Following the February 10, 2007 men's basketball game between Texas Tech and Oklahoma State, I introduced a new type of graph or chart to depict hot and cold shooting stretches at a glance. Within my system, the time of game, general distance (dunk/layup, longer two-point attempt, or three-point attempt), and hit/miss status of all of a team's shots can be easily grasped (only one team is depicted, and free throw attempts are not shown).

Now, whenever I see a game in which the same team goes through what seem to be pronounced hot and cold stretches, I consider plotting its shooting outcomes in my new format.

The Tennessee men's shots in an NCAA Sweet Sixteen game against Ohio State last Thursday night seem like another excellent collection to graph (game article). It was torrid shooting that allowed the Volunteers to take a 20-point just before halftime (17 at the intermission). But then some disastrous Tennessee shooting in the second half allowed the Buckeyes not only to chip away at the lead, but actually tie and pass the Vols; OSU led 72-68 with about 7 minutes remaining in the game and seemed to be taking control. The game's twists and turns were not yet complete, however, as Tennessee found its three-point groove again. The game was extremely tight the rest of the way, ending in an 85-84 Buckeye win.

Tennessee's shooting for the entire game is depicted in the graph below. I think the Vols' hot and cold periods clearly stand out, thus readily conveying the flavor of the contest from the Tennessee offensive perspective.

A question that I hear now and then about hot-hand research is whether a team's or an individual's run of success in one aspect of a game tends to be accompanied by success in other aspects. For the North Carolina Tar Heels, today's overtime loss to Georgetown in the final of the men's NCAA East Regional suggests can that different facets of basketball performance can actually operate in inverse fashion to each other.

UNC's field-goal shooting imploded in the second half and into the overtime period. Quoting from the above-linked game article:

Georgetown overcame an 11-point deficit in the second half, then ripped off 14 straight points in overtime to stun the top-seeded Tar Heels 96-84...

[The Hoyas] were helped by an amazing collapse from Carolina (31-7), which made only one of 23 field goal attempts, including its first 12 in overtime, over a 15-minute span after seemingly have [sic] the game in hand.

Yet, concurrent with their second-half slump on field-goal attempts, the Tar Heels hit a perfect 12-for-12 on free throws in the second half (not counting overtime), as shown on this play-by-play sheet.

***

Talking about clutch second-half free-throw shooting, Ohio State made each and every one of its last 20 attempts from the line in finishing off Memphis Saturday in the final of the men's South Regional (play-by-play sheet). Ordinarily, 20 straight made free throws would be a nice accomplishment, but not a blockbuster -- in January 2005, Wake Forest made 50 straight -- but the pressure of trying to make the Final Four raises the Buckeyes' impressiveness quotient in my mind.

From an Ohio State press release, these were the Buckeye players' FT percentages before the Memphis game (rearranged by me to go from highest to lowest, with the players who shot during the "Streak from the Stripe" denoted, naturally, in red).

Butler, Jamar...... .837
Lewis, Ron......... .762
Harris, Ivan........ .750
Lighty, David...... .688
Hunter, Othello.... .683
Conley Jr., Mike... .672
Cook, Daequan..... .667
Oden, Greg......... .640
Terwilliger, Matt... .621
[Two players, with only two attempts between them, are excluded from this list.]

The distribution of shots among the participating players was quite even, with each of them going to the stripe four times, except for Lewis (6) and Lighty (2). The Buckeyes' prior team FT%, weighted for number of shots during the streak, is thus:

4(.837) + 6(.762) + 2(.688) + 4(.672) + 4 (.667) = 14.652, divided by 20 = .733.

.733 raised to the 20th power then = .002.

Ohio State's perfect 20-of-20 from the stripe to close out the game is thus roughly a 1-in-500 phenomenon!

A couple of men's basketball items, neither involving the NCAA Division I tournament...

In the Division II final, defending champion Winona State had its 57-game winning streak snapped, as Barton College took this year's national title. The improbability of Barton's victory can be seen from the lead paragraph of the above-linked game article:

Anthony Atkinson scored 10 points in the final 39 seconds, including a layup at the buzzer, to give Barton College its first NCAA Division II title with a 77-75 victory over previously unbeaten and defending champion Winona State.

I watched the last couple minutes of the game on television and Winona State's combination of missed free throws, turnovers, and inopportune fouls really was not something you'd expect from a team that had won 57 straight games.

***

In NBA action, the Lakers' Kobe Bryant, whose offensive exploits have often prompted hot-hand analyses over the years, has now scored at least 50 points in four straight games. As noted in the linked article:

Only the late Wilt Chamberlain has had more 50-point games in a row than Bryant, hitting that mark in seven consecutive games during the 1961-62 season.

As shown in this Laker game-by-game log, the team has been quite streaky in recent weeks, starting February 8 against the Pistons:

at Detroit L 78-93
at Toronto L 92-96
at Cleveland L 90-99
New York L 106-107
Cleveland L 108-114
Portland L 108-112

Boston W 122-96
at Golden State W 102-85
at Utah W 102-94

Sacramento L 108-116
at Phoenix L 94-99
at Minnesota L (OT) 107-117
at Milwaukee L 90-110
at Philadelphia L 92-108
Dallas L 72-108
at Denver L 86-113

Portland W (OT) 116-111
Minnesota W 109-102
at Memphis W 121-119
at NO/Oklahoma City W 111-105

Some of the losing stretches appear to be associated with the Lakers' going on road trips, but they've also lost some home games and won some on the road. Many of you have probably seen highlights of Bryant's shooting in recent games, and some of the video clips are truly amazing. There's one, late in the game against Portland I believe, where he goes through a set of gyrations to shed the Blazer defenders before draining a three.

In tonight's NCAA men's action, Southern Cal and North Carolina traded big runs, with the Tar Heels coming out on top, 74-64, to advance to the Elite Eight (play-by-play sheet). USC outscored Carolina 15-0, expanding a 34-33 lead to 49-33 in a stretch that carried over from the first half into the second. A bit later, however, the Tar Heels one-upped the Trojans (or in this case, three-upped them) by staging an 18-0 spurt that transformed a 59-49 deficit into a 67-59 lead.

A similar scenario unfolded Thursday night in Ohio State's come-from-behind victory over Tennessee. I am preparing an elaborate graphical display of the Volunteers' shooting performance, which I should have up in the next few days.

As can be seen in my earlier entries below, I have spent the past four days following streak-related phenomena in the opening rounds of the men's (and to a lesser extent, women's) NCAA basketball tournaments. In many cases, I provided simple mathematical calculations for how likely a given streak was, given the prior (baseline) performance level of the player. Statistics, and numbers more generally, are not the only way to characterize streaks, however.

There is a growing trend toward representing statistical phenomena in a visual/graphical format. If you've read any game articles on ESPN.com, you'll see an example of such a graphical presentation, what ESPN calls a "Game Flow" graph (shown to the right of an article, midway down the page).

These Game Flow graphs contain two differently colored lines (or curves), one representing each team in a game. The horizontal axis depicts elapsed time and the vertical axis, each team's total score at a given point in time. The vertical distance between the two teams' curves at any instant represents the magnitude of lead for the team that's ahead.

If you don't want to figure out each team's score by comparing the height of the curve to the score labels on the vertical axis, you can click directly on the graph. Then, you can move the cursor left and right and see the actual score displayed for any point in time (this only works on the actual ESPN Game Flow graphs, and not on my examples below).

It is quite easy to detect team scoring runs and droughts from the ESPN Game Flow graphs. Shown below are two examples I created. In the top graph, the grey trianges depict scoring runs, as one team's point total is staying flat (a scoring drought), while the other team's total is rising concurrently (such triangles do not exist in the actual ESPN graphs, just in my examples). Such a run can help a team either pull away from its opponent or wipe out a deficit.

UCLA's first-round men's victory over Weber State, which featured Bruin spurts of 12-0, 9-0, and 14-0, provides sequences from an actual game that resemble my first example (link to article and Game Flow graph). If you have any difficulty picking out the three runs on the ESPN page, just look for the three flat line segments for Weber State (dark blue), located late in the first half and early and late in the second half.



My lower example presents an oversimplified situation in which no team scoring runs occur. Rather, one team gradually expands its lead over the other team, by accumulating small momentary scoring advantages. Team A might make a three-pointer, Team B might get a two, Team A might also then get a two, and Team B might not score on its next possession. Team A might then hit another three, followed by Team B making one free-throw out of two. Under this scenario, both teams' point totals increase in a relatively linear fashion, albeit with one team's score advancing with a sharper slope. The first-round men's game between Tennessee and Cal State Long Beach matches this latter scenario (link to article and Game Flow graph).

By coincidence, my Texas Tech faculty colleague Peter Westfall (who is in the Rawls College of Business Administration) published an early version of a Game Flow graph in 1990 in the American Statistician ("Graphical Presentation of a Basketball Game," Vol. 44, pp. 305-307). Instead of plotting two overlapping curves to represent the teams in a game and inviting readers to figure out the point difference at any moment, Westfall opted to plot only the difference score between the two teams (i.e., Team A's score minus Team B's) across time. He provides his reasoning for this choice in the article.

Tennessee exploded for a 34-0 run (from a Lady Vols' lead of 21-14 to 55-14) in a first-round NCAA game against Drake. I'll have to look at some materials I've compiled over the years, but this may be the longest run of shutting out an opponent in a game that I'm aware of.

***

Today closed out the first weekend of NCAA men's play, as the final slots in the Sweet Sixteen were claimed. A couple of games had streak-relevant developments.

UNLV's 74-68 upset of No. 2 seed Wisconsin in the Midwest Regional featured a number of swings (a 10-0 Rebel run in building a lead, a 16-2 Badger rally to get back in contention, and a 26-15 close-out by the Rebels to win the game).

As noted in one of my earlier entries below, UNLV's Kevin Kruger, the son of coach Lon, was 0-for-8 on three-point attempts in the Rebels' NCAA opener. The slump continued for a while in the Wisconsin game, but the young Kruger finally righted himself. Quoting from the above-linked game article:

Kevin Kruger shook off a shooting slump, connecting on three straight 3-pointers late in the second half Sunday as UNLV beat second-seeded Wisconsin 74-68 in the second round of the Midwest Regional...

Kruger was shooting just 1-of-15 in the NCAAs -- including 1-of-7 Sunday -- when he found the range. He tied it with a 3, then hit another to give the Runnin' Rebels the lead with just under six minutes left.

Elsewhere, Memphis scored the final 14 points of the game against Nevada, to extend a 64-62 lead to a 78-62 victory.

Evening

UCLA has just withstood a late 16-3 run by Indiana, which tied their men's NCAA tournament game at 49-49, to beat the Hoosiers, 54-49. This brings to an end an exciting day of action in the tournament.

Since I wrote about the Ohio State-Xavier overtime contest earlier today, two more games needed extra time (since I previously discussed a statistic about which team scored last in regulation to force overtime, I'll be mentioning that again).

In one, Virginia Commonwealth came back from a 19-point deficit (51-32) to Pitt with about 12 minutes remaining, suggesting that VCU would be the team with the momentum heading into OT (officially, though, Pitt scored the final points of regulation, making it 69-69, and also had a free-throwing opportunity to win in regulation). Ultimately, though, Pitt won the game in overtime, 84-79.

The other late game to go into overtime -- two OT's, actually -- was that between Vanderbilt and Washington State. Vandy had gone on a 12-0 run midway through the second half (play-by-play sheet), making four three-pointers to turn a 51-43 Cougar lead into a 55-51 Commodore advantage. It was a tight battle for the remainder of the second half, with the game tied at 55-55, 57-57, and 60-60 (the score at the end of regulation); WSU was the last team to score in regulation. The ultimate winner here was Vandy, 78-74.

One last game I want to discuss for today is that between Texas A&M and Louisville, won by the Aggies, 72-69. From a statistical perspective, probably the most interesting thing about this game was the free-throw shooting of Cardinals frosh guard Edgar Sosa. As summarized in the linked game article:

In the end, however, Sosa simply ran out of magic. After making 15 straight free throws, he missed two with 29.8 seconds left and the Cardinals trailing by one.

Coming into the game, Sosa's FT% was .676. Given this baseline, his probability of making 15 or more free throws in 17 attempts was .052 (online calculator). One way to look at this is that Louisville was fortunate to get as many points on Sosa's free-throwing as it did; however, with his having been perfect on 15 attempts going into the two late attempts, Cardinal fans probably were not expecting him to miss both.

Late afternoon

A basketball scenario that really seems to suggest the operation of momentum/streakiness is when one team stages a dramatic rally to send a game into overtime, then continues riding the wave to blow out the opponent in the extra period.

This, of course, is precisely the story of South region No. 1 seed Ohio State's comeback victory over No. 9 seed Xavier. Quoting from ESPN.com's game article:

Ohio State... was still down 61-52 with 2:54 left... [Xavier's Justin] Cage then made of one of two foul shots with 9.3 seconds left for a 62-59 lead, giving Ohio State its last chance. [Ron] Lewis came down and, with two defenders flying at him, swished the tying 3-pointer from several feet beyond the arc.

The Buckeyes then breezed through the overtime, winning 78-71. It would be hard to dispute that Xavier seemed demoralized and Ohio State, energized. But this was just one game. Are there more comprehensive statistics on overtime games?

Indeed there are, from NBA games, as displayed at the website 82games.com:

Q: How often does the team that comes from behind to send the game to overtime end up winning?

If we look at the team that scored last in the fourth quarter (thus being the final "come from behind team" since leads can go back and forth), they have won 44 of 79 overtime games, good for a mild 56% win rate which given the sample size doesn't suggest momentum plays a big role. In multi-overtime games, the team scoring last in an overtime period to send it on to another OT round, is only 5-9 (36%) and so if we combine the two (scoring last in the fourth quarter/overtime to send it to overtime/another overtime) the "momentum team" is just 49-44 (53%).

One can argue that the overall category of "scored last in the fourth quarter" is very broad, perhaps including some scenarios as dramatic as Ohio State's, but others that were not that dramatic, thus watering down a potential momentum effect. For example, if one team scored to tie a game with two minutes remaining in regulation and then neither team scored during the final two minutes, one would not expect the team that tied the score to necessarily get much of a lift in overtime.

Until more extensive analyses are done, however, our best current evidence seems to be that any carryover momentum of sending a game into overtime is fairly modest.

Late night

Several of the spurts detailed below occurred at the very beginning of games. Such bursts can also, of course, occur at the end of games. Virginia Tech did just that, scoring the final 12 points of the game against Illinois, as the Hokies edged the Illini, 54-52.

Early evening

Today's afternoon-session games continued to exhibit team scoring bursts, but one in particular had an interesting dimension.

In the Midwest region, No. 15 seed Texas A&M-Corpus Christi ran out to 10-0 and 19-4 leads over No. 2 seed Wisconsin.

The Badgers ultimately came back and won the game, 76-63, tripling their first-half scoring total (19) with 57 points in the second. That's not something you see every day!

What makes the early spurts by TAM-CC so unusual is that they were unleashed by an underdog (and a heavy one at that) against a favorite. In nearly all the runs detailed in my morning entry below, it was favored teams lighting up underdogs (the run by Michigan State, a 9 seed, against 8th seeded Marquette is an exception).

Having heavily favored teams explode against underdogs makes it difficult to distinguish between possible explanations for the runs. Is it simply that the favored teams overmatched their opponents on sheer talent, or did the favored teams exhibit runs of momentum.

When a heavy underdog goes on runs, as did TAM-CC against Wisconsin, I think we can safely rule out the explanation that TAM-CC was physically superior to the Badgers. If it can be shown that the Islanders had a pronouned tendency to go on scoring runs during this past season, then they could be labeled as a streaky team. Otherwise, their runs would likely be due simply to chance.

Notre Dame staged a 22-3 second-half rally against Winthrop to get back in the game after trailing by 20, but Winthrop pulled away again at the end to win, 74-64.

Virginia took a 19-2 lead over Albany. The linked article also discusses how UVa's J.R. Reynolds responded after some recent poor-shooting games and got "on a roll" today.

Although his team (coached by his father Lon) ultimately defeated Georgia Tech, 67-63, UNLV's Kevin Kruger went 0-for-8 on three-point attempts, the only shots from the field he tried all game. Looking at young Kruger's career statistics, he seems to have about a .36 shooting percentage from behind the arc (his four yearly percentages have been .360, .354, .403, and .365). Using .36 as a baseline success rate on threes (which translates into a failure rate of .64), we simply raise .64 to the 8th power to estimate Kruger's likelihood of missing eight straight threes (analoguous to estimating the probability of double sixes on dice by raising 1/6 to the 2nd power, for 1/36). The answer in Kruger's case is .028.

Morning

NCAA tournament action always seems to produce a lot of team scoring runs, and yesterday's opening day of men's play was no exception.

Butler went on a 17-0 second-half run en route to a win over Old Dominion.

Michigan State jumped out to a 14-0 lead over Marquette, the latter going nearly the first 10 minutes of the game without a point.

Pitt got off to a 13-0 lead over Wright State.

North Carolina darted off to a 22-3 lead over Eastern Kentucky. The Tar Heels' lead, which got up to 27, at one point got cut to four, but UNC pulled away again.

In the Georgetown-Belmont game, "The Hoyas reeled off 11 straight points as part of a 20-4 run, holding the Bruins without a field goal for 8 minutes..."

Another interesting development in this game was the 4-of-6 three-point shooting of Georgetown's Jessie Sapp. Again quoting from the game article:

Belmont coach Rick Byrd wanted Georgetown's shots to come from Sapp -- who entered shooting 28 percent from 3-point range -- but that plan backfired when the sophomore guard matched his previous career high from beyond the arc by halftime.

"I guess that shows you that at least we had a game plan," Byrd said. "It might not have been a good one, but we felt that they're obviously a better team than we are, and would win the game more often ... To [Sapp's] credit, he didn't look like a 28 percent 3-point shooter to me."

Using an online binomial calculator, the probability of a prior .28 three-point shooter making 4 (or more) out of 6 is .0557. Statisticians typically look for a probability of .05 or smaller to conclude that some occurrence is unlikely to be due to chance, so Sapp's performance nearly qualifies.

Ohio State scored the final 11 points of the first half against Central Connecticut State.

UCLA had spurts of 12-0, 9-0, and 14-0 against Weber State.

In Washington State's victory over Oral Roberts, "The Cougars started the second half 9-for-11 and used an 18-4 run to build a 44-32 lead..."

Louisville at one point led Stanford 41-13 and, finally, Vanderbilt clobbered George Washington early, going on a 15-1 run and leading 45-20 at the half.

***

In the NBA, the L.A. Lakers lost their seventh straight game last night, getting blown out by the Denver Nuggets. It is the first time in Phil Jackson's 16-year NBA coaching career that a team of his has had this long of a losing streak.

Last night featured some impressive shooting streaks, one in a college NIT game and the others in an NBA game.

In the NIT, Georgia senior Levi Stukes hit 8-of-9 on three-point attempts, as his team defeated Fresno State. The logic of how we could analyze Stukes's performance is identical to that I followed for Florida's Lee Humphrey earlier in the season. In this instance, I'll just present an abridged version for Stukes.

As should be familiar to regular visitors to this site, we need to start with a player's prior (i.e., baseline) probability of success at the given task, before the streaky performance in question. For three-point shooting, I think we can safely assign Stukes a prior probability of .400. As seen in his career statistics, other than his sophomore year when he shot .305, his other seasonal percentages from behind the arc range from .377-.415.

We then go to the trusty binomial calculator, and ask what the probability is of a long-term .400 three-point shooter hitting eight (or more) out of nine. The answer is .0038, around 4-in-1,000.

Last night's other hot shooting occurred in the showdown of the NBA's two best teams, the Phoenix Suns visiting the Dallas Mavericks. In this double-overtime thriller, won by the Suns 129-127, the torrid performances were as follows:

*Phoenix's Amare Stoudemire shot 16-of-19 from the floor.

*Dallas got some exceptional three-point shooting from Jerry Stackhouse (5-6) and Jason Terry (5-7). See the box score.

*The above-linked game article alluded to:

...the Mavs turn[ing] what had been a 16-point deficit into a 15-point lead going into the fourth quarter.

Phoenix rall[ying] by opening the fourth by making eight straight shots and 10-of-12.

And now we have NCAA tournament action starting!

As men's basketball conference tournaments have come to a close and it's on to the NCAA tournament (the "Big Dance"), I wanted to mention briefly a final few streaky performances.

In Saturday's Pacific-10 tournament final, quoting from this article, Oregon's Bryce Taylor "went 11-for-11 from the field, 7-for-7 from 3-point range and 3-for-3 from the line before being removed with 6 1/2 minutes left and the Ducks ahead by 40 points." (The 11-of-11 shooting from the floor includes the three-point attempts, so he ended up with 32 points.)

Today's ACC tournament final was a fierce battle between a highly ranked University of North Carolina team, which was definitely going to the NCAA tourney but wanted to keep its momentum going and improve its national seed, and a scrappy NC State squad, which could only make the NCAA field by completing a run of four straight upsets (the first three of which they had gotten) to nab the conference's automatic bid associated with winning the ACC tournament. The Tar Heels ultimately prevailed 89-80, but not without multiple attempts by the Wolfpack to claw back into the game.

Whether in spite of the pressure or because of it, both teams turned in exemplary performances at the free-throw line. NC State went 16-for-19 (.842), whereas UNC, a stellar 23-for-24 (.958).

According to this pre-ACC tourney stat sheet from the NC State athletics site, the Wolfpack came in with a .722 FT%. Using this binomial calculator, the probability of a .722 team hitting 16 (or more) of 19 free throws is .18; though an 18% probability is pretty small, it does not meet the conventional .05 cut-off for statistical significance.

Looking at a UNC stat sheet (also pre-ACC tourney), the Tar Heels' FT % was .705. The probability of a team that typically shoots free throws at such a clip now making 23 (or more) of 24 is .003 -- now that's significant!

Finally, today's Big 12 title game, won by Kansas 88-84 over Texas in overtime, featured a number of big runs. The Longhorns enjoyed an early 32-10 lead. However, the Jayhawks went on a 24-7 run late in the first half to get back in the game, and later unleased a 9-0 spurt.

With the major college basketball conferences' men's tournaments reaching their quarterfinal and semifinal rounds as of the close of play tonight, a number of streaky performances have occurred. Here's a quick list:

*Defending NCAA champion Florida, bouncing back from some recent difficulties, got off to 17-0 and 28-4 leads over Georgia in the SEC quarterfinals, en route to the next round.

*Pitt outscored Louisville 20-2 to start the second half of the teams' Big East semifinal contest tonight. Pitt won 65-59.

*A night earlier in Big East play, Georgetown took a 26-2 lead over Villanova, but won only 62-57.

*Texas super frosh Kevin Durant, a big man (6-9) with an amazing shooting range -- not unlike another Kevin, Garnett -- missed his first 12 shots tonight against Baylor in the Big 12 quarters, but came back to score 29 points. Paralleling Durant's fortunes, the Longhorns rallied from 20 down to beat the Bears.

I've been busy and haven't blogged on this page for a while. I, at least, want to mention some recent basketball streakiness-related developments, some of which I may return to in more detail at a later point.

Just tonight, in the championship game of the Big Ten women's tournament, the nation's No. 12-ranked Purdue darted off to a 21-0 lead over No. 5 Ohio State, en route to a 64-52 victory.

Also as of tonight, the Miami Heat's Jason Kapono, a 6-8 guard/forward swing player out of UCLA ('03), is leading the NBA in three-point shooting percentage at a .516 clip (96-186), similar to Steve Kerr's league record for a season of .524. The erstwhile Bruin also won the three-point shooting contest during All-Star Weekend a couple weeks ago. My friend Gregg in Los Angeles, with whom I attended UCLA in the early 1980s, has been keeping me posted on Kapono's hot shooting. The NBA three-point distance is 23 feet, 9 inches around the arc (22 feet along the baseline), compared to 19-9 all around the arc for men's college play, which makes Kapono's accomplishments all the more impressive.

On February 26, the Dallas Mavericks became the first team in NBA history to assemble three different winning streaks of at least 12 games in the same season, with a 110-87 win over Atlanta.

The Mavs' overall win-loss record is quite gaudy -- 47-9 after defeating the Hawks, and currently 50-7 -- which brings up an important issue. As discussed previously by Gabe Farkas, teams with outsized win totals and very few losses are bound to have long stretches of wins or, stated differently, few transition points between wins and losses. Be sure to take a look at his graph resembling St. Louis's Gateway Arch, which shows a curvillinear relation between teams' numbers of wins and their expected numbers of runs (uninterrupted streaks of wins or of losses).

The Dallas Morning News has a Mavericks Blog, on which statistical issues related to the team's winning ways are often discussed.