Transcript Lecture 4

CRIM 483
Chapter 7: Normal Curves &
Probability
Normal Curves & Probability
• The goal of statistics=estimate probability of an outcome—I.e., how
likely is it that a particular outcome will occur
• Probability=the likelihood that an event will occur
– Provides the basis for determining the degree of confidence we have in
stating that a particular outcome is “true”
– Basis of the normal curve and the foundation for inferential statistics
• One way to estimate the probability of an outcome is by using a
normal curve=visual representation of a distribution of scores
– Provides the basis for understanding the probability associated with any
possible outcome
– Also known as a bell curve
Normal Curve Characteristics
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Shape looks like a bell
Mean, median, mode are equal to one another
The mean represents the center of the distribution
– The shape is symmetrical
– 50% of the values lie above the mean and 50% of the values lie below this point
– Tails are asymptotic—tails come closer and closer to axis but never touch it
More Description
•
The Normal Curve represents a distribution of events/scores
– Few events occur at the tails of the distribution—low probability that they will
occur
– Most events occur within the middle—higher probability that they will occur
– Represents the basic nature of things—most distributions will naturally fall into
this shape
– E.g., intelligence, height
Distribution Patterns
•
In a any distribution of scores that follow a Normal Curve:
– Almost 100% of the scores will fit between -3 & +3 standard deviations from the
mean REGARDLESS of the value of the mean and standard deviation
Distribution of Scores
• A certain percentage of cases will fall between
different points on the axis
• Example: Mean=100; SD=10
% Within
+ Side
- Side
B/T Mean & 1 SD
34.13%
100-110
90-100
B/T Mean & 2 SD
13.59%
110-120
80-90
B/T Mean & 3 SD
2.15%
120-130
70-80
3 SD +
.13%
130+
Below 70
50% * 2
Percentages & Probabilities
• The percentages or areas under the curve
also represent probabilities of a certain
score occurring
• Convert the percentage to a probability:
– 42%/100= .42
• Convert a probability to a percentage:
– .42 * 100= 42%
Comparing Means Across
Groups: The Z-Score
• Each group has a distribution—but in their
original form, the groups are not comparable
• Each original score can be converted to a zscore, which is a standard score that can be
compared across groups
z=(x-mean)/s
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•
•
•
z=z-score
x=score
mean=mean of the distribution
s=standard deviation of the distribution
Z-Score Characteristics
• Z-scores are distributed in a bell curve
– Mean, median, mode=0
– SD=1
• Scores below the meannegative z-score
• Scores above the meanpositive z-score
• Z-scores provide a way to compare group
scores using a standard score
– Book Example: Page 127-128
Characteristics, Continued
• Given the standard distribution of scores
within a normal curve, the following
statements are true:
– 84% of the scores fall below z-score=1
– 16% of the scores fall above z-score=1
• The more extreme the z-score, the farther
it is from the mean
Comparing Distributions
Dist. 1 (mean=12; SD=2)
z Score
X-mean
X
0
0
12
1.5
3
15
-0.5
-1
11
0.5
1
13
-2
-4
8
1
2
14
0
0
12
0.5
1
13
0
0
12
-1
-2
10
Dist. 2 (mean=59; SD=14.5)
z Score
X-mean
X
0.55
8
67
-0.34
-5
54
0.41
6
65
-1.79
-26
33
-0.21
-3
56
1.17
17
76
0.41
6
65
-1.79
-26
33
-0.76
-11
48
1.17
17
76
Using z-Scores
• In distribution with mean of 100 and SD of 10 what is the probability
that any one score will be 110 or greater?
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–
–
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z-score(110)=1.00
Also know that 16% of the scores fall above this point
16%/100=.16
The probability that any one score will be 110 or greater=.16
• Table of z-scores and related probabilities is provided for less
straightforward z-score values (e.g., 1.28, .84, etc.)
• Using the z-score table—Table B1, Appendix B
– Provides a z-score for every possible point on the x-axis
– (Area between mean & z-score)+50=percent likelihood that the event
will fall below z-score
– To find the area that falls above the score you must make the following
computation:
• 100 – (the area provided in the chart+50%)
Applying z-Scores
• Finding the area between two z-scores
– E.g., the amount of area between a z score of
110 and 125 (mean=100; SD=10)
– This number corresponding to the area=the
probability that a score will fall between these
two scores of interest
• Find the area for each z-score and subtract it from
one another
Comparing the Area B/T Scores
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(110-100)/10=10/10=1.00
(125-100)/10=25/10=2.50
Area between mean and 1.00=34.13%
Area between mean and 2.50=49.38%
Area between 1.00 and 2.50=15.25%
– Thus, the probability that a score will fall
between 110 and 125 is .1525
• The probability that a score will fall under
1.00=.50 + .3413=.8413
Using Probabilities
• Ultimately, the point of statistics is to estimate
the probability of an outcome
• Can use z-scores to determine the probability of
an event occurring AND we can determine if it is
as likely, more likely, or less likely to occur than
what would be expected by chance