Probability DistributionA probability distribution describes the values and probabilities that a random event can take place. The values must cover all of the possible outcomes of the event, while the total probabilities must sum to exactly 1, or 100%. For example, a single coin flip can take values Heads or Tails with a probability of exactly 1/2 for each; these two values and two probabilities make up the probability distribution of the single coin flipping event.
Normal DistributionsNormal distributions, also called Gaussian distributions, are a family of probability distributions that have the same general shape. All normal distributions are symmetric and have bell-shaped density curves with a single peak.
The normal distributions are a very important class of statistical distributions. Many psychological measurements and physical phenomena (like noise) can be approximated well by the normal distribution.
Characteristics of Normal Distributions
The Standard Normal Distribution or Z Distribution
Z = X - m/s
Where X is a score from the original normal distribution, m is the mean of the original normal distribution, and s is the standard deviation of original normal distribution.
The standard normal distribution is sometimes called the Z distribution. A 'Z score' always reflects the number of standard deviations above or below the mean a particular score is. For instance, if a person scored a 70 on a test with a mean of 50 and a standard deviation of 10, then he scored 2 standard deviations above the mean. Converting the test scores to Z scores, an X of 70 would be:
Z = 70-50/10 = 2
So, a Z score of 2 means the original score was 2 standard deviations above the mean. Note that the Z distribution will only be a normal distribution if the original distribution (X) is normal.
Directions: Answer the following questions: