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So what is this solution of the problem Hamilton's line of thought is clearly described in a letter he wrote to his friend John T. Graves, which was later published in the Philosophical Magazine [264]. Here we will argue differently. To extend the connection between multiplication by i and rotation in two dimensions to three dimensions, we must generalize the two-dimensional complex plane to include a third axis, orthogonal to both the real line and the ordinary i-imaginary line, as shown in Figure C.3. To rotate a vector on the real line by 90 degrees to take it to this new axis, we multiply the vector by a new number which we will call j. So any vector on the third axis is a real number multiplied by j. Notice that this third axis must be an imaginary rather than real axis; that is,

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The center line and upper and lower control limits for a control chart for individuals are UCL CL LCL x x x mr 3 d2 x mr 3 1.128 3 mr d2 x 3 mr 1.128 (16-19)

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The procedure is illustrated in the following example. EXAMPLE 16-2 Table 16-3 shows 20 observations on concentration for the output of a chemical process. The observations are taken at one-hour intervals. If several observations are taken at the same time, the observed concentration reading will differ only because of measurement error. Since the measurement error is small, only one observation is taken each hour. To set up the control chart for individuals, note that the sample average of the 20 concentration readings is x 99.1 and that the average of the moving ranges of two observations shown in the last column of Table 16-3 is mr 2.59. To set up the moving-range chart, we note that D3 0 and D4 3.267 for n 2. Therefore, the moving-range chart has center line mr 2.59, LCL 0, and UCL D4mr 13.267212.592 8.46. The control chart is shown as the lower control chart in Fig. 16-11 on page 618. This control chart was constructed by Minitab. Because no points exceed the upper control limit, we may now set up the control chart for individual concentration measurements. If a moving range of n 2 observations is used,


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The control chart for individual concentration measurements is shown as the upper control chart in Fig. 16-11. There is no indication of an out-of-control condition. The chart for individuals can be interpreted much like an ordinary X control chart. A shift in the process average will result in either a point (or points) outside the control limits, or a pattern consisting of a run on one side of the center line. Some care should be exercised in interpreting patterns on the moving-range chart. The moving ranges are correlated, and this correlation may often induce a pattern of runs or cycles on the chart. The individual measurements are assumed to be uncorrelated, however, and any apparent pattern on the individuals control chart should be carefully investigated. The control chart for individuals is very insensitive to small shifts in the process mean. For example, if the size of the shift in the mean is one standard deviation, the average number of points to detect this shift is 43.9. This result is shown later in the chapter. While the performance of the control chart for individuals is much better for large shifts, in many situations the shift of interest is not large and more rapid shift detection is desirable. In these cases, we recommend the cumulative sum control chart (discussed in Section 16-10) or an exponentially weighted moving-average chart (Montgomery, 2001).

as multiplying by j twice, just like multiplying by i twice, must correspond to rotation by 180 degrees because the new axis is also at 90 degrees from the real line. So far, all we have done has been to apply the same old idea of the ordinary imaginary line to a third dimension. But we cannot just apply that idea in a straightforward way, simply because with three dimensions we end up with two imaginary

Table 16-3 Chemical Process Concentration Measurements Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 x Concentration x 102.0 94.8 98.3 98.4 102.0 98.5 99.0 97.7 100.0 98.1 101.3 98.7 101.1 98.4 97.0 96.7 100.3 101.4 97.2 101.0 99.1 mr Moving Range mr 7.2 3.5 0.1 3.6 3.5 0.5 1.3 2.3 1.9 3.2 2.6 2.4 2.7 1.4 0.3 3.6 1.1 4.2 3.8 2.59

A random variable is sometimes de ned as a function of one or more random variables. The CD material presents methods to determine the distributions of general functions of random variables. Furthermore, moment-generating functions are introduced on the CD

Brougharn Bridge" is properly referred to as Broome Bridge. narned after a local family (see Graves's biography of Hamilton [267]).

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