The Magnus effect: why pitches move

February 19, 2009 • Training

When a ball spins, it creates an envelope of air around it called the boundary layer. This boundary layer moves with the ball whether it spins forward or backward or sideways. The interaction of this boundary layer with the surrounding air results in an outside force that changes the path of the baseball. This is the Magnus effect.

Named for German scientist Heinrich Magnus, this effect is a principle of fluid dynamics that describes the lift created by the spin of an object that is moving through a fluid (gas or liquid).

To better understand lift, here is a brief look at how airplane wings create lift. The shape of airplane wings causes air to move faster over the top of the wings than it moves beneath the wings. The faster moving air results in lower air pressure above the wing and greater air pressure beneath the wing. The greater air pressure pushes the wing up; this is lift.

HOW SPIN CREATES LIFT

The spin of the ball dictates the rotation of the boundary layer. When the ball has back-spin, like a fastball, the boundary layer under the baseball shoots air forward into the air that is trying to move around the baseball. The opposing air flows result in slower air movement and higher air pressure underneath the baseball.

On top of the ball, the boundary layer shoots air backward in the same direction as the air that is trying to move around the baseball. These air flows compliment each other and combine to create faster air movement and lower air pressure on top of the baseball.

The combination of slower air movement under the ball and faster air movement over the ball creates lift that opposes gravity - a "rise". The Magnus effect, in this case, acts just like an airplane wing.

For a curveball, the top-spin is like turning that wing upside-down. The opposing air flows are now on top of the baseball, and the complimentary air flows are on bottom. Here, the Magnus effect creates lift that compliments gravity - a drop.

With a tilted spin axis, the Magnus effect creates a tilted lift. A left tilt adds right-to-left movement when the pitch has back-spin and left-to-right movement when the pitch has top-spin. A right tilt has the opposite effects.

When a pitch spins perfectly sideways, like a screwball or a sweeping curveball, the Magnus effect does not create a "rise" or drop. Instead, it creates sideways lift. Viewed from the top, clockwise spin results in left-to-right lift, and counter-clockwise spin results in right-to-left lift.

Curveball - Forward Spin
Curveball - RHP Perspective
Sweeping Curve - Side Spin
Sweeping Curveball - RHP Perspective
Screwball - Side Spin
Screwball - RHP Perspective

MAGNUS EFFECT ON OTHER PITCHES

The Magnus effect is greatest when the ball's spin axis is perfectly perpendicular to the velocity of the baseball. As the spin axis turns (or yaws, if you're into that sort of thing) from perpendicular to parallel to the baseball's velocity, the Magnus effect decreases accordingly. Likewise, the magnitude of the Magnus effect increases as the spin axis moves from parallel to perpendicular to the baseball's velocity.

Gyroball

When the ball's spin axis is perfectly parallel to its velocity, the Magnus effect is null, barring crosswinds. In this case, the ball spins like a bullet - clockwise for righties and counter-clockwise for lefties when viewed from the pitcher's perspective - and no part of the boundary layer opposes or compliments the surrounding air flow.

A pitch with this spin is called a gyroball, and despite what was widely reported when Daisuke Matsuzaka came to the states, the null Magnus effect makes this the straightest pitch that can be thrown.

Wind tunnel studies have shown that this type of spin results in a smaller wake behind the ball. A smaller wake means less wind resistance which means a gyroball does not slow down as much as a fastball does on its way to the plate.

Slider

A slider is intended to have glove-side lift, but Pitch-f/x data suggests that sliders move less than any other commonly thrown pitch. On Pitch-f/x charts, sliders are usually grouped around or very near to the chart's origin where zero horizontal movement meets zero vertical movement. This suggests that most sliders spin like gyroballs. I tend to agree.

Good sliders, though, will have a spin that is somewhere between that of a curveball and that of a gyroball. Such spin will create the sliding movement and, depending on the degree of tilt, a varied amount of additional drop for the pitch.

Cut fastball

If a slider's spin is between that of a curveball and that of a gyroball, the spin of a cut fastball should be between that of a fastball and that of a gyroball. Where a slider ideally has some top-spin, a cut fastball has a large amount of back-spin.

The combination creates lift nearly identical to a fastball, but because the spin axis is turned slightly to the pitcher's glove-side, it also has glove-side run.

Split-finger fastball

Split-finger fastballs can be thrown with one of two different spins. The first spin is simply a slower back-spin than a normal fastball that creates less lift than a normal fastball would. When thrown at nearly the same speed as a normal fastball, the split-finger fastball appears to drop due to the smaller lift.

The second spin is actually top-spin. This is the ideal spin for an effective split-finger fastball because the forward tumble creates a drop like a curveball. The velocity of the pitch is similar to a fastball, but the spin is like a curveball albeit with a much slower rotation.

When top-spin is present in this pitch, it is sometimes called a forkball. A forkball is usually held with a deeper grip than a split-finger fastball, but the two pitches are practically identical give or take a couple of ticks on the radar gun.

Sinker

Some sinkers spin like reverse-cut fastballs, and some sinkers spin like reverse-sliders. Most are somewhere in between. A power sinker, like the one thrown by Brandon Webb, spins almost like a screwball but with fastball velocity.

The same rules that apply to cut fastballs and sliders also apply to sinkers. The difference is that cut fastballs and sliders have glove-side lift while sinkers have arm-side lift.

THE ROLE OF SEAMS

The 108 stitches on a baseball grab the air around the ball and create a larger boundary layer than a ball with no seams would create. The horseshoe shape all around the baseball allows a pitcher to throw just about any pitch as a two-seam pitch, a four-seam pitch, or something that isn't quite either of those (a three seamer?). Most sliders fall into the third category.

A four-seam pitch spins on an axis that allows four seams to influence the boundary layer. The four seams are evenly spaced (balanced) around the baseball. This symmetry creates a stable and relatively predictable Magnus effect.

A two-seam pitch, though, spins on an axis that unbalances the seams even though all four seams still influence the boundary layer. This axis puts a seam loop on either side of the ball, leaving the two connecting seams close together on one side of the ball.

With the axis turned slightly to the left or the right, one of the seam loops moves toward the point of pressure (where the ball breaks through the surrounding air and experiences the greatest wind resistance), and the other seam loop moves away from it. This axis exaggerates the Magnus effect of the seam that moves toward the point of pressure, and reduces the Magnus effect of the seam that moves away.

The dominant seam, because of its almost circular shape, creates a point of nearly constant friction as it pushes boundary layer air almost directly into the air breaking across the point of pressure. When the seam catches that angle just right, the baseball will dart left or right depending on which seam is dominant.

CLOSING THOUGHTS AND OTHER NOTES

I've talked a lot about how a pitch spins and why it moves the way it does, but I haven't yet touched on the magnitude of the Magnus effect. The obvious part is that greater movement is due to a greater Magnus effect. The not so obvious part is how to increase the Magnus effect to create even more movement. The simple answer is to give the ball more spin.

The faster a ball spins, the greater the resulting Magnus effect will be.  Squeezing just one extra rotation out of a pitch can have dramatic results on the pitch's movement.

You may have noticed that I didn't talk about the knuckle ball at all. Well, the knuckle ball doesn't spin, so it has no Magnus effect. A knuckle ball's movement is strictly an aerodynamics issue where the seams cause immediate disruption in the surrounding air flow rather than through a boundary layer. On the pitch's way to the plate, chaos theory takes over and the knuckle ball waivers as the seams catch air and unpredictably change the path of the ball.

Finally, release angles play a sizable role in creating "hidden" movement. For example, if a pitcher releases the ball two feet outside of the rubber, it has to move roughly 3 1/2 feet to reach the opposite corner of the plate.  Sliders and curveballs with glove-side lift will look like they are moving nearly 4 feet as they cross that corner, even though they only break about 3 to 5 inches.


Texas Rangers Win-Curve Part II: Playoff Probability

February 4, 2009 • Analysis

This is Part II of a series that examines the Texas Rangers 2009 revenue outlook in a rough version of the framework laid out by Vince Gennaro in his fantastic book Diamond Dollars. Check out the Offline Reading list for other great reads.

In Part I, Texas Rangers Win-Curve Part I: Wins vs. Attendance, I walked through a model for predicting 2009 home attendance based on the team's on-field success as measured by wins.

Part II aims to add another piece to the puzzle by determining a team's chances of making the playoffs for a given number of wins.

WHAT EVERYONE KNOWS

Two types of teams make to the playoffs: 3 division champions and 1 wild card team.

The more games a team wins, the better its chances are for making it into the playoffs by either method.

In reality, for a given number of wins, a team will either make it to the playoffs or not. There are only two outcomes: 'yes' and 'no'.

MODELING THE DATA

Because there are only two outcomes, the data can be modeled with a logistics curve. The curve is created by a generalized binomial regression. Basically, using an independent variable (wins), it determines the probability that the dependent variable (team makes the post-season) is true.

I gathered 11 years of historical data for the American League in its current alignment - since Tampa Bay's inaugural season in 1998.

I ran regressions for each division and for the American League as a whole.

THE RESULTS

Wins vs Post-season Probability - American League 2009
Wins vs Post-season Probability - American League 2009

One hypothesis that I was eager to test was that for teams in smaller divisions, like the 4-team AL West, the odds of winning the division (and therefore the odds of making the playoffs) are greater than for teams in a 6-team division like the NL Central.

I tested this hypothesis by comparing the curves for each of the three divisions against the American League curve. Essentially, all 4 curves are the same but shifted either to the left or to the right.

The AL West curve, surprisingly, is shifted right, meaning it is harder to make the playoffs in the AL West than in the AL as a whole. The AL Central curve is shifted left, and the AL East curve showed a right shift approximately equal to the shift in the AL West curve.

At 92 wins, an AL team has had a 66.96% chance to make the playoffs. The AL West, AL Central, and AL East have had 62.69%, 78.24%, and 61.94% chances, respectively, at the 92-win level.

Since it has been easier to make the playoffs in the 5-team AL Central than it has been in the 4-team AL West, the hypothesis does not hold up. The difference between the AL West and AL East was barely noticeable.

AL West
AL West
AL Central
AL Central
AL East
AL East

TEXAS RANGERS POST-SEASON PROBABILITY

The two curves that apply to the Rangers, the AL curve and the AL West curve, are fairly similar. At 80 wins, the AL curve shows a 0.35% chance, and the AL West curve shows a 0.24% chance.

In what appears like it could be a weak division in 2009, 85 wins might be enough to get the Rangers in. Historically, though, 85 wins has resulted in only a 4.70% chance on the AL curve and an even smaller 3.58% chance on the AL West curve.

If the Rangers make the improbable jump from 79 wins to 95 wins, the AL curve gives them a 90.88% chance of making the post-season, while the AL West curve gives them an 89.59% chance.

Based on the 2009 outlook, if any AL West team can get to 95 wins, it should win the division handily. One team reaching that level would have a fair amount of shock value by itself, but if two teams hit the 95-win mark, it would be absolutely stunning.

APPLYING THE POST-SEASON EFFECT

When a team makes the post-season, the fan response typically includes increases in season ticket sales, television ratings, and merchandise sales. This post-season effect has a tangible benefit on team revenue for current and future seasons.

According to Gennaro's model, a net present value (NPV) is calculated for the post-season effect. For each win, the NPV is multiplied by the post-season probability for that win total, and the resulting value is added to that point on the win-curve.

In Part III, the dual focus will be on turning attendance figures into attendance dollars and assigning a value to the post-season effect.


Brandon McCarthy: Notes on weight gain

January 30, 2009 • Training
Texas Rangers RHP Brandon McCarthy. (Source: nyrangers3019, rootzoo.com)
Texas Rangers RHP Brandon McCarthy. (Source: nyrangers3019, rootzoo.com)

Several days ago, it was reported by several local media outlets that Brandon McCarthy choked down 7,000 calories a day this off-season to add about 25 pounds to his slender frame. This weight gain has a lot of fans excited about the positive impact it could have on his 2009 season.

The expectation is that McCarthy will be stronger and more durable going into 2009 than he has ever been before, but while he may be stronger, he may not necessarily be any more durable.

At 7,000 calories a day, there is virtually no chance that all 25 pounds are added muscle, and even if it is all muscle, that isn't necessarily a good thing.

Since coming to the Rangers before the 2007 season, McCarthy has battled a stress fracture in his right scapula, a strained flexor tendon in his flexor-pronator mass, and a strained flexor tendon in his right middle finger.

GROUND ZERO

McCarthy's injury trouble started with a stress fracture in his right scapula in July 2007. A stress fracture occurs when a bone "flexes" repeatedly - an action obviously not meant for bones. The injury indicates that the bone is being bent, stretched, or pulled by some unnatural movement, but it is not out of the question that natural movement can cause it as well.

Following his 2007 shoulder injury, McCarthy spent the off-season working hard to get stronger for 2008. Word had it that he was up about 15 to 20 pounds. Without a doubt, his focus was on staving off another shoulder injury.

The nature of a stress fracture indicates (but does not guarantee) that a mechanical flaw is responsible. The other two injuries are indicative of a lack of physical fitness.

When throwing a pitch, the arm and hand have to be strong enough to overcome the inertia created by the rest of the body. If the inertia is stronger than the bones and soft tissue, the unfit tissues tend to break down.

THE WEAK LINKS

Strains occur when muscles attempt to handle larger loads than what they are capable of handling.

McCarthy's body got stronger, and thanks to both his increased mass and increased strength, he was creating more intense loads for his elbow, wrist, hand, and fingers. His forearm was not ready for the increased load, and his forearm flexor tendon suffered a very serious strain.

Pitching with a compromised flexor tendon puts the ulnar collateral ligament at serious risk for strains and ruptures, and it's usually pretty painful.

After spending several months strengthening and conditioning his forearm, McCarthy finally returned to the mound. With a stronger forearm, the next injury occurred at the next weak link in the kinetic chain, the flexor tendon of his right middle finger.

DOING THE MATH

If you've followed what I've said so far, you can see how it's important for a pitcher to be strong from his toes to his finger tips.

A pitcher's arm must be strong enough and conditioned to handle the loads generated by the rest of his body, otherwise, even someone with anatomically perfect mechanics can suffer muscle strains, fatigue, tendinitis, or worse.

Richard Durrett of the Dallas Morning News wrote the following on January 20, 2008 during the Rangers pitching mini-camp:

McCarthy said he's worked hard to strengthen not only the area that has proved bothersome, but also the areas around that.

To me, this is far more encouraging than reading that he gained 25 pounds. Perhaps now, his arm is ready to take advantage of the leverage his 6' 7" frame is capable of generating. If it is, McCarthy is primed for a break out season.


Eric Hurley: Hamstrings, rotator cuffs, and Mark Connor

January 26, 2009 • Analysis
Eric Hurley. (Source: Jason Cole, LoneStarDugout.com)
Eric Hurley. (Source: Jason Cole, LoneStarDugout.com)

It has become somewhat fashionable to blame Mark Connor for Eric Hurley's shoulder problems - a torn rotator cuff and frayed labrum. Without a full blown analysis of Hurley's arm action, though, it is also hard to say with certainty that his mechanics were responsible.

Given the nature of these injuries, though, Hurley's mechanics and hamstring are far more likely than Connor's teachings to be the cause of Hurley's shoulder injuries.

ANATOMY

Like any soft tissue in the body, rotator cuff muscles and tendons are torn gradually over time as stress creates micro-tears that build up and compound. There are exceptions, of course, but most of them involve severe external trauma like violent collisions and power lifting.

In pitching, the rotator cuff contracts most powerfully during the deceleration phase as it tries to keep the humerus from twisting and flying out of socket.  When the arm moves across the body, the head of the humerus becomes an obstacle to this contraction.  This forces the muscles to contract "around a corner" which adds more tension to the muscle than it can create on its own.

A frayed labrum is an early stage SLAP (superior labrum from anterior to posterior) lesion. Later stage SLAP lesions are commonly referred to simply as "torn labrums". The lesions are caused by the compressive force and friction created when the long head of the biceps brachii contracts and pulls directly on the glenoid labrum in an unnatural manner.

Certain arm actions, most notably transverse hyperabduction of the shoulder (scap-loading), can position the head of the humerus as an obstacle to the contraction of the biceps creating extra tension on the labrum where the long head of the biceps attaches.

Since part of the long head of the biceps merges with the labrum, SLAP lesions can sometimes be misdiagnosed as biceps tendinitis.  This is was the reported initial diagnosis for Hurley's shoulder injury on July 30, just three days after his final start of 2008.  On August 1, the Rangers reported that it was, in fact, shoulder soreness.

ATTACK OF THE HAMSTRING

Hurley was cruising along fairly well before he injured the hamstring of his left leg - his landing leg.

The hamstring of the landing leg experiences an eccentric contraction as the upper body moves forward over the waist. A negative change in the muscle's flexibility can decrease the amount of trunk flexion and/or shoulder rotation that occurs during a pitch. Since the body is less engaged in the deceleration of the arm, the shoulder handles more of the load than it would with normal hamstring flexibility.

Limited trunk flexion or shoulder rotation can cause the throwing shoulder's forward movement to stop early, even though the arm tries to continue moving toward the plate. The force of this action slings the arm across the body and moves the head of the humerus into the path of the muscular contraction as described above.

Hamstrings are notoriously slow-healing muscles, and flexibility can be compromised for a long period even after the muscle is fully functional.

Hamstring injuries will not always lead to shoulder injuries, but they represent a huge risk factor for someone already dealing with a weakened shoulder.

WAS HURLEY'S SHOULDER ALREADY WEAKENED?

Terry Clark (left), his mustache, and Michael Schlact. (Source: Jason Cole, LoneStarDugout.com)

The answer to this question simply has to be, "Yes."

In 234 minor league innings over the last two seasons, Hurley worked primarily with Rick Adair, Terry Clark, and Andy Hawkins - one of whom has one of the greatest mustaches in baseball.  All three coaches are extremely well regarded; none of them is Mark Connor.

Mark Connor was Hurley's primary pitching coach for about 32 innings, all in 2009. (Hurley had a 7.1-inning rehab start in Frisco near the end of that 32-inning span.) 32 innings is simply not enough to tear the healthy rotator cuff of a professional pitcher - someone whose rotator cuff should be exceptionally strong and well conditioned.

Barring severe external trauma, his shoulder must have been compromised before reaching the Majors and long before he hurt his hamstring.

SUMMARY

Hurley likely began damaging his shoulder well before his injuries became apparent. One can argue about the inevitability of a major tear, but excluding an external traumatic event, Hurley's mechanics are the most likely cause of the injuries.

When his hamstring started giving him trouble, his body compensated for that injury, effectively placing more (too much) stress on a rotator cuff that was, in all likelihood, already damaged.

Rotator cuffs simply don't tear suddenly enough to blame Connor for the injury.

Why Hurley was allowed to start that last game (and remain in it for as long as he was) is a different matter entirely.


Texas Rangers Win-Curve Part I: Wins vs Attendance

January 23, 2009 • Analysis

Fans respond to winning, but different fan bases respond differently. Fan response is most easily measured by a team's revenue stream, the largest factor of which is home attendance - essentially a measurement of demand. It follows that if one can understand the relationship between wins and attendance, then one can reasonably predict revenue at different win levels.

When plotted on a wins versus revenue graph, the function that predicts these points is called a win-curve. (NOTE: The win-curve is actually disjoint. Since modern-era Major League Baseball games do not end in ties, there are no fractional wins. The line itself serves strictly as an illustration because win totals will always be integers.)

The concepts discussed here are elaborated on in far greater detail in Vince Gennaro's Diamond Dollars - a book that I highly recommend if you find that this article sparks your interest (check the Offline Reading list). Mr. Gennaro developed win-curves for all 30 teams as a part of his research into these relationships. Using 37 years of historical data, I attempted to build my own for the Texas Rangers.

This is Part I: Wins vs. Attendance.

PREPARING THE DATA

Each team's win-curve is different. The trick to building an accurate win-curve is identifying trends in the relevant market. In Mr. Gennaro's model, he used a 50-50 weighted average of the previous year's wins and the current year's wins, and he compared that to a per-game average of the current year's home attendance.

He accounted for a "new franchise halo", overall industry growth, new stadium effects, and work stoppages. Due to the small sample size, these effects were identified and generalized for Major League Baseball as a whole, rather than for individual teams.

Based on the assumption that each market behaves differently, it is unreasonable to assume these effects will be the same from market to market, but they do represent a reasonable approximation.

In my attempts to recreate Mr. Gennaro's work, I struggled to capture these effects. Without doing similar studies for the other 29 teams, I tried to find other ways to account for these other attendance factors.

Wins

As Gennaro surely did, I played with several different values for my wins variable. Each was represented as a winning percentage to help adjust for seasons of different lengths. Here is the list of my various definitions for wins:

  • Gennaro's average of previous wins and current wins
  • Separate variables for previous wins and current wins
  • Average wins of the three most recent seasons
  • Separate variables for the wins of the three most recent seasons

For each definition of wins, I tried different weights. In nearly every model, the only significant variable from the group was current wins, though the previous wins variable was significant in a few. In the final model, I chose current winning percentage as my wins variable.

"New Franchise Halo"

This effect was minimal at best with the Rangers. In their first two years, 1972 and 1973, the Rangers averaged fewer than 4,000 fans per home game. Early on, the Rangers weren't very good, but after several years in Arlington, attendance climbed to well above 15,000 fans per game. This is counter to the typical new franchise halo, where a team sees an early boom that tapers off after a few seasons.

The Metroplex area has only had one Major League Baseball franchise, so there is only one from which to develop a model, leaving too few samples to effectively quantify the "new franchise halo" for Dallas/Fort Worth.

Industry Growth

To measure industry growth, a simple counting variable was added - a value of 0 in 1972, up to 36 in 2008. Effectively, this functioned the same as a "Years in Area" variable. In the final model, this variable is not significant - p = 0.255 (approx.) - but its inclusion in the model resulted in a smaller standard error and better R-square and Adjusted R-square values than when it was excluded.

New Stadium Effect / Work Stoppages

The Rangers are a unique franchise in this aspect. In 1994, the year they opened the only new stadium in their history, the MLBPA went on strike and the World Series was canceled. The strike continued into 1995, dramatically reducing and effectively negating the new stadium effect that was experienced prior to the strike. As a result, in every regression that was run, 1994 was a positive outlier and 1995 was a negative outlier.

Until 2008, these two seasons were the only outliers in every single model tested.

In 1996, the Rangers began the winningest period in the franchise's history, and in each of the models I ran, the results suggested that this winning period was responsible for the high attendance experienced during that period rather than the new stadium. The coincidence is somewhat striking, and in actuality, it was probably a combination of the two that resulted in the high attendance averages.

I added a value for stadium age (in years) to try and capture the new stadium effect, but even after figuring in Arlington Stadium's prior existence as Turnpike Stadium (making it 7 years old in 1972), the variable failed to be significant.

Other Variables

I tried several other variables to help build a better statistical model. Though none turned out to be significant, the following variables were included at some point during testing:

  • Years since playoff appearance
  • Made playoffs (1 or 0)

I also tried to include TMR's Fan Cost Index, but I was only able to find data for the 18 most recent of the 37 seasons. The lack of sample data for this variable resulted in its exclusion from the tested models.

THE FINAL MODEL

When narrowing my model down to the most relevant sample data, I greatly simplified my thinking. Instead of trying to identify individual factors that affect attendance, I realized that, historically, all of this information already existed as a single variable: attendance. It is definitely not the perfect solution, but last season's attendance is the most significant indicator of attendance for the current season.

Based on the models I ran, none was able to predict the huge drop off experienced in 2008. Something that I think might be responsible is the price of gasoline and the increased reliance on gas-guzzling vehicles. Not only did families have less disposable income to spend on baseball games, but the trip to the games became more expensive. Without an effective public transportation solution as an alternative means to get to the ballpark, the mostly commuter fan-base spent their money elsewhere.

After including the previous year's attendance in my models, its significance was immediately apparent, but two other variables remained viable: current winning percentage and growth factor (the counting variable discussed above). Logically, this makes sense.

With this three-variable model, three seasons stood out as dramatic outliers: 1994 (new stadium), 1995 (strike), and 2008 (transportation cost).

It is statistically questionable to eliminate outliers, but in this case, I think it makes sense. I understand that this brings the study as a whole into question, but I'm going to run with it anyway.

In 2009, the Rangers will not be moving into a new stadium; there is no labor conflict on the horizon; and for now, the price of gasoline has returned to a reasonable level.

Because another spike in gasoline prices is possible, 2008 was not removed from the data set. 1972, 1981, and 1995 were all removed because of work stoppages, and 1994 was removed because of the new stadium. (The years that followed still used accurate previous year attendance, so the effects of these events were carried forward to future years even though the immediate effects were not a part of the model.)

The final model included data from 1973 through 2008, skipping over 1981, 1994, and 1995. The dependent variable, of course, was current season average home attendance. The independent variables were the previous season's average home attendance, the current season's winning percentage, and the growth factor described above.

CONCLUSIONS

The model says that for 2009, Texas Rangers home attendance can be estimated within 2,646 attendees per game using a chosen win level, the 2008 attendance per game (24,021), and the growth factor (37).

The graph below represents the relationship between wins and attendance for 2009.

winsattendance2009.gif

The yellow dot marks the 2008 average home attendance, and the red dot marks last season's win total. According to the model, the Rangers should see an increase in attendance over last season for as few as 69 wins.

I think the transportation cost will be a huge factor going into the 2009 season, since the most viable method of getting to the ballpark is to drive.

RELEVANT STATISTICS NOTES

For the final model, the R-square value is 0.904, the adjusted R-square is 0.894, with a standard error of 2,646.

The independent variables have the following p-values: previous attendance average < 0.000002, winning percentage < 0.0008, and growth factor < 0.255.

As discussed above, the removal of the growth factor variable resulted in smaller R-square values and a higher standard error, so it was left in the final model.

IN PART II

In Part II, I will tackle the topic of post-season probability at different win levels. Combined with this article, it will be possible to start turning these numbers into dollars.