Greenwood press

STANDARD DEVIATION
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chart or histogram will appear to be bell-shaped. In this type of distribution, there
are some handy “rules of thumb” that use standard deviation to describe the
spread of data. In a normally distributed set of data, about 68 percent of it is con-
tained within one standard deviation of the mean (as shown in the figure below),
95 percent within two standard deviations, and 99.7 percent within three standard
deviations.
The normal curve has two points of inflection where the curve changes from con-
cave-downward, to concave-upward. These are located at ±1 standard deviation
units. The point of inflection at +1 standard deviation is shown in the figure.
The rules of thumb for a normal distribution stop at ±3 standard deviations
from the mean, because almost all of the data is trapped by those limits. That is
not enough for the management goal of “six sigma” quality adopted by many
American businesses. In such cases, the goal is to have fewer than 3.4 defects per
million products. The six sigma, or 
6σ, is chosen because 99.99966 percent of
the cases in a normal distribution fall within six standard deviations of the mean.
If that proportion represents defect-free products, then the remainder, 0.00034
percent, represent defects. Such high-quality control standards at six sigma will
likely reduce the number of defects in a product, but at a high cost when an in-
spection fails. Reducing the standards to 99.7 percent defect-free products will
likely save the company money in the long run, unless the company is dealing
with personal health and safety issues. Physicists use a five-sigma criterion in
determining whether a subatomic particle has been revealed. They think that only
a five-sigma result, indicating a 99.99995 percent chance that the result can be
reproduced, is trustworthy and can survive the test of time.
The rules of thumb are often used by manufacturers to design clothing and
furniture that will sell to the broadest audience. For example, an automobile
manufacturer developing an automobile for potential female customers might
design the driver’s seat to fit the heights of most women. To make the greatest
profit, the seat must be as standard as possible. The heights of American women
are normally distributed with a mean of 64 inches, with a standard deviation of
2.5 inches. If the manufacturer has its designers work on a seat that will be com-
fortable for women from 59 to 69 inches tall (two standard deviations above and
below the mean), then the rule of thumb says that the seat would be appropriate
for 95 percent of the women. 
In medical quality-control testing it is difficult to evaluate the effectiveness
of a medical instrument, because many medical measurements such as blood