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Are OT goals more really frequent than regulation goals?
14 years ago
Fair warning. There will be a ton of mathy nerd talk ahead. So, for those without a strong background in mathematics, statistics, and probability, it may be best to skip this entry.
Now, as a sports fan, I know that despite new rules that have come into place since the lockout, goals in the NHL are still a rare commodity. During the first season after the lockout, goals skyrocketed to a still-meager 6.2/game. Since there, teams have become more defensive minded, reducing the number of goals scored throughout the course of a season. In the past two seasons, goals have hovered between the pre-lockout era of 5.1/game and the 2005-06 season average.
One of the major changes was the new overtime rules during the regular season. If the two teams were tied after the first 60 minutes of play, they would play in a 5-minute overtime in which each team would have 4 skaters on the ice as opposed to the 5 skaters per team used in regulation. If the teams were still tied, a shootout would take place between the two teams.
Being down a skater changes a lot about a hockey game including the speed of the game itself and the chemistry of the players on the ice. But, does this result in a difference in goals scored? Since most teams sacrifice a forward during overtime, some would say a team is less likely to score in overtime. However, with more open ice and a faster pace, some would say a team is more likely to score in overtime. But is either of these really true? Let's find out.
First off, we need to find not only the average number of goals scored in regulation but the frequency of them. Hopefully, the two are closely related.
Over the last two seasons, 13,632 goals have been scored in 2,460 regular season games in the NHL. But, if we use 5.5 goals/game, we are misrepresenting the number of goals scored in regulation since those 13,632 goals include overtime and the "plus-one" goals added to the final after a shootout.
Now, all overtime award one goal and always one goal to the total for a game. So, all we have to do is find the number of games that went into overtime, and subtract that from the total number of goals. With 698 overtime games in the past two seasons, the math is easy: 13,632 - 698 = 13,034 regulation goals. This changes the average number of regulation goals to 5.3 goals/game.
As stated earlier in this entry, goals in the NHL are rare. There is a probability designed for situations in which occurrences are rare, but must be at least 0 and can be any non-negative integer. This distribution is called a Poisson distribution, or the Poisson law of small numbers. But first, we have to test this distribution to see if the number of goals scored in the 60 minutes of regulation fits this distribution.
Now, I don't have the time to go thru all 1,230 games this past season. However, we can look at games that went into overtime tied at 0-0. According to Poisson distribution, the probability of any given game going into overtime tied at 0-0 is e^(-5.3), better read as 0.5% or 1/200. Using this information, we can expect 6.14 games will be scoreless after 60 minutes each season. Let's test this.
Our null hypothesis is that 0.5% of games will be scoreless after 60 minutes. Our alternative hypothesis is that our null hypothesis is incorrect. To reject the null hypothesis, we will use an alpha of 5%, which translates into a 95% confidence interval.
During the 2010-11 season, 1,230 regular season games were played. 8 of them went into overtime tied 0-0.
New Jersey at Buffalo (Oct 13, 2010)
Calgary at Nashville (Oct 19, 2010)
Pittsburgh at St. Louis (Oct 23, 2010)
Buffalo at Ottawa (Dec 4, 2010)
Tampa Bay at Washington (Jan 4, 2011)
Los Angeles at Minnesota (Feb 1, 2011)
Ottawa at Toronto (Feb 19, 2011)
New Jersey at Pittsburgh (Mar 25, 2011)
Using normal approximation, let's test to see if our difference is enough to reject the idea.
z = (8 - 6.14) / sqrt[(1230)(.005)(.995)] = 1.86/2.46 = 0.76. This result converts into a p-value of 44.72% which is much greater than our 5% alpha. With this information, we accept the idea that regulation goals per NHL game follow a Poisson distribution.
Note: A Goodness of Fit test would be a better measure of whether regulation goals per NHL game follow a Poisson distribution. However, that would require gathering the number of regulation goals scored in all 1,230 games which I frankly do not have time for.
Using this knowledge that the number of regulation goals scored follows a Poisson distribution, we can assume that the first regulation goal scored in a game will follow an exponential distribution. In this case, it assumes that on average, fans will have to wait until there is 8:41 left in the 1st period to see the first goal of the game. Exponential distribution is also "memoryless", so if a goal is not scored in the first period, exponential distribution assumes that on average, the first goal will be scored with 8:41 left in the 2nd period.
To see if there won't be goal scored in the first 60 minutes of a game, we just do some simple integration using an exponential distribution with a mean of 11.32 (60/5.3) from a range of t=60 to t=infinity. This is measuring the probability that the first goal of a game is scored after 60 minutes. The result is e^(-5.3), better read as 0.5% or 1/200. Does this look familiar? It should, it's the same result we got using Poisson distribution.
Since an exponential distribution is "memoryless", we can use it to measure the probability of scoring a goal in the first 5 minutes of a game or in the first 5 minutes of overtime. But, does the frequency of scoring change in overtime with the fatigue factor and the fact that both teams are minus one skater?
Our null hypothesis is that no, it doesn't change. Our 1st alternative hypothesis is that the fatigue and reduced number of skaters lead to more goals. Our 2nd alternative hypothesis is that the fatigue and reduced number of skated lead to fewer goals. Our alpha is set to 5%, which translates to a 95% confidence interval.
Using our average of 5.3 goals per 60 minutes of regulation, we find the probability of the first goal being scored in the first 5 minutes of overtime. The final result is 1 - e^(-5/11.32), better read as 35.7%. So, our null hypothesis states that 35.7% of games that go into overtime will also go into a shootout.
This past season, the NHL started keeping track of wins obtained in regulation as opposed to overtime and shootouts. This information was used as one of the tiebreakers for seeding the Stanley Cup Playoffs. But we can use it to see how many games result in overtime and how many result in a shootout. Using data from the 2010-11 season, we see that 297 games went into overtime while 149 of those games went into a shootout.
According to the null hypothesis, the expected number of those 297 overtime games going into a shootout is 191. The fact that in reality, only 149 went into a shootout sets off warning lights. But is the difference significant enough? Let figure out the z-score.
z = (191 - 149) / sqrt[(297)(.357)(.643)] = 42/8.25 = 5.09. This result converts into a p-value of 0.000018% which is much, much, MUCH less than our 5% alpha. With this information, we reject the idea fatigue and 4-on-4 skaters results in the same number of goals, accepting our first alternative hypothesis that MORE goals are scored because of the overtime format.
Now, for some fun, if regulation was played with 4 skaters on each side, how would that change the number of goals scored per game? Granted, we only have information for the first 5 minutes of 4-on-4 hockey, so how accurate this will be may be debated. But, let's find out anyway.
Our information was taken assuming that half of the time, a goal will be scored in the first 5 minutes of 4-on-4 hockey. We have all but one of our parameters needed to find how many goals would be scored in 60 minutes of 4-on-4 hockey. Plugging these numbers into an exponential distribution with an unknown mean, have -e^(5x) + 1 = 149/297 in which x^-1 is the new mean. Simple algebra from this point on results in the answer we were looking for. If the NHL switched to 4-on-4 hockey in regulation, the average number of regulation goals per game would greatly increase from the current 5.3/game to a new 8.4/game.
Whether or not a change like this would be good or bad for the NHL is a whole new debate entirely.
Now, as a sports fan, I know that despite new rules that have come into place since the lockout, goals in the NHL are still a rare commodity. During the first season after the lockout, goals skyrocketed to a still-meager 6.2/game. Since there, teams have become more defensive minded, reducing the number of goals scored throughout the course of a season. In the past two seasons, goals have hovered between the pre-lockout era of 5.1/game and the 2005-06 season average.
One of the major changes was the new overtime rules during the regular season. If the two teams were tied after the first 60 minutes of play, they would play in a 5-minute overtime in which each team would have 4 skaters on the ice as opposed to the 5 skaters per team used in regulation. If the teams were still tied, a shootout would take place between the two teams.
Being down a skater changes a lot about a hockey game including the speed of the game itself and the chemistry of the players on the ice. But, does this result in a difference in goals scored? Since most teams sacrifice a forward during overtime, some would say a team is less likely to score in overtime. However, with more open ice and a faster pace, some would say a team is more likely to score in overtime. But is either of these really true? Let's find out.
First off, we need to find not only the average number of goals scored in regulation but the frequency of them. Hopefully, the two are closely related.
Over the last two seasons, 13,632 goals have been scored in 2,460 regular season games in the NHL. But, if we use 5.5 goals/game, we are misrepresenting the number of goals scored in regulation since those 13,632 goals include overtime and the "plus-one" goals added to the final after a shootout.
Now, all overtime award one goal and always one goal to the total for a game. So, all we have to do is find the number of games that went into overtime, and subtract that from the total number of goals. With 698 overtime games in the past two seasons, the math is easy: 13,632 - 698 = 13,034 regulation goals. This changes the average number of regulation goals to 5.3 goals/game.
As stated earlier in this entry, goals in the NHL are rare. There is a probability designed for situations in which occurrences are rare, but must be at least 0 and can be any non-negative integer. This distribution is called a Poisson distribution, or the Poisson law of small numbers. But first, we have to test this distribution to see if the number of goals scored in the 60 minutes of regulation fits this distribution.
Now, I don't have the time to go thru all 1,230 games this past season. However, we can look at games that went into overtime tied at 0-0. According to Poisson distribution, the probability of any given game going into overtime tied at 0-0 is e^(-5.3), better read as 0.5% or 1/200. Using this information, we can expect 6.14 games will be scoreless after 60 minutes each season. Let's test this.
Our null hypothesis is that 0.5% of games will be scoreless after 60 minutes. Our alternative hypothesis is that our null hypothesis is incorrect. To reject the null hypothesis, we will use an alpha of 5%, which translates into a 95% confidence interval.
During the 2010-11 season, 1,230 regular season games were played. 8 of them went into overtime tied 0-0.
New Jersey at Buffalo (Oct 13, 2010)
Calgary at Nashville (Oct 19, 2010)
Pittsburgh at St. Louis (Oct 23, 2010)
Buffalo at Ottawa (Dec 4, 2010)
Tampa Bay at Washington (Jan 4, 2011)
Los Angeles at Minnesota (Feb 1, 2011)
Ottawa at Toronto (Feb 19, 2011)
New Jersey at Pittsburgh (Mar 25, 2011)
Using normal approximation, let's test to see if our difference is enough to reject the idea.
z = (8 - 6.14) / sqrt[(1230)(.005)(.995)] = 1.86/2.46 = 0.76. This result converts into a p-value of 44.72% which is much greater than our 5% alpha. With this information, we accept the idea that regulation goals per NHL game follow a Poisson distribution.
Note: A Goodness of Fit test would be a better measure of whether regulation goals per NHL game follow a Poisson distribution. However, that would require gathering the number of regulation goals scored in all 1,230 games which I frankly do not have time for.
Using this knowledge that the number of regulation goals scored follows a Poisson distribution, we can assume that the first regulation goal scored in a game will follow an exponential distribution. In this case, it assumes that on average, fans will have to wait until there is 8:41 left in the 1st period to see the first goal of the game. Exponential distribution is also "memoryless", so if a goal is not scored in the first period, exponential distribution assumes that on average, the first goal will be scored with 8:41 left in the 2nd period.
To see if there won't be goal scored in the first 60 minutes of a game, we just do some simple integration using an exponential distribution with a mean of 11.32 (60/5.3) from a range of t=60 to t=infinity. This is measuring the probability that the first goal of a game is scored after 60 minutes. The result is e^(-5.3), better read as 0.5% or 1/200. Does this look familiar? It should, it's the same result we got using Poisson distribution.
Since an exponential distribution is "memoryless", we can use it to measure the probability of scoring a goal in the first 5 minutes of a game or in the first 5 minutes of overtime. But, does the frequency of scoring change in overtime with the fatigue factor and the fact that both teams are minus one skater?
Our null hypothesis is that no, it doesn't change. Our 1st alternative hypothesis is that the fatigue and reduced number of skaters lead to more goals. Our 2nd alternative hypothesis is that the fatigue and reduced number of skated lead to fewer goals. Our alpha is set to 5%, which translates to a 95% confidence interval.
Using our average of 5.3 goals per 60 minutes of regulation, we find the probability of the first goal being scored in the first 5 minutes of overtime. The final result is 1 - e^(-5/11.32), better read as 35.7%. So, our null hypothesis states that 35.7% of games that go into overtime will also go into a shootout.
This past season, the NHL started keeping track of wins obtained in regulation as opposed to overtime and shootouts. This information was used as one of the tiebreakers for seeding the Stanley Cup Playoffs. But we can use it to see how many games result in overtime and how many result in a shootout. Using data from the 2010-11 season, we see that 297 games went into overtime while 149 of those games went into a shootout.
According to the null hypothesis, the expected number of those 297 overtime games going into a shootout is 191. The fact that in reality, only 149 went into a shootout sets off warning lights. But is the difference significant enough? Let figure out the z-score.
z = (191 - 149) / sqrt[(297)(.357)(.643)] = 42/8.25 = 5.09. This result converts into a p-value of 0.000018% which is much, much, MUCH less than our 5% alpha. With this information, we reject the idea fatigue and 4-on-4 skaters results in the same number of goals, accepting our first alternative hypothesis that MORE goals are scored because of the overtime format.
Now, for some fun, if regulation was played with 4 skaters on each side, how would that change the number of goals scored per game? Granted, we only have information for the first 5 minutes of 4-on-4 hockey, so how accurate this will be may be debated. But, let's find out anyway.
Our information was taken assuming that half of the time, a goal will be scored in the first 5 minutes of 4-on-4 hockey. We have all but one of our parameters needed to find how many goals would be scored in 60 minutes of 4-on-4 hockey. Plugging these numbers into an exponential distribution with an unknown mean, have -e^(5x) + 1 = 149/297 in which x^-1 is the new mean. Simple algebra from this point on results in the answer we were looking for. If the NHL switched to 4-on-4 hockey in regulation, the average number of regulation goals per game would greatly increase from the current 5.3/game to a new 8.4/game.
Whether or not a change like this would be good or bad for the NHL is a whole new debate entirely.
And I'm seeing you used very ugly formulas! they bring me back memories from university classes :S