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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 individual 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 easily 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 dif cult to identify any speci c patterns in the variability, 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

intended application, so he decides to consider an alternative design with a greater wall thickness, 1 8 inch Eight prototypes of this design are built, and the observed pull-off force measurements are 129, 137, 128, 139, 142, 132, 135, and 131 The average is 134 Results for both samples are plotted as dot diagrams in Fig 1-3, page 3 This display gives the impression that increasing the wall thickness has led to an increase in pull-off force However, there are some obvious questions to ask.

First, Trudy lists the polynomials f o ( x ) , which are of length N / 2 . We identify these polynomials with the length-N vectors formed by appending N / 2 zeros. This requires

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For instance, how do we know that another sample of prototypes will not give different results Is a sample of eight prototypes adequate to give reliable results If we use the test results obtained so far to conclude that increasing the wall thickness increases the strength, what risks are associated with this decision For example, is it possible that the apparent increase in pull-off force observed in the thicker prototypes is only due to the inherent variability in the system and that increasing the thickness of the part (and its cost) really has no effect on the pull-off force Often, physical laws (such as Ohm s law and the ideal gas law) are applied to help design products and processes We are familiar with this reasoning from general laws to speci c cases.

But it is also important to reason from a speci c set of measurements to more general cases to answer the previous questions This reasoning is from a sample (such as the eight connectors) to a population (such as the connectors that will be sold to customers) The reasoning is referred to as statistical inference See Fig 1-4 Historically, measurements were obtained from a sample of people and generalized to a population, and the terminology has remained Clearly, reasoning based on measurements from some objects to measurements on all objects can result in errors (called sampling errors) However, if the sample is selected properly, these risks can be quanti ed and an appropriate sample size can be determined In some cases, the sample is actually selected from a well-de ned population The sample is a subset of the population.

steps and is done in the following manner. The f o ( x ) polynomials are stored in bins based on the first k coordinates of f o ( x )* h(x) (mod 4 ) . To form the tins, divide the interval [0,y - 1 into subintervals of length 2e and call this 1 set of subintervals I, that is, the set I is composed of the subintervals

For example, in a study of resistivity a sample of three wafers might be selected from a production lot of wafers in semiconductor manufacturing Based on the resistivity data collected on the three wafers in the sample, we want to draw a conclusion about the resistivity of all of the wafers in the lot In other cases, the population is conceptual (such as with the connectors), but it might be thought of as future replicates of the objects in the sample In this situation, the eight prototype connectors must be representative, in some sense, of the ones that will be manufactured in the future Clearly, this analysis requires some notion of stability as an additional assumption.

For example, it might be assumed that the sources of variability in the manufacture of the prototypes (such as temperature, pressure, and curing time) are the same as those for the connectors that will be manufactured in the future and ultimately sold to customers..

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