The Normal Distribution

When people first began constructing histograms, a particular shape occurred so often that people began to expect it. Hence, it was given the name normal distribution. The normal distribution is symmetric (you can fold it in half and the two halves will match) and unimodal (single peaked). It is what psychologists call the bell-shaped curve. Every statistics course spends lots of time discussing it. Text books have long chapters about it and many exercises involving it. The normal distribution is important. Statistics couldn't function without it!

Some statistical techniques demand that individual data items follow a normal distribution, that is, that the population histogram from which the sample is drawn have a normal shape. When the data are not normal, the results from these techniques will be unreliable. We've already seen cases where reporting a mean and SD can give a misleading mental picture and it would be better to replace or supplement them with percentiles. Not every distribution is normal. Some are far from it. We'll discuss the importance of normality on a technique-by-technique basis.

When normality of the individual observations is essential, transformations such as logarithms can sometimes be used to produce a set of transformed data that is better described by a normal distribution that the original data. Transformations aren't applied to achieve a specific outcome but rather to put the data in a form where the outcome can be relied upon. If you ran up the stairs to be on time for a doctor's appointment, you wouldn't mind waiting to have your blood pressure measured if a high reading might result in treatment for hypertension. Values obtained after running up a flight of stairs are unsuitable for detecting hypertension. The reason for waiting isn't to avoid a diagnosis of high blood pressure and necessary treatment, but to insure that a high reading can be believed. It's the same with transformations.

Comment: Professor Herman Rubin of Purdue University provides an additional insight into the statistician's fascination with the normal distribution (posted to the Usenet group, 3 Oct 1998 08:07:23 -0500; Message-ID: 6v57ib$

Normality is rarely a tenable hypothesis. Its usefulness as a means of deriving procedures is that it is often the case, as in regression, that the resulting procedure is robust in the sense of having desirable properties without it, while nothing better can be done uniformly.

Comment: The probability that a normally distributed quantity will be within a specified multiple of standard deviations of its mean is the same for all normal distributions. For example, the probability of being within 1.96 SDs of the mean is 95%. [More often than not, people make casual use of 2 in place of the exact 1.96.] This is true whatever the mean and SD.

Comment: In the old days (B.C.: before computers) it was important to learn how to read tables of the normal distribution. Almost every analysis required the analyst to translate normal values into probabilities and probabilities into normal values. Now that computers do all the work, it is possible to be a good data analyst without ever having looked at a such a table.

Copyright © 2000 Gerard E. Dallal