Vol. 6, No. 1
from Professor Cleary
Amazingly, Simsack drops the right name this time. The behavior of averages in this case is indeed related to the central limit theorem, which in turn helps explain how control charts work.
The easiest way that I have found to demonstrate the central limit theorem is with the use of Quality Gamebox. Using Quality Gamebox, and taking a sample size of 5, the following results ensue:
Parent Population Distribution of sample means (Sampling distribution) n=5
Examining the distribution of sample means derived from the single mode parent population, the shape is predictable. The base is tighter and the shape is normal looking.
The most interesting result lies in the appearance of the distribution of sample means from the bi-modal parent population. Although the base is wider than the one from the single mode population, the shape is again normal-looking. Now it is possible to describe the central limit theorem.
A typical textbook explanation indicates that when one averages many samples of the size n and places them into a distribution of sample means, that distribution of sample means will have a mean close to the mean of the population. It will have the shape of a normal distribution and the variability of the standard deviation of the population divided by the square root of the sample size. My students can memorize this, but often do not really comprehend what it means until it is broken down into three pieces, indicated below.
Taking many samples of the size n, the averages of these samples form a distribution in which:
Note: It is equal to the standard deviation of the population divided by the square root of the size of the sample used to create the distribution of sample means:
Understanding the central limit theorem means knowing the basis upon which control charts work.
The diagram below shows the upper control limit and the lower control limit of a control chart, at the end of the normal distribution, or plus and minus three sigma on a normal distribution. Plus and minus three sigma captures 99.73 percent of the area under normal distribution. So if the process is in control, or stable, the probability of a point being out of control is only .27 percent. This is the basis for Rule #1, any point outside the control limits is a signal that the process is out of control. There is evidence that the process has changed.
Next month, we will take this to the next step, to see how the central limit theorem is used as a basis for some basic out-of-control rules.
Deviation is, of course, using Quality GameBox from PQ Systems to make his point clear. To see how Quality GameBox works
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