Some Misconceptions About The Normal Distribution

As part of a Six Sigma training course, practitioners are introduced to arguably the most important probability distribution in statistics: the normal distribution.

Statistical procedures are often based upon the assumption that data collected for an analysis are drawn from a normal distribution.

This article discusses three misconceptions regarding the use of the normal distribution in theory and practice:

1. Something is wrong if the distribution is non-normal
2. The larger the sample size, the closer it approximates a normal distribution
3. Capability estimates do not depend on normality

The Normal Distribution

A normal distribution can be described solely by the arithmetic mean () and standard deviation (s). These parameters may be estimated by the sample mean (x) and sample standard deviation (S) respectively.

A Normal Distribution is typically expressed in statistical shorthand as

For example, a normal distribution with a mean of 12 and standard deviation of 5 is written N(12, 25).(1)

The probability density function (PDF) for a normal distribution is:

It is alleged that Abraham de Moivre was the first to propose the normal distribution, in a supplement to a letter dated November 12, 1733.

However, the name of Carl Friedrich Gauss is more closely associated with the normal, or Gaussian, distribution. The movement away from the term Gaussian occurred during the first part of the 20th century. According to Helen M. Walker, the origin of the term normal is obscure, though the Encyclopedia of Statistical Sciences states that C. S. Peirce may have been the first to use it in the mid 1870s.

Misconception 1:

Something is wrong if the distribution is non-normal

Often, distributions other than the normal are more appropriate for a given set of data. In particular, when a naturally occurring boundary exists (e.g., zero, with cycle time data), the assumption of normality may not be sensible because the normal distribution has positive probability throughout the entire real number line (i.e., from negative to positive infinity).

Some Six Sigma practitioners are encouraged to discover why the data are nonnormal and to continue to look for explanations until normality is obtained. This may be poor advice and frustrate the investigator because, despite best efforts, the assumption of normality frequently cannot (reasonably) be obtained.

The misunderstanding may be due to an unwarranted inference from the name of the distribution itself. Six Sigma practitioners, especially those new to statistical theory, may believe that it is normal to see such a distribution in practice. Though the normal distribution may be a reasonable assumption for many processes, it is not reasonable for all processes.

Furthermore, practitioners are occasionally led to believe that an approximately normal distribution implies that a process is in statistical control. Again, the inference is not valid n through control charting a practitioner can assess the stability of a process.(4)

Misconception 2:

The larger the sample size, the closer it approximates a normal distribution.

This misconception may be due to a misunderstanding of the central limit theorem (CLT). As discussed by Robert V. Hogg and Johannes Ledolter, the CLT may be stated as follows:

If X is the mean of a random sample X, X 2, Xn from a distribution with mean and finite variance s2 > 0, then the distribution of approaches a distribution that is N(0,1) as n becomes large.(5)

In other words, when sampling from such a distribution (normal or otherwise), as the sample size increases, the distribution of X gets closer to a normal distribution. When sampling from a normal distribution, the distribution of will, necessarily, be normal.

Of course, none of this implies that when larger samples are taken from a non-normal distribution the underlying distribution itself becomes normally distributed.