Module 13: Introduction to Normal Probability Distributions

Key Concepts

• Z-Score: A standardized score that translates the distance between an x-value and the mean into standard deviation units. It represents how many standard deviations an x-value is from the mean.

• Standard Normal Distribution: A normal probability distribution of z-scores with a mean of 0 and a standard deviation of 1.

Understanding Standardization

• Transformation to Z-Scores: Every x-value in a normal probability distribution can be translated into a z-score. This moves the distribution to have a mean of 0 and a standard deviation of 1.

• Example: Comparing normal probability curves:

  • Foot Lengths: Normal probability curve with a mean of 11 inches and a standard deviation of 1.5 inches.
  • Standard Normal Curve: Mean of 0 and a standard deviation of 1.

Key Properties

• Mean and Standard Deviation:

  • Mean: The z-score is 0 for the mean of x.
  • Standard Deviation: Z-score of 1 for values one standard deviation above the mean, and -1 for values one standard deviation below.

• Area Under the Curve: Both the normal probability distribution and the standard normal distribution have an area under the curve equal to one.

Using Z-Scores

• Finding Probabilities:
  • The probability of x being greater or less than a certain value is equal to the probability of the z-score being greater or less than the corresponding z-score.
  • Example: For x = 13, z = 1.33, the probability for x > 13 is the same as for z > 1.33.

Historical Context

• Use of Z-Scores: Historically, statisticians used z-scores and standard normal curves to simplify probability calculations due to the limitations of technology and reliance on tables.

• Current Practice:

  • Modern technology, like statistical software, eliminates the need for standardization to find probabilities.
  • Tools like StatCrunch allow direct input of mean and standard deviation to calculate probability, bypassing the z-score calculation.

Educational Perspective

• Despite technological advances, statistics education still teaches the use of the standard normal curve, even though it's often redundant in practice due to available tools.