1. Chapter 7: Normal Curves & Probability
CRIM 483
Chapter 7: Normal Curves & Probability

2. 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

3. Normal Curve Characteristics
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

4. 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

5. 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

6. 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%
90-100
B/T Mean & 2 SD
13.59%
80-90
B/T Mean & 3 SD
2.15%
70-80
3 SD +
.13%
130+
Below 70
50% * 2

8. 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%

9. 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 z-score, which is a standard score that can be compared across groups
z=(x-mean)/s
z=z-score
x=score
mean=mean of the distribution
s=standard deviation of the distribution

10. 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

11. 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

12. Comparing Distributions

13. 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?
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%)

14. 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

15. Comparing the Area B/T Scores
( )/10=10/10=1.00
( )/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= =.8413

17. 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