1. Probability and the Normal Curve
Statistics for Political Science
Levin and Fox Chapter 5

2. Probability
Probability refers to the relative likelihood of occurrence of a particular outcome or event.
The probability associated with an event is the number of times that event can occur relative to the total number of times any event can occur.
We use a capital P to indicate probability.
Probability varies from 0 to 1.0 although percentages rather than decimals may be used to express levels of probability.

3. Probability The Probability Spectrum:
A zero probability indicates that something is impossible.
Probabilities near zero (like .05 or .10) imply very unlikely occurrences. A probability of 1.0 constitutes certainty.
High probabilities like .90, .95, or .99 signify very probable or likely outcomes.

4. Probability = Number of times the outcome or event can occur
Probability of an outcome or event =
Total number of times any outcome or event can occur

5. Probability
A Probability Distribution: Similar to a frequency distribution, except it is based on theory (probability theory), rather than what is observed in the real world (empirical data).
Event
Probability
Return to Prison
.875
Stay out of Prison
.125
Total
1.00

6. The Difference Between a Probability Distribution and a Frequency Distribution
A frequency distribution records how often an event occurs. It is based on actual observations.
A probability distribution records the likelihood that an event is to occur. It is based on theoretical assumption of what should happen.

7. New Symbols for Probability
(mu)
The mean of a probability distribution σ
(sigma)
The standard deviation of a probability distribution σ²
(sigma square)
The variance of a probability distribution
NOTE: Xbar, s², s represent the mean, variance and standard deviation of observed data.

8. Probability
Probability distributions can take on different shapes, just like frequency distributions.
Let’s take a look at the normal curve, which is a theoretical or ideal model obtained from a mathematical equation rather than actual research situations.

9. The Normal Curve
The normal curve can be used (1) for describing distributions of scores, (2) interpreting standard deviations, and (3) making statements of probability.

10. Features of the Normal Curve
It is symmetrical. If we were to fold it, we would have equal halves.
It is unimodal. It has only one peak or point of maximum frequency.
The point at the middle of the curve is where the mean, mode, and median coincide.
The tails fall off gradually and extend indefinitely in both directions, getting closer to the baseline but never reaching it.

11. Probability Normal Curve: Accuracy?
Is the normal curve an accurate representation of how variables are distributed in the real world?
A Normal World?
If social and phenomena were normally distributed, than the following would be true:
Human weight: range from 5 to 6 feet, with few below 5 or above 6.
IQ: range from 85 to 115, with few below 85 or above 115.

12. Probability
But, some variables do not conform to the theoretical notion of the normal distribution.
Often, our distributions are not normal but are skewed negatively or positively, and may be bi- or multi-modal.

13. The Area Under the Normal Curve
The standard deviation is the distance from the mean and the point on the baseline just below where the S-shaped portion of the curve shifts direction.

14. Probability
To employ the normal distribution to solve problems, we must acquaint ourselves with the area under the normal curve: the area that lies between the curve and the base line containing 100% of all the cases in any given normal distribution.

15. 100%