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```Statistics [1/2,3/2]
The Essential Mathematics
Standard Error
• What standard deviation is to an
individual (relative to a population
mean), standard error is to a sample
mean (relative to a population mean)
• standard deviation/sqrt(n)
• All parameters have a standard error
associated with them...we use them to
“normalize” statistical tests
Short Exercise
• What is the mean of {1,2,3,4,5}?
• Now, let’s take all possible triplets:
• {1,2,3}, {1,2,4}, {1,2,5}, {1,3,4}, {1,3,5},
{1,4,5}, {2,3,4}, {2,3,5}, {2,4,5}, {3,4,5}
Short Exercise
•
•
•
•
What is the mean of {1,2,3,4,5} = 3
•
Std. Dev (sample) = 1.58114
Now, let’s take all possible triplets:
2, 7/3, 8/3, 8/3, 3, 10/3, 3, 10/3, 11/3, 4
•
•
Mean = 3, Std. Dev (sample) = .60858
Maximum offset: 1 (was originally 2)
Message: having a group reduces the Std.
Dev, hence we have standard error
Statistical Tests
• Null hypothesis: A hypothesis of no
change
• Alternate hypothesis: A hypothesis of
change
• All stats tests assume “no change from
something”...the goal is to prove
otherwise...
Common Tests
• Skewness and Kurtosis
• Z-Test / T-Test
• ANOVA / F-Test
• Correlation Test
Standard Error
Skewing
Who’s Skewed?
Standard Error
Skewing
Who’s Kurtic?
•
•
•
Central Limit
Theorem
Population distribution X
Take n (large) random samples and
compute the mean of the samples
The distribution of these random sample
means (independent of X) will follow the
Gaussian distribution, hence we call it
Normal
Normal Distribution
Z-Test
•
•
Assumes normality
•
Either you know it should be normal, or
you have enough of a sample size to use
the Central Limit Theorem
(observed - mean)/(std. dev / sqrt(n))
•
This equation is generalized for sample
means of sample size n (individual is n =
1)
Example
• A group of 9 people takes an IQ test.
The population is known to follow a
normal distribution with average score
of 100 on the same test with a standard
deviation of 15. The group of 9
averaged a score of 105. Should we
assume that this group differs from the
population of test takers?
Calculation
• (sample mean - population mean) = 5
• (std. dev)/sqrt(9) = 5
• z = 5/5 = 1
• What does this 1 mean?
Generalization
• An arbitrary Gaussian distribution down
to a Gaussian distribution with mean 0
and standard deviation 1
• It’s a value that helps us find another
value
p-value
• Every statistical test has a p-value
• The probability that other observations
(less than it) have already occurred
• In other words, how extreme the
observation is relative to others of its
kind
• z = 1 links to a p-value of .8414 (or
.1586)
• Not something very extreme
a-level
•
•
•
•
Every statistical test has an alpha level
The level at which you reject the null
hypothesis in favor of the alternate
hypothesis
This defines how you handle the p-value
Otherwise known as Type 1 Error (false
rejection probability)
T-test
• A test for when normality cannot be
assumed
• Behaves just like a z-test, but has a
different distribution to work from
• Degrees of freedom
ANOVA
• A way to test whether or not there is a
difference based upon some factor in a
study
• Partitions variance into sources and
uses the ratio as the determining factor
One-Way ANOVA
SS
Betwee
n
Within
Total
df
k-1
SSTSSB
MS
p
SSB/df MSB/M FdfB,dfW(
B
SW
f)
SSW/df
k(n-1)
W
kn-1
F
Two-Way ANOVA
SS
df
A
a-1
B
b-1
AB
Within
Total
MS
SSA/df
A
SSB/df
B
F
p
MSA/MSW
FA,W(fA)
MSB/MSW
FB,W(fB)
SSAB/dfA MSAB/MS
(a-1)(b-1)
F
AB,W(fAB)
B
W
SSAB
nab-1
SSW/dfW
Example ANOVA
Switch vs. LHP
Switch vs. RHP
Example
• It has always been said that hitter of the
opposite hand as the pitcher throws will
succeed at a higher rate
• Does this claim hold water?
Example
• Managers frequently set their lineups on
the principle that they do not want lefthanded hitters back-to-back because a
left-handed specialist (almost always an
LHP) can be used to get consecutive
outs, yet righties are frequently stacked
without concern.
• Are these managers paranoid, or is there
some merit to this?
Sample Set
•
•
30 of the top 75 qualifying hitters for MLB
batting titles in 2012 were selected
•
•
•
Top 10 right-handed hitters
Top 10 left-handed hitters
Top 10 switch hitters (both left and right)
Average against LHP and average
against RHP was recorded for each of
these 30 hitters
Let’s check it out!
Correlation Test
• I got an r-value from a regression that I
performed
• What does it tell me?
• Long story short, it depends on the
sample size
Correlation Test
Statistic
H0: correlation (r) = p
HA: correlation is <,> that
Interesting Picture
Positively
Correlated, but
could be perfect
positive model
Not
Correlated,
but could be
perfect positive
model
Not
No
Correlated
Clue
Not
Correlated,
but could be
perfect negative
model
Negatively
Correlated, but
could be perfect
negative model
Positively
Correlated
Negatively
Correlated
What did we learn?
•
When dealing with correlation studies, make sure you
have at least 13 observations
•
•
•
•
You can disassociate no correlation from the
possibility of a perfect model at this sample size (at
95% confidence)
With more confidence, you will need more
observations to achieve this
A little correlation goes a long way in large samples
With small samples, more correlation is required to
make a claim
Assignment
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•
Given definition of outliers for a population:
•
•
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25% - 1.5(IQR)
75% + 1.5(IQR)
Determine what the z-scores of the minimum outliers on
either side would be
I will send you an ANOVA table:
•
Tell me the factorial environment
•
•
•
A has a levels
B has b levels
How many subjects per block n
```
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