Probability Distributions

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Transcript Probability Distributions

Probability Distributions
What proportion of a group
of kittens lie in any selected
part of a pile of kittens?
Probability Distributions
Sometimes we want to know the chances that something will
occur?
For example:
1.
What are the odds that I will win the lottery?
2.
What are my chances of getting an A?
3.
If a person is young, what are the chances that he or
she will be in poverty?
4.
What chances do poor people have of graduating
from college?
To answer questions such as these, we turn to probability.
Probability Distributions
Probability:
Out of all possible outcomes, the proportionate expectation of a given
outcome. Values for statistical probability range from 0 (never) to 1
(always) or from 0% chance to 100% chance.
For example:
12 of 25 students in an engineering class are women. The probability
that a randomly selected student in that engineering class will be a
woman is 12/25 = .48 or 48%.
13
12
F
M
Probability Distributions
What is the probability that a student will get a C in Statistics?
What about a C or Higher?
10
5
0
3
F
5
D
12
C
7
B
5
A
Probability Distributions
What is the probability that a student will get a C in Statistics?
12/32 = .375
What about a C or Higher?
24/32 = .75
10
5
0
3
F
5
D
12
C
7
B
What is the probability that a person in the class got a grade? 32/32 = 1
5
A
Probability Distributions
Empirical probability distribution:
All the outcomes in a distribution of
research results and each of their
probabilities—what actually happened
The probability distribution of a variable
lists the possible outcomes together with
their probabilities
Probability Distributions
What is the probability that a student will get a C in Statistics?
12/32 = .375
What about a C or Higher?
24/32 = .75
.375
.219
0
F
D C B
A
P=1
100% of
cases
P=.25 or 25%
F
D C B
A
Empirical Rule
Many naturally occurring variables have bell-shaped
distributions. That is, their histograms take a
symmetrical and unimodal shape.
When this is true, you can be sure that the empirical rule
will hold.
Empirical rule: If the histogram of data is approximately bell-shaped, then:
1.
About 68% of the cases fall between Y-bar – s.d. and Y-bar + s.d.
2.
About 95% of the data fall between Y-bar – 2s.d. and Y-bar + 2s.d.
3.
All or nearly all the data fall between Y-bar – 3s.d. and Y-bar + 3s.d.
Empirical Rule
Empirical rule: If the histogram of data is approximately bell-shaped, then:
1.
About 68% of the cases fall between Y-bar – s.d. and Y-bar + s.d.
2.
About 95% of the cases fall between Y-bar – 2s.d. and Y-bar + 2s.d.
3.
All or nearly all the cases fall between Y-bar – 3s.d. and Y-bar + 3s.d.
Body Pile: 100% of Cases
s.d.
15
15
15
s.d.
15
M = 100
55
70
85
s.d. = 15
115
130
145
+ or – 1 s.d.
+ or – 2 s.d.
+ or – 3 s.d.
Probability Distributions
The Normal Probability
Distribution
A continuous probability
distribution in which the
horizontal axis
represents all possible
values of a variable and
the vertical axis
represents the
probability of those
values occurring.
Values are clustered
around the mean in a
symmetrical, unimodal
pattern known as the
bell-shaped curve or
normal curve.
Probability Distributions
The Normal Probability Distribution
No matter what the actual s.d. () value is, the
proportion of cases under the curve that corresponds
with the mean ()+/- 1s.d. is the same (68%).
The same is true of mean+/- 2s.d. (95%)
And mean +/- 3s.d. (almost all cases)
Because of the equivalence of all
Normal Distributions, these are often
described in terms of the Standard Normal Curve
where mean = 0 and s.d. = 1 and is called “z”
Probability Distributions
The Normal Probability Distribution
No matter what the actual s.d. () value is, the
proportion of cases under the curve that corresponds
with the mean ()+/- 1s.d. is the same (68%).
The same is true of mean+/- 2s.d. (95%)
And mean +/- 3s.d. (almost all cases)
68%
Because of the equivalence of all
Normal Distributions, these are often
described in terms of the Standard Normal Curve
where mean = 0 and s.d. = 1 and is called “z”
68%
Z = -3 -2 -1 0 1 2 3
Z=-3 -2 -1 0
1
2
Z = # of standard deviations away from the mean
3
Probability Distributions
Converting to z-scores
To compare different normal curves, it is
helpful to know how to convert data
values into z-scores.
It is like have two rulers beneath each normal
curve. One for data values, the second
for z-scores.
 = 100
IQ
Values
55
Z-scores -3
 = 15
70
85 100
-2
-1
0
115
1
130
2
145
3
Probability Distributions
Converting to z-scores
Z=Y–

Z = 100 – 100 / 15 = 0
Z = 145 – 100 / 15 = 45/15 = 3
Z = 70 – 100 / 15 = -30/15 = -2
Z = 105 – 100 / 15 = 5/15 = .33
 = 100
IQ
Values
55
Z-scores -3
 = 15
70
85 100
-2
-1
0
115
1
130
2
145
3
Probability Distributions
Engagement Ring Example:
Mean cost of an engagement ring
is $500, and the standard
deviation is $100.
Z = 500 – 500 / 100 = 0
Z=Y–

Z = 200 – 500 / 100 = -300/100 = -3
Ring
Values
Z = 550 – 500 / 100 = 50/100 = .5
 = 100
 = 15
200 300 400 500
Z-scores -3
Z = 600 – 500 / 100 = 100/100 = 1
-2
-1
0
600
700
800
1
2
3
Probability Distributions
Engagement Ring Example:
Mean cost of an engagement ring
is $500, and the standard
deviation is $100.
Now, use the empirical rule…
What percentage of people will
be above or below my preferred
ring price of $300?
Ring
 = $500
2.5%
Values
 = $100
200 300 400 500
Z-scores -3
2.5%
68%
-2
-1
0
600
700
800
1
2
3
Probability Distributions
Comparing two distributions by Z-score
Imagine that your partner didn’t get you a ring, but took you on a trip to express their
love for you. You could convert the trip’s price into a ring price using z-scores.
Your trip cost $2,000. The average “love trip” costs $1,500 with a s.d. of $250. What
is the equivalent ring price?
Trips
Rings
200 300 400 500
-3
-2
-1
0
600
700
1
2
800 750 1000 1250 1500
3
-3
-2
-1
0
1750 2000 2250
1
2
3
Probability Distributions
Comparing two distributions by Z-score
Your trip cost $2,000. The average “love trip” costs $1,500 with a s.d. of $250. What
is the equivalent ring price?
What percentage of persons got a trip that cost less than yours?
Trips
Rings
200 300 400 500
-3
-2
-1
0
600
700
1
2
800 750 1000 1250 1500
3
-3
-2
-1
0
1750 2000 2250
1
2
3
Probability Distributions
Comparing two distributions by Z-score
What about ACT versus SAT scores?
SAT
ACT
15
-3
18
-2
21
-1
24
27
30
33 400
0
1
2
3
-3
600 800 1000
-2
-1
0
1200 1400 1600
1
2
NOTE: This is a helpful process, but can be illogical at times. Remember that you are
comparing scores on a “population base” or percent of people above or below
each score. Is it logical to compare SAT score to self-esteem this way? No.
3
Probability Distributions
How to use a z-score table. (I could use
some z z z z’s).
F-N&L-G Appendix B has reports from the
literal measurements of area under
normal curves. The table gives you the
percent of values above, below, or
between particular z-scores (# of s.d.s
away from the mean).
Left column = z (out to two decimals)
Second column is area—proportion of
distribution—from mean to z
Right column is area—proportion of
distribution—from z to the end of the line.
Can work in reverse to find z-scores too.
Other tables will use different layouts, online
you can get automatic answers without
using a table.
Probability Distributions
Theoretical probability distribution:
The proportion of times we would expect to get a
particular outcome in a large number of trials—what
would happen if we had the time to observe it.
Q: Why are these important?
A: Sociologists usually get only one chance to draw a
sample from a population. Therefore, if we know what
kind of variation in measurement we would see if we
repeatedly sampled (theoretically), we can judge the
chance that numbers produced by our sample are
accurate (this will make sense later).
Probability Distributions
Theoretical probability distribution:
The number of times we would expect to get a particular outcome in a large number of
trials.
For Example: Let’s say the mean GPA at SJSU is 2.5.
Randomly take 100 SJSU students’ GPAs.
Record it.
Now, take 100 more SJSU students’ GPAs.
Record that.
Now, repeat the above.
Record again.
Now, lather, rinse, repeat.
Again.
Again. And on and on.
What might you see?
Probability Distributions
Theoretical probability distribution:
The number of times we would expect to get a particular outcome in a large
number of trials.
50% of samples would have
a mean GPA greater than
2.5
1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9
= a sample’s mean
2.5 = Overall Mean