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is the average run length of the X chart when the process is in control. That is, even if the process remains in control, an out-of-control signal will be generated every 370 points, on the average. Consider the piston ring process discussed in Section 16-4.2, and suppose we are sampling every hour. Thus, we will have a false alarm about every 370 hours on the average. Suppose we are using a sample size of n 5 and that when the process goes out of control the mean shifts to 74.0135 millimeters. Then, the probability that X falls between the control limits of Fig. 16-3 is equal to P 373.9865 X 74.0135 when 73.9865 74.0135 Pc 0.0045 0.5 P 3 6 Z 04 74.01354 74.0135 74.0135 Z d 0.0045

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Table 16-6 Average Run Length (ARL) for an X Chart with 3-Sigma Control Limits Magnitude of Process Shift 0 0.5 1.0 1.5 2.0 3.0 ARL n 1 370.4 155.2 43.9 15.0 6.3 2.0 ARL n 4 370.4 43.9 6.3 2.0 1.2 1.0

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That fateful day, the Google Guys capitalized on the mysterious power of links. Although a webmaster can easily manipulate everything (word choice, keyword placement, internal links, and so on) on his or her own website, it is much more difficult to influence inbound links. This natural link profile acts as an extremely good metric for identifying legitimately popular pages.

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That is, the control chart will require two samples to detect the process shift, on the average, so two hours will elapse between the shift and its detection (again on the average). Suppose this approach is unacceptable, because production of piston rings with a mean diameter of 74.0135 millimeters results in excessive scrap costs and delays nal engine assembly. How can we reduce the time needed to detect the out-of-control condition One method is to sample more frequently. For example, if we sample every half hour, only one hour will elapse (on the average) between the shift and its detection. The second possibility is to increase the sample size. For example, if we use n 10, the control limits in Fig. 16-3 narrow to 73.9905 and 74.0095. The probability of X falling between the control limits when the process mean is 74.0135 millimeters is approximately 0.1, so p 0.9, and the out-of-control ARL is ARL 1 p 1 0.9 1.11


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Thus, the larger sample size would allow the shift to be detected about twice as quickly as the old one. If it became important to detect the shift in the rst hour after it occurred, two control chart designs would work:

Table 16-6 provides average run lengths for an X chart with 3-sigma control limits. The average run lengths are calculated for shifts in the process mean from 0 to 3.0 and for sample sizes of n 1 and n 4 by using 1 p, where p is the probability that a point plots outside of the control limits. Figure 16-18 illustrates a shift in the process mean of 2 .

where I j) is the eigenstate of z with eigenvalue + 1 (upper atomic state), 11) is the eigenstate of az with eigenvalue -1 (the lower atomic state), and In) is the n-photon state. In this basis, the matrix elements of the two terms on the right-hand side of (8.124) are given by (8.130) and where Pnp,m q ==

+ 2



NOTE Google s PageRank was actually named after its creator, Larry Page. Originally, the algorithm was named BackRub after its emphasis on backlinks. Later, its name was changed to PageRank because of its connections to Larry Page s last name and the ability for the algorithm to rank pages. Larry Page s original paper on PageRank, The Anatomy of a LargeScale Hypertextual Web Search Engine, is still available online. If you are interested in reading it, it is available on Stanford s website at http://infolab.stanford.edu/~backrub/google.html. It is highly technical, and I have used it on more than one occasion as a sleep aid. It s worth noting that the original PageRank as described in this paper is only a tiny part of Google s modern-day search algorithm.

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