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Psyc 235:
Introduction to Statistics
http://www.psych.uiuc.edu/~jrfinley/p235/
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Independent vs. Dependent
Events
• Independent Events: unrelated events that
intersect at chance levels given relative
probabilities of each event
• Dependent Events: events that are related in
some way
• So... how to tell if two events are independent or
dependent?
 Look at the INTERSECTION: P(AB)
• if P(AB) = P(A)*P(B) --> independent
• if P(AB)  P(A)*P(B) --> dependent
Random Variables
• Random Variable:
 variable that takes on a particular numerical
value based on outcome of a random experiment
• Random Experiment (aka Random Phenomenon):
 trial that will result in one of several possible
outcomes
 can’t predict outcome of any specific trial
 can predict pattern in the LONG RUN
Random Variables
• Example:
• Random Experiment:
 flip a coin 3 times
• Random Variable:
 # of heads
Random Variables
• Discrete vs Continuous
 finite vs infinite # possible outcomes
• Scales of Measurement
 Categorical/Nominal
 Ordinal
 Interval
 Ratio
Data World vs. Theory World
• Theory World: Idealization of reality (idealization
of what you might expect from a simple
experiment)
 Theoretical probability distribution
 POPULATION
 parameter: a number that describes the population.
fixed but usually unknown
• Data World: data that results from an actual
simple experiment
 Frequency distribution
 SAMPLE
 statistic: a number that describes the sample (ex:
mean, standard deviation, sum, ...)
So far...
• Graphing & summarizing sample distributions
(DESCRIPTIVE)
• Counting Rules
• Probability
• Random Variables
• one more key concept is needed to start doing
INFERENTIAL statistics:
SAMPLING DISTRIBUTION
Binomial Situation
• Bernoulli Trial
 a random experiment having exactly two possible
outcomes, generically called "Success" and "Failure”
 probability of “Success” = p
 probability of “Failure” = q = (1-p)
Examples:
Coin toss: “Success”=Heads
p=.5
Heads
Tails
Robot Factory:
“Success”=Good Robot
p=.75
Good Robot
Bad
Robot
Binomial Situation
• Binomial Situation:
 n: # of Bernoulli trials
 trials are independent
 p (probability of “success”) remains constant
across trials
• Binomial Random Variable:
 X = # of the n trials that are “successes”
Binomial Situation:
collect data!
Population:
Bernoulli Trial:
one coin toss
Outcomes of all possible coin tosses
(for a fair coin)
Success=Heads
p=.5
Let’s do 10 tosses
n=10 (sample size)
Sample:
X=
....
Binomial Random
Variable:
X=# of the 10 tosses
that come up heads
(aka Sample Statistic)
Binomial Distribution
p=.5, n=10
0.30
0.25
probability
0.20
0.15
This is the
SAMPLING DISTRIBUTION
of X!
0.10
0.05
0.00
0
1
2
3
4
5
# of successes
6
7
8
9
10
Sampling Distribution
• Sampling Distribution:
Distribution of values that your sample
statistic would take on, if you kept
taking samples of the same size, from
the same population, FOREVER
(infinitely many times).
• Note: this is a THEORETICAL
PROBABILITY DISTRIBUTION
Binomial Situation:
collect data!
Population:
Bernoulli Trial:
one coin toss
Outcomes of all possible coin tosses
(for a fair coin)
Success=Heads
p=.5
0.3
Let’s do 10 tosses
n=10 (sample size)
0.25
Sampling Distribution
probability
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
8
9
10
# of successes
Sample:
X=
3
5
6
....
Binomial Random
Variable:
X=# of the 10 tosses
that come up heads
(aka Sample Statistic)
Binomial Situation:
collect data!
Population:
Bernoulli Trial:
one coin toss
Outcomes of all possible coin tosses
(for a fair coin)
Success=Heads
p=.5
0.3
0.25
Sampling Distribution
probability
0.2
0.15
Let’s do 10 tosses
n=10 (sample size)
0.1
0.05
0
0
1
2
3
4
5
# of successes
Sample:
X=
3
6
7
8
9
10
Binomial Random
Variable:
X=# of the 10 tosses
that come up heads
(aka Sample Statistic)

Binomial Formula
P(X  k)  P(exactly k many successes)
specific # of
successes you
could get
probability
of success
n k
nk
P(X  k)   p (1 p)
k 
Binomial
Random
Variable
combination
called the
Binomial Coefficient
n 
n!
 
k  k!(n  k)!
specific #
of
failures
probability
of failure
Binomial Formula
0.3
0.25
Sampling Distribution
probability
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
8
9
10
# of successes
3
p(X=3) =
Hmm... what if we had gotten X=0?...
pretty unlikely outcome... fair coin?
Remember this idea....
p=.5
n=10
More on the Binomial
Distribution
• X ~ B(n,p)
Expected Value
and Variance for X~B(n,p)
X  np
these are the
parameters for
the sampling
distribution of X
 X2  np(1 p)
Standard Deviation :  X  np(1 p)
Ex:
# heads in 5 tosses of a coin:
# heads in 5 tosses of a coin:
X~B(5,1/2)
Expectation
2.5
Variance Std. Dev.
1.25
1.12
Let’s see some more
Binomial Distributions
• What happens if we try doing a different #
of trials (n) ?
• That is, try a different sample size...
Binomial Distribution, p=.5, n=5
0.35
0.3
probability
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
# of successes
4
5
Binomial Distribution, p=.5, n=10
0.3
0.25
probability
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
# of successes
6
7
8
9
10
Binomial Distribution, p=.5, n=20
0.2
0.18
0.16
probability
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0
1
2
3
4
5
6
7
8
9
10
11
# of successes
12
13
14
15
16
17
18
19
20
Binomial Distribution, p=.5, n=50
0.12
0.1
0.06
0.04
0.02
# of successes
50
48
46
44
42
40
38
36
34
32
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
0
probability
0.08
Binomial Distribution, p=.5, n=100
0.09
0.08
0.07
0.05
0.04
0.03
0.02
0.01
# of successes
99
96
93
90
87
84
81
78
75
72
69
66
63
60
57
54
51
48
45
42
39
36
33
30
27
24
21
18
15
12
9
6
3
0
0
probability
0.06
Whoah.
• Anyone else notice those DISCRETE
distributions starting to look smoother as
sample size (n) increased?
• Let’s look at a few more binomial
distributions, this time with a different
probability of success...
Binomial Robot Factory
• 2 possible outcomes:
Good Robot
90%
Bad Robot
10%
You’d like to know about how many BAD robots you’re likely to get
before placing an order... p = .10 (... “success”)
n = 5, 10, 20, 50, 100
Binomial Distribution, p=.1, n=5
0.7
0.6
probability
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
# of successes
4
5
Binomial Distribution, p=.1, n=10
0.45
0.4
0.35
probability
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
# of successes
6
7
8
9
10
Binomial Distribution, p=.1, n=20
0.3
0.25
probability
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
8
9
10
11
# of successes
12
13
14
15
16
17
18
19
20
Binomial Distribution, p=.1, n=50
0.2
0.18
0.16
0.12
0.1
0.08
0.06
0.04
0.02
# of successes
50
48
46
44
42
40
38
36
34
32
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
0
probability
0.14
Binomial Distribution, p=.1, n=100
0.14
0.12
0.08
0.06
0.04
0.02
# of successes
99
96
93
90
87
84
81
78
75
72
69
66
63
60
57
54
51
48
45
42
39
36
33
30
27
24
21
18
15
12
9
6
3
0
0
probability
0.1
Normal Approximation
of the Binomial
If n is large, then
X ~ B(n,p)
{Binomial Distribution}
can be approximated by a NORMAL DISTRIBUTION with
parameters:
  np
  np(1 p)
0.3
0.25
probability
0.2
0.15
0.1
0.05
0
Normal Distributions
• (aka “Bell Curve”)
• Probability Distributions of a Continuous
Random Variable
 (smooth curve!)
• Class of distributions, all with the same overall
shape
• Any specific Normal Distribution is characterized
by two parameters:
 mean: 
 standard deviation: 
different
means
different
standard
deviations
Standardizing
• “Standardizing” a distribution of values results in
re-labeling & stretching/squishing the x-axis
• useful: gets rid of units, puts all distributions on
same scale for comparison
• HOWTO:
 simply convert every value to a:
Z SCORE:
z
x 

Standardizing
• Z score:
z
x 

• Conceptual meaning:
 how many standard deviations from the mean
a given score is (in a given distribution)
• Any distribution can be standardized
• Especially useful for Normal
Distributions...
Standard Normal Distribution
• has mean: =0
• has standard deviation: =1
• ANY Normal Distribution can be converted
to the Standard Normal Distribution...
Standard
Normal
Distribution
Normal Distributions &
Probability
• Probability = area under the curve
 intervals
 cumulative probability
 [draw on board]
• For the Standard Normal Distribution:
 These areas have already been calculated for
us (by someone else)
Standard Normal Distribution
So, if this were a Sampling Distribution, ...
Next Time
• More different types of distributions
 Binomial, Normal
 t, Chi-square
F
• And then... how will we use these to do
inference?
• Remember: biggest new idea today was:
 SAMPLING DISTRIBUTION