up to chance. Hot-handed players’ shots contained fewer sequences of consecutive scores than expected by chance, the result of a small number of especially long scoring runs.

As volleyball fans know, however, there is a third category of outcome for spike attempts, namely the ball is dug up (or otherwise kept in play) by the defense, and the rally continues. As I told the reporter, I definitely think those hit attempts should have been included in the analyses, but they apparently were unavailable in the data set the authors received. Hitting errors were very rare in the data, so balls kept in play may have been a better measure than errors of unsuccessful spike attempts.

(Cross-posted with VolleyMetrics.)

I heard on ESPN Radio the other morning that Creighton University's Doug McMermott has been hitting at a .580 clip on three-point attempts so far this season. The sophomore's high success rate behind the arc is surprising for a couple of reasons. Typically, guards are the main outside shooters and McDermott is a 6-7 forward. Also, his percentage from long-distance a season ago was much lower (though still good) at .405 (47-for-116)

To see if there were any interesting trends this season, I plotted his game-by-game three-point shooting percentages (below), based on his seasonal log. The size of each data point corresponds to his number of three-point attempts in a given game (see graph's legend) and opponents are listed along the horizontal axis. You may click on the graph to enlarge it.


First off, McDermott doesn't attempt that many threes, just 50 (of which he's made 29) in 14 games. In his most recent outing, against Drake, McDermott didn't try any treys, despite playing 32 minutes. Twice, against Northwestern and Wichita State, he didn't miss from behind the arc, but he was only 2-for-2 in each of these games.

His steadiest stretch of hot three-point shooting occurred in Games 3-8. In each of these contests (except vs. San Diego State), McDermott shot within a range of .667-.750. Accounting for number of attempts, his most impressive game was against St. Joseph's (5-of-7, .714).

Otherwise, he's had a lot of ups and downs, but not any prolonged stretches of poor three-point shooting.

Creighton and McDermott are next in action on Saturday at Bradley.

Happy New Year! Something new for 2012 is that I've started a Twitter feed and will use it from now on to "tweet" brief statements about streaky sports performances. For more in-depth analyses, I will still be writing here on the blog.

Follow me at: http://twitter.com/#!/alanreifman.

Christmas Day was Opening Day in the NBA this season, thanks to the owners' lockout. Among the marquee match-ups was a Chicago Bulls visit to play the L.A. Lakers. For most of the second half, Chicago's shooting showed the rust of a longer-than-usual off-season. However, the Bulls caught fire in the closing minutes to pull out a stunning 88-87 victory.

Below is a shot-sequence chart for Chicago, based on the play-by-play sheet. I haven't done one of these in a long time, but it seemed fitting for the Bulls' second half. (You may click on the graphic to enlarge it.)


For roughly the first 20 of the 24 second-half minutes, Chicago's missed shots (blue) greatly outnumbered its made baskets (red). As shown in the legend at the bottom of the chart, longer line segments indicate greater distances of shot attempts. For the stretch highlighted in pale yellow, the Bulls missed 21 of 22 field-goal attempts. However, the Bulls came back to score 18 points in just the final 4:13 of the game, which was enough to win.

Not only did the Denver Broncos' six-game winning streak -- powered by quarterback Tim Tebow's frantic rallying of the team -- come to an end today. So did two other major streaks in the National Football League.

Denver lost 41-23 to New England. Tebow worked as feverishly as ever to bring the Broncos back, but Denver's defense allowed the Patriots to score enough to maintain comfortable leads.

The Green Bay Packers, who entered today's play with a 13-0 record, lost to Kansas City, 19-14.

Finally, at the other end of the spectrum, the previously 0-13 Indianapolis Colts got their first win of the season, with a 27-13 victory over Tennessee.

American football fans are abuzz with what has been termed "Tebow-mania" or "Tebow Fever." The reference is to Denver Broncos' quarterback Tim Tebow, whose unorthodox playing style, his "mixing faith with football" (in the words of one writer), and his team's winning ways have garnered him great attention.

As seen in the Broncos' game-by-game log, Denver won in Tebow's first start, October 23 at Miami, lost the next week to Detroit, and then went on a still-ongoing six-game winning streak. Amanda Rykoff, writing in an ESPN-W "roundtable discussion" captures the essence of Tebow-mania:

...I'm not going to try to explain why he's 7-1 as a starter for the Broncos this year. Or why Denver has been able to come from behind late in the fourth quarter in five of those seven victories. Or how the Broncos managed to win what seemed to be an absolutely unwinnable game on Sunday against the Bears, trailing 10-0 with two minutes left in regulation.

The blog "This Given Sunday" has a detailed summary of each Broncos/Tebow win during the stretch.

The improbable nature of many Bronco wins has led probability analysts to estimate the likelihood of Denver's comeback-heavy winning streak. The key element is the win probability in each game, based on historical data. If you click here to get to the Advanced NFL Stats website and then scroll down to the graph with all the wavy lines, you'll get an idea of win probability. First, you would select one of the lines, corresponding to how your team is doing with regard to score (e.g., leading by 7, trailing by 3). Once you have your line, then follow it along the horizontal axis corresponding to how many minutes remained in the game. For example, a team trailing by 7 entering the fourth quarter tends to win about 10% of the time.

Using a chart like the one above, analysts would then find the win probabilities for Denver in each of its victories, given the depth of Denver's dire circumstances in a given game (i.e., when it trailed the most with the least time remaining). One can then multiply the probabilities together to get an overall probability estimate of the Broncos' streak. (This is analogous to calculating the probability of rolling double-sixes with a pair of dice by multiplying 1/6, which is the probability of a six on one die, by 1/6, the probability of a six on the other die, to yield 1/36.)

Using this methodology, ESPN's Statistics and Information Blog estimates a probability of Denver winning its last six straight at "approximately one in 137,000. The odds are better that a flipped coin comes up heads 17 consecutive times."

"This Given Sunday" (the blog cited above) calculates the probability of the Broncos winning the seven games with Tebow as starter (ignoring the one loss). The verdict: "the odds of the Broncos winning all seven games from their lowest odds in each particular game situation [are] 1 in 27 million."

If you're going to use seven as the win total, you must take into account the one loss, in my view. The seven game-specific probabilities from "This Given Sunday" are as follows:

.01, .15, .58, .17, .18, .14, .01

If you throw out the high and low values, the Tebow-led Broncos typically faced around a .15 probability of winning in many of the games during the stretch. Using an online tool known as a Binomial Probability Calculator, we can ask the question: For a team that faced only a .15 probability of winning each game, what is the likelihood of that team winning seven or more out of eight games? The answer, given these assumptions, is .00001 or 1-in-100,000.

As many writers have acknowledged, Tebow obviously should not get sole credit for Denver's winning stretch. However, his contribution appears to be great. One metric is quarterback efficiency ratings, which attempt to boil down many passing statistics (e.g., completion rate, yardage gained, touchdowns, interceptions) into a single number.

ESPN's Statistics and Information Blog notes that Tebow has the highest score (96.3) on one such metric, the Total QBR, of all NFL quarterbacks this season in the final 7 minutes of the fourth quarter (using a certain minute-mark rather than, say, the fourth quarter as a whole seems a little arbitrary; would Tebow still lead if we used the final 8 minutes or 6 minutes?).

Along with Denver's defense, which has had to shut down opposing offenses, another key figure in the Broncos' recent success is kicker Matt Prater. Without his hot foot, the winning streak would be over. According to Adena Andrews's commentary in the aforementioned ESPN-W roundtable:

Prater, who was recognized as the AFC special teams player of the week after his performance against the Bears on Sunday (a 59-yard field goal to tie the game, and then a 51-yarder for the overtime victory), has hit 28 of the 29 career field goals he has attempted in the fourth quarter or overtime.

One final factor to consider is that, as unusual as the Broncos' stretch appears, maybe in the larger historical scheme it is not so unexpected. The National Football League has been around for roughly 90 years. Initially, the league had around 10 teams, and increased over the years to the teens and mid-twenties (with the NFL-AFL merger), and continued to expand to the present 32 teams. The number of games per team per season has increased from roughly 10-12 in the early years to the present 16.

Let's say that, in a given season under the modern schedule, each team would have 10 opportunities to begin a winning stretch for six games (e.g., right from the start, beginning after Week 1, beginning after Week 2, etc.). Once Week 11 had gone by, of course, it would no longer be possible to start a six-game winning streak.

As a simplification, let's say further that within each of the most recent 30 seasons, there were 300 opportunities for a six-game winning streak (roughly 30 teams X 10 opportunities); that for each of the prior 30 years, there were 200 opportunities; and for the first 30 years of the NFL, there were 100 annual opportunities. That yields roughly 18,000 opportunities. Considering the above likelihood estimate of the Broncos' winning six straight in the comeback fashion they did (i.e., 1-in-137,000), it seems that Denver's recent feat goes beyond the ordinary course of events.

Penn State's run of consecutive NCAA women's volleyball championships, which reached four last year, is now over. The Nittany Lions were eliminated this evening in three straight games by UCLA in the Sweet 16 round.

Today, I did my first-ever book signing, in the bookstore of my home university, Texas Tech. Several people, mainly friends and colleagues, came by. Thanks to everyone who came by to chat and/or buy the book!

Tiger Woods yesterday ended his streak of 26 golf tournaments without a win. For most of Woods's career, it seemed the only long streaks he would record would be of the winning variety. But, times have changed.

Last night on the NHL Network's highlight show, it was pointed out that the Detroit Red Wings have been very streaky in their wins and losses thus far this season. Based on the team's ESPN.com game log, I created the following graphic of Detroit's streaks of wins (red) and losses (grey). If you want to see the little notations for games decided during a five-minute overtime (OT) period or a post-overtime shoot-out (SO), you probably need to click on the graphic to enlarge it.


As can be seen, the Red Wings won their first five games, then lost their next six, and so forth. One of the statistical methods of detecting streakiness is known as the runs test. A "run" is an uninterrupted sequence of entirely wins or entirely losses. The fewer runs a team has, the stronger the evidence of streakiness. After 24 games, Detroit has only five runs. Based on an online runs-test calculator, into which I typed a 1 for each win and a 0 for each loss, the Red Wings' number of runs was fewer than would be expected by chance (with a significance of .00082 for those of you with some statistical training).

In this type of analysis, one must be careful to check if the team's schedule contained stretches of easy or difficult opponents, which could inflate the amount of apparent streakiness. This may be the case to some extent for Detroit, but not totally. During the Wings' string of six losses, two were to Columbus and Calgary, both of which today are in last place in their respective divisions. Conversely, during its current seven-game winning streak, Detroit has beaten some of the league's better teams, such as defending Stanley Cup champion Boston, Los Angeles, and Buffalo. The November 25 win at Boston, in fact, snapped the Bruins' 10-game winning streak.