Page 17 - Applied Statistics And Probability For Engineers

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1-1 THE ENGINEERING METHOD AND STATISTICAL THINKING 3

Statistical methods are used to help us describe and understand variability. By variability,
we mean that successive observations of a system or phenomenon do not produce exactly the
same result. We all encounter variability in our everyday lives, and statistical thinking can
give us a useful way to incorporate this variability into our decision-making processes. For
example, consider the gasoline mileage performance of your car. Do you always get exactly the
same mileage performance on every tank of fuel? Of course not—in fact, sometimes the mileage
performance varies considerably. This observed variability in gasoline mileage depends on
many factors, such as the type of driving that has occurred most recently (city versus highway),
the changes in condition of the vehicle over time (which could include factors such as tire
inflation, engine compression, or valve wear), the brand and/or octane number of the gasoline
used, or possibly even the weather conditions that have been recently experienced. These factors
represent potential sources of variability in the system. Statistics gives us a framework for
describing this variability and for learning about which potential sources of variability are the
most important or which have the greatest impact on the gasoline mileage performance.
We also encounter variability in dealing with engineering problems. For example, sup-
pose that an engineer is designing a nylon connector to be used in an automotive engine
application. The engineer is considering establishing the design specification on wall thick-
ness at 3 32 inch but is somewhat uncertain about the effect of this decision on the connector
pull-off force. If the pull-off force is too low, the connector may fail when it is installed in an
engine. Eight prototype units are produced and their pull-off forces measured, resulting in the
following data (in pounds): 12.6, 12.9, 13.4, 12.3, 13.6, 13.5, 12.6, 13.1. As we anticipated,
not all of the prototypes have the same pull-off force. We say that there is variability in the
pull-off force measurements. Because the pull-off force measurements exhibit variability, we
consider the pull-off force to be a random variable. A convenient way to think of a random
variable, say X, that represents a measurement, is by using the model

X (1-1)

where is a constant and is a random disturbance. The constant remains the same with every
measurement, but small changes in the environment, test equipment, differences in the indi-
vidual parts themselves, and so forth change the value of . If there were no disturbances,
would always equal zero and X would always be equal to the constant . However, this never
happens in the real world, so the actual measurements X exhibit variability. We often need to
describe, quantify and ultimately reduce variability.
Figure 1-2 presents a dot diagram of these data. The dot diagram is a very useful plot for
displaying a small body of data—say, up to about 20 observations. This plot allows us to see eas-
ily two features of the data; the location, or the middle, and the scatter or variability. When the
number of observations is small, it is usually difficult to identify any specific patterns in the vari-
ability, although the dot diagram is a convenient way to see any unusual data features.
The need for statistical thinking arises often in the solution of engineering problems.
Consider the engineer designing the connector. From testing the prototypes, he knows that the
average pull-off force is 13.0 pounds. However, he thinks that this may be too low for the

3
= inch
32
1 inch
12 13 14 15 12 13 14 15 = 8
Pull-off force Pull-off force
Figure 1-2 Dot diagram of the pull-off force Figure 1-3 Dot diagram of pull-off force for two wall
data when wall thickness is 3/32 inch. thicknesses.