A look into the how the 2010 year of the pitchers measured up, the odds, and in reality what it meant on a historical baseball pperspective.

1. This presentation will be taking a look at one of the many real
life use for statistics, Sports, and what some basic and
advanced options can tell us when trying to use statistics to
decide a not so straight forward question. Using SSPS as our
backbone along with already printed statistical analysis we will
take a look at how things like z-scores, standard deviations,
selective cases, and logistic regressions, can help us obtain a
solid foundation to answer a hazy question.
Memorable or just plain Amazing:
A Statistical Look at the 2010 MLB Season otherwise known as
“The Year of the Pitcher”
By: KC Burgos
Math 449
nov. 1, 2011

2. What Exactly Are We Looking To Answer?
• 2010 MLB Season has been classified as the
“Year of the Pitcher”
• Seven No-Hitters (Six official)
• What exactly defines a great pitching year?

3. List of No-Hitters in 2010
• April 17th, Ubaldo Jiménez vs. Atlanta Braves (Final Score 4-0)
• May 9th, Dallas Braden vs. Tampa Bay Rays (Final Score 4-0)
• May 29th, Roy Halladay vs. Florida Marlins (Final Score 1-0)
• June 2nd, Armando Galarraga* vs. Cleveland Indians (Final Score 3-0)
• June 25th, Edwin Jackson vs. Tampa Bay Rays (Final Score 1-0)
• July 26th, Matt Garza vs. Detroit Tigers (Final Score 5-0)
• October 6th, Roy Halladay vs. Cincinnati Reds (Final Score 4-0)

4. Armando Galaragga’s No-Hitter
• Armando Galaragga’s No-Hitter is officially
recognized by MLB as a one-hitter.
• The 27th out

5. Defining Parameters for our Question
• Many ways to answer what makes a season the
“Year of the Pitcher”
• Comparing Seasons and Stats
• No-Hitter Probabilities

6. Comparisons: Variables
• IP – Innings Pitched
• R – Runs
• ER – Earned Runs
• H – Hits
• BB – Base on Balls (Walks)
• ERA – Earned Run Average
• WHIP – Walks and Hits per Inning Pitched

7. Computation Variables
• IP – 123.1 = 123 1/3, 453.2 = 453 2/3
• ERA – (9*ER)/IP
• WHIP – (H + BB)/IP

8. Comparisons: The Data
• 2010 Season
• 30 Teams
• Specifically Looking at Pitching
• Other Possible Variables We Could Have Used

9. Comparisons: The Data

10. Comparisons: Descriptives
MLB
NL
AL

11. Comparisons: Yearly Data
• Compare it to Data from 1969-2009
• Why 1969?

12. Comparisons:
Yearly Data
• Special Years:
1969
1977
1981
1993
1994
1995
1998

13. Comparisons: Normaliity
If we can have, or make, our values match a normal curve then we
can easily use these already derived percentages of where each
standard deviation should fit.

14. Comparisons: Normality and Homoscedasticity
MLB ERA 1969-2009

15. Comparisons: Normality and Homoscedasticity
MLB WHIP 1969-2009

16. Comparisons: Normality and Homoscedasticity
AL ERA 1969-2009

17. Comparisons: Normality and Homoscedasticity
AL WHIP 1969-2009

18. Comparisons: Normality and Homoscedasticity
NL ERA 1969-2009

19. Comparisons: Normality and Homoscedasticity
NL WHIP 1969-2009

20. Comparisons: Fischer’s Method
Fischer’s Method tells us that if the
skewness value is twice its standard error
in either direction then the curve is
severely skewed. A distribution is
considered normal if it meets
-1.96 <
𝑺
𝑺𝑬𝑺
< 1.96

21. Comparisons: Fischer’s Method
It should also be noted that this applies
for Kurtosis. However I focus here more
on the skewness of data, having already
taken kurtosis into account.
-1.96 <
𝑲
𝑺𝑬𝑲
< 1.96

22. Comparisons: Yearly Data Descriptives

23. Comparisons: Doing the Math
With Normality Assumed ERA and WHIP
should be within 3 Standard Deviations
from the mean.

24. Comparisons: Doing the Math
Lower 3 Standard Deviations
1969-2009 (1) (2) (3)
MLB ERA 4.0522 3.67005 3.2879 2.90575
MLB WHIP 1.3673 1.32313 1.27896 1.23479
AL ERA 4.31734 3.7317 3.29 2.8483
AL WHIP 1.3836 1.32916 1.27472 1.22028
NL ERA 3.91 3.54707 3.18414 2.82121
NL WHIP 1.35 1.30743 1.26486 1.22229

25. Comparisons: Doing the Math
z-Scores
z-Scores can be used to determine an
exact distance from the mean a
number ‘X’ is in terms of the standard
deviation.
z =
𝑋−𝜇
𝜎

26. Comparisons: Doing the Math
1969-
2009 (1) (2) (3)
MLB
ERA
4.0522 3.67005 3.2879 2.90575
MLB
WHIP
1.3673 1.32313 1.27896 1.23479
AL ERA 4.31734 3.7317 3.29 2.8483
AL WHIP 1.3836 1.32916 1.27472 1.22028
NL ERA 3.91 3.54707 3.18414 2.82121
NL
WHIP
1.35 1.30743 1.26486 1.22229
2010 MLB ERA = 4.0743
2010 MLB WHIP = 1.3473
2010 AL ERA = 4.1384
2010 AL WHIP = 1.3463
2010 NL ERA = 4.0183
2010 NL WHIP = 1.3482
Z-Scores
2010 MLB ERA: z = .05783
2010 MLB WHIP: z = -.452796
2010 AL ERA: z = -.4051166
2010 AL WHIP: z = -.685158
2010 NL ERA: z = .298405
2010 NL WHIP: z = -.0422833

27. Comparisons: What Does It Tell Us?
Since all of the Sample Means are within one standard
deviation of the Parent Mean then we can say that 2010
was a relatively normal pitching year with respect to
post-1969 pitching years in terms of the variables ERA
and WHIP.

28. Comparisons: Defining a Smaller Interval
When looking at a scatter plot
graph of MLB ERA between 1969
and 2009 we can see a unusual
increase. So let’s apply a R2 Linear
Fit Line.
Although helpful a R2 Linear Fit
Line doesn’t quite give us a great
description of what’s truly going
on. So instead let us use a a R2
LOESS Fit Line.
LOESS (Locally Weighted
Scatterplot Smoothing) looks
more specifically at subsets
within our values and produces a
curve to fit them.

29. Comparisons: Defining a Smaller Interval
We can now see an interesting
curve in our later values.
We want to start a subinterval
at that point. It is the last year
at which ERA would go down
the following year before
dramatic increases.
The year turns out to be 1991

30. Comparisons: 1991-2009 Data Descriptives
Like before we are going to want to determine skewness and
homoscedasticity. However, thanks to Fischer’s method, we can just use
the descriptives. As for Homoscedacity, they all passed.
We can see that we have skewness issues for two variables.

31. Comparisons: 1991-2009 Data Descriptives
Although there are certainly methods to deal with skewness to retrieve a
normal distribution, instead we can see that each league has a statistic
that is regarded as normal.
WHIP
Therefore we will focus on those three across the board.

32. Comparisons: Doing the Math 2
Like before we will determine the three standard deviations away from
the mean and use them to compare to the 2010 numbers
1991-2009 (1) (2) (3)
MLB WHIP 1.3996 1.36352 1.32744 1.29136
AL WHIP 1.4214 1.37659 1.33178 1.28697
NL WHIP 1.3783 1.3406 1.3029 1.2652

33. Comparisons: Doing the Math 2
1991-
2009 (1) (2) (3)
MLB
WHIP
1.3996 1.36352 1.32744 1.29136
AL
WHIP
1.4214 1.37659 1.33178 1.28697
NL
WHIP
1.3783 1.3406 1.3029 1.2652
2010 MLB WHIP = 1.3473
2010 AL WHIP = 1.3463
2010 NL WHIP = 1.3482
z-Scores
2010 MLB WHIP: z = -1.4496
2010 AL WHIP: z = -1.67597
2010 NL WHIP: z = -.798408

34. Comparisons: So what does this tell us???
That even when looking at a recent interval of years (19
between 1991-2009) that last year pitching wise, in terms
of WHIP, was at most within a range of two standard
deviations away from the MLB, AL, and NL, averages over
those 19 years (2010 MLB WHIP: z = -1.4496, 2010 AL
WHIP: z = -1.67597, 2010 NL WHIP: z = -.798408).
Therefore last year was above average, however, was not
so far off that it could be considered extremely rare or an
outlier. So once again not exactly the “Year of the
Pitcher”

35. Is There An Easier Way???
Comparisons Cons:
• Too many numbers
• Sample Sizes
• Normality

36. Logistic Regression
Logistic Regression – When we want an odds ratio for a
question that does not have a continuous answer (i.e.
yes or no questions), a Logistic Regression can give you
that equation based on the factors you give.
P(event) =
1
1+𝑒−(𝑎+𝑏1 𝑋1+𝑏2 𝑋2+ … +𝑏 𝑖 𝑋 𝑖)
Odds =
𝑃 𝑒𝑣𝑒𝑛𝑡
1+𝑃 𝑒𝑣𝑒𝑛𝑡
ln(odds) =𝑎 + 𝑏1 𝑋1 + 𝑏2 𝑋2 + … + 𝑏𝑖 𝑋𝑖
a = Y-Intercept
b = regression coefficient
X = factors

37. Logistic Regression: Assumptions
• DOES NOT ASSUME Linearity
• DOES NOT ASSUME Normal Distribution
• DOES NOT ASSUME Homoscedasticity
• DOES NOT ASSUME Normality of Residuals
• Does Assume Sample Representativeness
• Does Assume Levels of Measurement
• Does Assume no Multicollinearity (exist if VIF > 10)

38. Logistic Regression: Behind the Scenes
Suppose we want to predict a variable Y from X where our data is
described as: (x1 , y1) … (xn , yn)
We can look at two possibilities.
If Y has a Normal Distribution, with mean and variance 2.
If Y has only two possible values 0 and 1.

39. Linear Regression
If Y has a Normal Distribution, with mean and variance 2.
Then it has a probability density function (pdf) of
𝑒
−
𝑦−𝜇 2
2𝜎2
2𝜋𝑟
Suppose = a + bx
The Likelihood function is
𝑒
−
𝑦1−(𝑎+𝑏𝑥1) 2
2𝜎2
2𝜋𝑟
∗ ⋯ ∗
𝑒
−
𝑦 𝑛− 𝑎+𝑏𝑥 𝑛
2
2𝜎2
2𝜋𝑟
=
𝑒
− 𝑖=1
𝑛
𝑦 𝑖− 𝑎+𝑏𝑥 𝑖
2
2𝜎2
2𝜋
𝑛
2 𝑟 𝑛

40. Linear Regression
If Y has a Normal Distribution, with mean and variance 2.
Maximizing this likelihood function is the same as minimizing
Q =
𝑖=1
𝑛
𝑦𝑖 − 𝑎 + 𝑏𝑥𝑖
2
This can be done by setting derivatives with respect to a and with
respect to b equal to zero.
𝑑𝑄
𝑑𝑏
=
𝑖=1
𝑛
2[𝑦𝑖 − 𝑎 + 𝑏𝑥𝑖 ](−𝑥𝑖)
𝑑𝑄
𝑑𝑎
=
𝑖=1
𝑛
2 𝑦𝑖 − 𝑎 + 𝑏𝑥𝑖 (−1)

41. Linear Regression
If Y has a Normal Distribution, with mean and variance 2.
𝑑𝑄
𝑑𝑏
=
𝑖=1
𝑛
2[𝑦𝑖 − 𝑎 + 𝑏𝑥𝑖 ](−𝑥𝑖)
𝑑𝑄
𝑑𝑎
=
𝑖=1
𝑛
2 𝑦𝑖 − 𝑎 + 𝑏𝑥𝑖 (−1)
Solving these equations leads to the solutions
𝑏 =
𝑖=1
𝑛
(𝑥𝑖 − 𝑥) 𝑦𝑖 − 𝑦 2
𝑖=1
𝑛
(𝑥𝑖 − 𝑥)2
𝑎 = 𝑦 − 𝑏 𝑥

42. Logistic Regression
If Y has only two possible values 0 and 1.
It’s probability density function (pdf)
𝑝 𝑦
1 − 𝑝 1−𝑦
, y = 0, 1
Where p is the probability that y = 1
and 1 − p is the probability that y = 0
Suppose 𝑝 = 𝑎 + 𝑏𝑥
This Doesn’t Work!
It allows the possibility that p < 0 or p > 1.
Let’s Try Odds!

43. Logistic Regression
If Y has only two possible values 0 and 1.
𝑜𝑑𝑑𝑠 =
𝑝
1−𝑝
= 𝑎 + 𝑏𝑥.
However, This does not work either since it allows the possibility that
𝑝
1−𝑝
< 0.
Let’s try this then
log
𝑝
1−𝑝
= 𝑎 + 𝑏𝑥
This will work since log() can be any real number
𝑝
1 − 𝑝
= 𝑒 𝑎+𝑏𝑥
⇒ 𝑝 =
𝑒 𝑎+𝑏𝑥
1 + 𝑒 𝑎+𝑏𝑥
=
1
1 + 𝑒− 𝑎+𝑏𝑥
⟹ 1 − 𝑝 =
1
1 + 𝑒 𝑎+𝑏𝑥

44. Logistic Regression
If Y has only two possible values 0 and 1.
The likelihood function is
𝑒 𝑎+𝑏𝑥1
1 + 𝑒 𝑎+𝑏𝑥1
𝑦1
1
1 + 𝑒 𝑎+𝑏𝑥1
𝑦1
∗ ⋯ ∗
𝑒 𝑎+𝑏𝑥 𝑛
1 + 𝑒 𝑎+𝑏𝑥 𝑛
𝑦 𝑛
1
1 + 𝑒 𝑎+𝑏𝑥 𝑛
𝑦 𝑛
p 1 p
Now finding the derivatives with respect to a and b and setting these equal is
very difficult! Typically this function is maximized by using iterative procedures
from numerical analysis.

45. Logistic Regression: Our Use
• What made 2010 noticeable to begin with?
• NO HITTERS!!!!
• Step 1, Find our Variables

46. Logistic Regression: Variables and Coding
• Games Started (GS)
• Winning Percentage of Games Started (WinPer)
• Earned Run Average (ERA)
• Complete Game Percentage of Games Started
(CGPer)
• Strikeouts per Nine Innings Pitched (Kper9)
• Batting Average Against (BAA)
• Innings Pitched (IP)
• Hits per Nine Innings Pitched (Hper9)
• Shutout Percentage of Games Started (ShoPer)
• No Hitter Achieved (Coded 1 = yes, 0 = no)

47. Logistic Regression: Preparation
Before Throwing Our Variables into the
regression we can actually eliminate ones we
will not need using a Independent t-Test to
see which variables are statistically significant
between those who threw a no-hitter and
who didn’t.

48. Logistic Regression: Preparation

49. Logistic Regression: Preparation
But HOLD IT You still haven’t checked
for Multicollinearity!!!
Actually I have, it just turns out the two variables from the
independent t-Test fit.

50. Logistic Regression: Recap
What We Know
• The two variables being input are CGPer and ShoPer
• Both fit all assumptions
LET’S DO THIS THING!

51. Logistic Regression: Equation
So what do we get?
ln(odds of Throwing a No Hitter) = -2.505 + 38.772 * (ShoPER)

52. Logistic Regression: Using It!
Take a look at the ln(odds) for the six pitchers who threw a no hitter
(taking into account Stats before their 2010 season)
• Roy Halladay ln(odds) = -2.505 + 38.772 * (.05226) = -.47878
• Ubaldo Jimenez ln(odds) = -2.505 + 38.772 * (0) = -2.505
• Dallas Braden ln(odds) = -2.505 + 38.772 * (0) = -2.505
• Armando Galarraga ln(odds) = -2.505 + 38.772 * (0) = -2.505
• Edwin Jackson ln(odds) = -2.505 + 38.772 * (.00909) = -2.15256
• Matt Garza ln(odds) = -2.505 + 38.772 * (.02326) = -1.60316

53. Logistic Regression: Using It!
Probability Table
Pitcher ln(odds) P(no-hitter)
Roy Halladay -.47878 0.382540
Ubaldo Jimenez -2.505 0.075508
Dallas Braden -2.505 0.075508
Armando Galarraga -2.505 0.075508
Edwin Jackson -2.15256 0.104092
Matt Garza -1.60316 0.167540

54. Logistic Regression: One Last Comparison
Let’s make one last comparison. Let’s compare the average ln(odds) of
the previous six pitchers to throw no hitters with the previous six.
-1.647795
Of our six pitchers we will find only one, The Philadelphia Phillies’ Roy
Halladay, is above the average at -.47878. In fact he is above the
ln(odds) average of everyone who threw a no hitter in our database,
-1.173541089.
As for our five others, the average ln(odds) of those pitcher in our
database who didn’t throw a no-hitter is -1.575964583. All of our
other five pitchers fall short of this mark, the closest being Matt Garza
at -1.60316.

55. Logistic Regression: What Does it Mean?
It means four of the six pitchers who threw a no-hitter had, in laymen’s
terms, PULLED A MIRACLE OUT OF THEIR HAT!
Statistically, it means that four of our six pitchers not only beat the
odds, they destroyed them.
3 of our pitchers, (Ubaldo Jimenez, Dallas Braden, and Armando
Galarraga) beat the odds by nearly one having never even thrown a
shutout before their no-hitters.
The other two (Edwin Jackson and Matt Garza) showed odds more
likely to not throw a no-hitter even having previously thrown shutouts.
***Although not directly assessed in this analysis, both Dallas Braden
and Armando Galarraga pitched Perfect Games, exponentially rarer***

56. Logistic Regression:

57. Conclusion
What does the data tell us?
Overall pitching was average when we use the yearly averages and
standard deviations from 1969-2009 (2010 MLB ERA: z = .05783, 2010 MLB
WHIP: z = -.452796, 2010 AL ERA: z = -.4051166, 2010 AL WHIP: z = -.685158,
2010 NL ERA: z = .298405, 2010 NL WHIP: z = -.0422833), and only slightly
improved when reducing our interval to 1991-2009 and focus in on WHIP (2010
MLB WHIP: z = -1.4496, 2010 AL WHIP: z = -1.67597, 2010 NL WHIP: z = -
.798408).
The pitchers who made this season memorable with their no-hitters
showed that only one of them any sign that a no-hitter could be coming (Roy
Halladay, ln(odds) = -.47878). As for the other five, all of them showed odds less
than the average ln(odds) of those pitchers who didn’t throw no hitters (144,
average ln(odds) = -1.575964583). Three of the five actually had the lowest
possible odds according to our ln(odds) (Dallas Braden, Armando Galarraga,
Ubaldo Jimenez, ln(odds) = -2.505)

58. Conclusion
So what should we call the 2010 MLB season?
“The Comeback Year of the Pitcher”

59. Special Thanks
Dr. Sprechini – Advisor
David Brown – Peer Reviewer

60. Bibliography
• Abu-Bader, Soleman. Advanced & Multivariable Statistical Methods For Social
Science Research with a complete SSPS guide. Chicago, Ill: Lyceum Books,
2010.
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19.
• Bauman, Mike. Halladay stood out in Year of the Pitcher. MLB News.
http://mlb.mlb.com/news/article.jsp?ymd=20101116&content_id=1611319
0&vkey=news_mlb&c_id=mlb
• Forman, Sean, Justin Kubatko. www.baseball-reference.com
• McCarthy, David, David Groggel, and John A. Bailer. "Career Pitching Statistics and
the Probability of Throw a No-Hitter in MLB: A Case-Control Study." Chance
23.3 (2010): 25-35.
• Schmotzer, Brian, Patrick D. Kilgo, and Jeff Switchenko. "'The Natural'? The Effect
of Steroids on Offensive Performance in Baseball." Chance 22.2 (2009): 21-
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• Wikipedia Contributors. No-Hitter. Wikipedia, The Free Encyclopedia,
http://en.wikipedia.org/wiki/No-hitter#Major_League_Baseball_no-hitters