Unit 5: The Normal Distribution
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"Everybody believes in the normal approximation, the experimenters because they think it is a mathematical theorem, the mathematicians because they think it is an experimental fact!"
G. Lippman, quoted in D'Arcy Thompson's On Growth and Form.
• The normal distribution is a mathematical model that idealizes distributions of variables that are symmetric and unimodal. However, keep in mind that there are other distributions that can model symmetric and unimodal distributions.
• The normal distribution is merely one of many distributions, all of which are used as idealized summaries of distributions in populations. The area under a distribution between any two values represents the proportion of a population that lies between those two values.
• The standard unit is a useful and fundamental way of comparing observations from two different populations. In essence, we use the standard deviation as a basic unit of measurement that replaces the natural units a variable was first measured in. The z-score is the number of standard units (standard deviations) an observation is from the mean.
• Empirical Rule: About 95% of the data are within two standard deviations of the mean for a symmetric unimodal distribution. About 68% are within one standard deviation. About 99.7% (almost all) are within three standard deviations. This turns out to be exactly true for the normal distribution.
• Fewer populations than you may think or your book may suggest actually have distributions that are well approximated by the normal distribution. However, it is of fundamental importance for statistical analyses because of a result called the Central Limit Theorem, which we discuss in a later unit.
"Normality is a myth; there never has, and never will be, a normal distribution", Geary, R.C. in Biometrika, "Testing for Normality", Volume 34, 1947 (p. 241)