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The following
is an excerpt from Chapter 4 of The
Quality Engineering Handbook by Thomas
Pyzdek, © Quality Publishing. It may be ordered from the Quality
Publishing Order Form.
SPC Basics
- Terms and Concepts
Distributions
A central concept in statistical
process control is that every measurable phenomenon is a statistical distribution.
In other words, an observed set of data constitutes a sample of the effects
of unknown common causes. It follows that, after we have done everything
to eliminate special causes of variations, there will still remain a certain
amount of variability exhibiting the state of control. Figure IV.2 illustrates
the relationships between common causes, special causes, and distributions.
Figure
IV.2.Distributions.
From Continuing
Process Control and Process Capability Improvement, p. 4a. Copyright
1983 by Ford Motor Company. Used by permission of the publisher.
There are three basic
properties of a distribution: location, spread, and shape. The location
refers to the typical value of the distribution, such as the mean. The
spread of the distribution is the amount by which smaller values differ
from larger ones. The standard deviation and variance are measures of
distribution spread. The shape of a distribution is its pattern—peakedness,
symmetry, etc. A given phenomenon may have any one of a number of distribution
shapes, e.g., the distribution may be bell-shaped, rectangular-shaped,
etc.
Central
limit theorem
The central limit theorem
can be stated as follows:
Irrespective of the
shape of the distribution of the population or universe, the distribution
of average values of samples drawn from that universe will tend toward
a normal distribution as the sample size grows without bound.
It can also be shown that
the average of sample averages will equal the average of the universe
and that the standard deviation of the averages equals the standard deviation
of the universe divided by the square root of the sample size. Shewhart
performed experiments that showed that small sample sizes were needed
to get approximately normal distributions from even wildly non-normal
universes. Figure IV.3 was created by Shewhart using samples of four measurements.
Figure
IV.3.Illustration of the central limit theorem.
From Economic
Control of Quality of Manufactured Product, figure 59. Copyright
© 1980.
Used by
permission of the publisher, ASQC Quality Press. Milwaukee, Wisconsin.
The practical implications
of the central limit theorem are immense. Consider that without the central
limit theorem effects, we would have to develop a separate statistical
model for every non-normal distribution encountered in practice. This
would be the only way to determine if the system were exhibiting chance
variation. Because of the central limit theorem we can use averages of
small samples to evaluate any process using the normal distribution. The
central limit theorem is the basis for the most powerful of statistical
process control tools, Shewhart control charts.
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