The Cost of a Win in MLB

The amount of money in Major League Baseball is astonishing. The New York Yankees are worth $4.6 billion. The Los Angeles Dodgers, Boston Red Sox, Chicago Cubs, and San Francisco Giants are all worth over $3 billion each. The smallest valuation, the Miami Marlins, is worth $1 billion. The average annual salary for a player in 2019 was $4.36 million, but players like Mike Trout, Nolan Arenado, and Manny Machado were making upwards of $30 million.

My question is simple: Are these highest paid players worth the money?

The most common way to determine this has been to use Wins Above Replacement (WAR). According to Baseball-Reference.com, the idea behind WAR is to see “how much better a player is than a player that would typically be available to replace that player.” Players are compared to averages in a complex variety of ways, ultimately focusing on runs contributed offensively and saved defensively. One win is estimated to equal about ten runs, so WAR values are presented with decimals. As a scale to refer the values to, Baseball-Reference.com lists < 0 as replacement level, 0-2 as reserve, 2+ as starter, 5+ as All-Star quality, and 8+ as MVP quality.

A difficult aspect becomes assigning a dollar value per WAR. This value is different for pitchers versus position players, as well as starting pitchers versus relief pitchers. Younger players with high WARs can also throw off the value since, as Dave Cameron of FanGraphs puts it, “players with zero to six years of service time have an artificially depressed salary due to not being able to qualify for free agency.” The value changes year to year thanks to inflation. Matt Schwartz, another writer for FanGraphs, states how $/WAR can change for a player within a contract from year to year. He says that, “Unsurprisingly, players decline over time. As a result, the $/WAR in a deal’s first year is generally higher than the $/WAR in later years.” Dave Cameron used money spent on free agents along with a weighted average of their win values to create a value of $4.5 million/win for the 2008 season. John Edwards used Cameron’s formula along with FanGraphs community research to determine the $/WAR values for starting pitchers, position players, and relief pitchers from 2006 to 2017. He determined that the cost of a win is $4.2 million for starting pitchers, $5.7 million for position players, and $10.9 million for relief pitchers.

We can use these numbers, and the help of FanGraphs’ Contract Estimation Tool, to determine the value of certain players. The Tool factors in a 5% inflation rate for the first five years of the contract as well as an aging curve. The aging curve adds 0.25 WAR/year for players from 18 to 27, 0 WAR/year for players from 28 to 30, subtracts 0.5 WAR/year for players from 31 to 37, and subtracts 0.75 WAR/year for players over 37.

The values are up to 2017 so let’s look at some of the top free agents from that year.

Let’s start with Eric Hosmer. He was coming off his best year in terms of WAR, in which he played first base at 4.1 WAR for the Kansas City Royals. He truly stepped up in a contract year, and got paid for that level of play. The San Diego Padres signed him to an 8 year, $144 million contract. Using the Contract Estimation Tool and setting the $/WAR to start at $6 million, the estimated value of the contract is $171.1 million. Based on this model, the Padres got a deal for a talented player. However, the model also assumes the player will maintain that WAR and only have slight decreases with age. If we look at how Hosmer actually played in 2018 and 2019, we see a different story. According to Baseball-Reference.com, Hosmer’s WAR in 2018 fell to 1.4, and it fell even farther to -0.3 in 2019. With a WAR of 1.4 and a $6 million/WAR ratio, the Padres should have only had to pay Hosmer $8.4 million in 2018. Much less than the $18 million he was actually owed and the $24.6 million he was expected to make using the Tool.

FanGraphs’ Contract Estimation Tool

Another big free agent from 2017 was starting pitcher Yu Darvish. He spent 5 solid seasons with the Texas Rangers then had a partial season stint with the Los Angeles Dodgers, before he was signed by the Chicago Cubs for 6 years and $126 million. The minimum $/WAR available on the Tool is $5.5 million, so even though John Edwards calculated the cost of a win for starting pitchers to be about $4.2 million, we will use $5.5 million. The Tool estimated a contract for Yu Darvish to be valued at $77.5 million. Darvish struggled in 2018 and also had a shortened season due to an elbow injury. His WAR dropped to -0.1. After an OK year in 2019, he posted a 3.3 WAR. Contracts for starting pitchers are hardly ever based solely on $/WAR. Teams value starters highly and most age well enough to not have as steep drop-offs over the years.

FanGraphs’ Contract Estimation Tool

Picking on all of the 2017 free agents would be interesting, but I want to check on one of the best players in the game with the largest contract in MLB history, Mike Trout. He has played for the Los Angeles Angels for his whole career, which began in 2012, and has been an MVP level player every year since then. He signed a 12 year, $428 million contract extension with the Angels in 2018. His “worst” year was 2017 when he was forced to sit out almost 50 games with a thumb injury, and posted a WAR of 6.6. Using that value, the Contract Estimation Tool still predicted a 12 year contract for him to be worth $453.7 million. When I used his career average WAR of 9.0, his 12 year contract value rose to $655.7 million. I then changed the Tool to say he ages well, meaning his WAR would not decrease with age as much, and his 12 year contract value jumped to $706.8 million. Seeing as he posted WAR values of 10.2 and 8.3 in 2018 and 2019, respectively, the contract estimations with the higher WAR seem more accurate.


Assigning the proper value to player contracts to determine the of wins they are worth is impossible. Teams pay players for much more than wins. They can pay an old veteran more than his expected $/WAR if they want some added leadership in the clubhouse. They can pay a relief pitcher more than his expected $/WAR if they are trying to make a playoff push. Teams can invest in a player to become the face of the franchise and improve fan engagement. A popular, but low WAR player may drive ticket sales better than an unknown, high WAR prospect. The variables that go into $/WAR are constantly changing. In order to properly analyze a players worth, many factors must be considered. The answer to my “simple” question is more complex than I could’ve imagined.

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The Impact of Weather on Mendenhall’s Football Team

Teeth are chattering, hands and toes are becoming numb and large amounts of hot chocolate are being consumed as football fans watch their teams play on a cold fall night. While the fans are freezing, they wonder if the cold is impacting the football players. Rain and cold are two weather conditions that football players face when playing in a game. The question I am trying to analyze is whether environmental factors like temperature and precipitation make a difference on offensive performance in a football game.

Coach Mendenhall took over the University of Virginia football team in 2016 and before UVA he was the coach for Brigham Young University for eleven years. Mendenhall is a student of the game of football and uses an adaptive style of coaching that incorporates the current resources that are available to his program. During his first two year at UVA, Mendenhall primarily used a run and gun offense, which is a type of offense that has a receiver suddenly changing positions by running left or right, parallel to the line of scrimmage, just before the ball is snapped. The traditional tailback runs with a pocket passer. Once Bryce Perkins was recruited to play for the University of Virginia, the offense changed to primarily a run-pass offense because of Perkin’s mobility. Teams playing against UVA had to account for not only the tailback runs but also the quarterback runs and passes. 

I was curious to examine the impact of temperature and weather on the offense while Mendenhall was the coach for both UVA and BYU. I analyzed six years of Mendenhall’s coaching career with two years at BYU and four years at UVA. Overall, Mendenhall won 55% of these games and lost 45% of them. For each game I looked at temperature, weather, total passing yards, passing yards attempted, passing yards completed, total rushing yards, average per rush, touch downs, fumbles, interceptions, total points and whether the team won or lost. I used temperature and weather (rain or clear) as the variables to compare the offense statistics to in order to see if there was a correlation.

The first set of data I analyzed was the impact of temperature on passing and rushing yards. 

When the temperature is below 25 degrees Fahrenheit, both rushing yards and passing yards are negatively affected. The offense was not as effective in achieving yards when the temperature was extremely cold. As temperature increases, both passing and rushing yards per game increase. Between temperatures 26 to 100 degrees, there is not a large difference between passing and rushing yards, however, in general the offense tends to have more passing yards than rushing yards. 

The next set of data was the impact of rain on passing and rushing yards.

When it was raining, the offense ran the ball more and when the weather was clear with no rain the offense passed the ball more. Before starting this project I hypothesized that the offense would rush the ball more when it was raining due to a higher chance of mistakes in the rain with less visibility and grip on the ball. The offense had to adapt to the weather conditions in order to have the best chance of winning.

I also looked at the impact of temperature on fumbles and interceptions (turnovers).

Interceptions were minimally affected by temperature and remained relatively constant. The offense had the greatest number of fumbles per game in cold weather. As the temperature increased, the average fumbles per game decreased. Overall there is a correlation between cold temperatures and number of fumbles.

The final set of data I analyzed was the impact of rain on fumbles and interceptions (turnovers).

When it was clear outside there were more interceptions compared to when it was raining. This makes sense because when the weather is clear the offense is passing the ball more. Greater passing leads to more opportunities for interceptions. There is less of a difference between rainy weather and clear weather with fumbles. However, there were slightly more fumbles per game when it was clear compared to when it was raining.

The weather and temperature are two factors that affect the style of play.  When it is raining outside, offense tends to rush the ball more. When it is clear outside, offense tends to pass the ball more. It will be interesting to see if these statistics hold up going forward as Mendenhall continues his time at UVA. 

Data from all the games

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Basketball Players Shoot Free Throws Better at Home: Fact or Fiction?

North Carolina guard, Cameron Johnson, hits the ground with a smack after a nasty foul from Duke competitor, Zion Williamson. Johnson steps up to the foul line as the ref passes him the ball, eager to earn his team two “gimme” points. 

As the camera pans out behind Johnson, the sea of craziness behind the hoop is electric. Duke’s student section, the Cameron Crazies, is buzzing with energy and excitement, dressed head to toe in royal blue. Hands wave in every direction, heads bob up and down arrhythmically, and thundering screams take over the arena. 

“BRICK. BRICK. BRICK. BRICK.”

Johnson performs his free throw ritual: two dribbles, flip the ball back to himself, one dribble, align his feet, elbows up. Shoot. 

BANK. The ball deflects off the rim as the Cameron Crazies erupt in screams of celebration. Hoots, hollers, and high-fives between the students are payment for their efforts. Their hard work has paid off! They have successfully distracted the free throw shooter!

Johnson tries to clear his mind, and again steps up to the line. The Crazies return to their shenanigans and commotion, determined to produce the same result. Two dribbles, flip the ball back to himself, one dribble, align his feet, elbows up. Shoot. 

Swish.


Free throw distractions: they come in all shapes and sizes, with individual student sections priding themselves on their uniqueness in the art of diversion.

But do they really work? Do players really shoot free throws better at home?


The Importance of Free Throws

It is almost impossible to overemphasize the importance of free throws, both to players and teams. If a player only makes four baskets per game, but can add four free throws to their total, they are now a double-figure scorer. A stellar free throw shooting team can add large amounts of points to the board from the foul line, creating a very difficult obstacle for opponents to overcome. 

Free throws are also notorious for making or breaking a well-fought game, especially in the world of college basketball. For instance, take the semifinal game of the 2019 March Madness tournament. Virginia and Auburn had battled hard the entire game, with Auburn pulling ahead at a score of 62-60. With six-tenths of a second left on the clock, Kyle Guy was fouled from behind the three point line. Three free throws, 70,000 fans on their feet inside U.S. Bank Stadium, and six-tenths of a second left. 

Guy made all three free throws to send the Cavaliers to the national championship game with a 63-62 victory over Auburn. Virginia went on to win the championship – for the first time in program history. Without ice running through Guy’s veins while standing on the free throw line, this would not have been possible. 


The Conventional Wisdom

The conventional wisdom of the general population is yes — free throw shooters can be distracted and teams shoot free throws better at their home courts. 

People believe this to be true due to a variety of reasons. 

At home, a player has the familiarity factor. They are practicing on their courts and shooting free throws on the same exact hoops. A coach can make their team shoot 50 free throws per practice to build up repetition on these nets. After practicing so many, the motion of taking a free throw can turn into muscle memory. If shot continually on the same basketball hoop, some believe muscle and visual memory could work together to improve accuracy in this same location.

Others believe shooting free throws better at home to be thanks to the home court atmosphere. Players enter a game with a familiar arena, packed to the brim with their fans, peers, friends, and family. Players recognize mentally that they are in a place of support, with the majority of attendees rooting for them and hoping the best for them. This could provide an extra edge of confidence necessary to consistently put the ball through the net while standing on the foul line. 

The largest contributing variable to the assumption of shooting free throws better at home is a team having their own student section. When shooting free throws at home, the background behind the net is full of still students with their hands raised gracefully in their air for good luck. They are as silent as can be, while they internally pray for a point scored. They do not frantically wave their hands and arms, shout obscenely, nor jump up and down erratically. There is no air of distraction and a player is able to focus on their one goal: making the shot. 

The opposing team, however, does not receive quite the same luxury. The passion and hard work of the “sixth man” on the home team is thought to take a large toll on the opposition. In college, high school, and NBA basketball, the sixth man is the fans attempting to influence the game by cheering and chanting for their team of choice. Or… relentlessly aiming to distract and throw-off the away team. 

Sixth man clubs began in college basketball, where deep crowds of students gather to chant for their team and torment the visiting team. Many wear matching shirts and outfits to appear even more numerous and unified. The sixth man sections are well known for intimidation tactics to distract the opposing team and their creativity in chants. Some well-recognized sixth man groups include Duke’s ‘Cameron Crazies’, Virginia’s ‘Hoo Crew’, and Michigan State’s ‘Izzone’.

A student section, decked out in their school’s colors, demonstrates the usage of props to distract the opposing team.
Image credit: Creative Commons

The Truth

While all these beliefs held by the common basketball spectator are logical and understandable, they are simply not true.

With the assumption of players shooting free throws better at home being so widely accepted, I wanted to look at the raw statistics of free throw percentages to see if this was a myth or legend. More specifically, I focused on the 15 teams composing the ACC in their 2018-2019 season: UVA, Boston College, Clemson, Duke, Florida State, Georgia Tech, Louisville, Miami, UNC Chapel Hill, NC State, Notre Dame, Pitt, Syracuse, Virginia Tech, and Wake Forest. 

The free throw percentages for all home, away, and neutral games, per team, were collected from the first 20 games of the 2018-2019 season. A neutral game was defined as a game played between two teams at a location not favoring one team or the other. For example, a neutral game played between UVA and Duke could be played in the Florida State arena. Neither team is close to home, and thus it is assumed that one team is not benefitting more than the other thanks to “home court advantage”

The average home, away, and neutral free throw percentages were calculated for each team by simple mean calculations: m = (sum of terms) / (number of terms). The overall averages from the 2018-2019 ACC season were determined by taking each team’s individual averages, combining them, then finding the overall mean.

Overall home, away, and neutral free throw percentages for teams in the ACC. Statistics were calculated from the first 20 games of the 2018-2019 season.

From my calculations, the overall home, away, and neutral free throw shooting percentages (FT%) were as follows, respectively: 0.7272, 0.7061, and 0.7373. At first glance, the distracting spectators at away games seem to have succeeded, with the away FT% being the lowest.

It is true that 0.727 (home FT%) is greater than 0.706 (away FT%)… but is this statistically significant? Statistical significance is defined as the likelihood that a relationship between two or more variables is caused by something other than chance. In order to test for significance between these two values, a t-test was performed. 

A t-test allows for comparison between the average values of two data sets (in this case, home versus away free throw percentages) and determine if the difference is significant. For more background on running a t-test, please see Statistics Solutions. The null hypothesis for this analysis was that there would be no statistical difference between the home and away free throw percentages. i.e., m1= m2. A two-tailed independent t-test (heterogeneous) was utilized because two different groups were assessed by the same measure (how many free throws were made per game). A two-tailed T-test was run at the 0.05 p-value level. 

The calculated p-value from this data set was .218423, which is greater than the p-value level of 0.05. Thus, the results are not significant and we fail to reject the null hypothesis / the null hypothesis is accepted. 

In normal language, this means that based on my analysis, the assumption that players shoot free throws better at home is not true, simply fiction, and a myth. 

It should be recognized that the average neutral free throw percentage was the highest overall, but it also proved to be not significant after statistical analysis.


Taking a Closer Look at Virginia – An Outlier

The acceptance of the null hypothesis (that there is no significant difference between home and away free throw percentages) was for all the ACC teams as a whole, on average, in their 2018-2019 season. However, when looking at the data on a more individual basis, this does not always hold true.

After running a t-test for the University of Virginia, home of the Wahoos, it proves to have a statistically significant difference between home and away free throw percentages. A p-value of 0.003539 was obtained, which is lower than the p-value level of 0.05, thus we reject our null hypothesis. 

So why are the Cavaliers such worse free throw shooters on the road? Why do they thrive on their home court in John Paul Jones (JPJ) arena? 

This could be contributed to a variety of factors outside the scope of this analysis, such as decibel level, size of student section, or key players’ shooting percentages. 

However Ty Jerome, star guard for the Cavaliers who helped bring home the 2019 national championship, has a theory of his own for why the Hoos thrive in free throw shooting at home. He contributes the excellence at home to coach Tony Bennett’s intense free throw shooting practice agenda. 

“We had a free throw ladder,” said Jerome. “Every single day after practice, the two people near each other on the ladder, which was a wooden board that had a list of everybody’s name, would play against each other in a free throw competition.” 

The competition consisted of each contestant shooting 30 free throws, and whoever made the most of out 30 was declared the winner. If the winner was underneath on the ladder, he would jump up. If the person that won was above his opponent on the ladder, he would stay where he was and play the person above the following day. The reward for being at the top of the ladder was being excused from cardio training for that day.

“I think the way we practiced free throws made us so good at them at home,” said Jerome. “Not a day went by where I wasn’t shooting at least 30 free throws in the JPJ. We were all motivated to make every single shot so we wouldn’t have to do sprints. The ladder allowed for some friendly competition and continuous repetition of free throw shooting.”

Ty Jerome in deep concentration as he follows through on a free throw.
Image credit: Ty Jerome

Key Takeaways

The common assumption of most basketball spectators is that teams shoot free throws better at their home court. A variety of reasons are thought to contribute to this, such as familiarity factor, supportive fans, and minimal distraction. While this assumption is both understandable and widely accepted, it is simply false. 

Based on an analysis of the 15 teams composing the ACC in their 2018-2019 season, there is no significant difference between home and away free throw shooting percentages.

Student sections across the nation will choose to ignore this data, and relentlessly continue to try to distract away teams. They will decorate themselves in their school’s colors, bring props, yell at the top of their lungs, and full-heartedly celebrate when an opponent misses a free throw. They will believe that they truly are the sixth man on the court.

Statistical analysis proves that there is no difference between home and away free throw percentages for teams in the ACC, but the hearts of dedicated fans will still choose to believe otherwise.

A student section dressed in their school’s colors to show their support of the team.
Image credit: Creative Commons
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