SOME ADDITIONAL TESTS FOR CONTROL CHARTS

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First, consider the probability of a false reading, that is, an observation falling beyond the three sigma limits entirely due to chance rather than to a real change in the process

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Assuming that the observations are normally distributed, an assumption justi ed for the sample mean if the sample size is moderate, then the probability of an observation outside the three sigma limits is 00027 Such observations are then very rare and when one occurs, we are unlikely to conclude that the observations occurred by chance alone However, if 50 observations are made, the probability that at least one of them is outside the three sigma limits is 1 (1 00027)50 = 0 12644 This probability increases rapidly, as the data in Table 115 indicate Here n represents the number of observations

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Table 115 n 50 100 200 300 400 500 600 700 800 900 1000 Probability 0126444 0236899 0417677 0555629 0660900 0741233 0802534 0849314 0885011 0912252 0933039

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So an extreme observation becomes almost certain as production continues, although the process in reality has not changed Minitab offers eight additional tests on the data; we will describe four of them here, namely, tests 1, 2, 5, and 6When these tests become signi cant, Minitab indicates this by putting the appropriate number of the test on the control chart This explains those numbers on the chart in Figure 118 In general, since the probabilities of these events are quite small, the following situations are to be regarded as cautionary ags for the production process The calculation of the probabilities involved in most of these tests relies upon the binomial or normal probability distributions The default values of the constant k in each of these tests can be changed easily in Minitab

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Test 1 Points more than k sigma units from the centerline We have used k = 3, but Table 116 shows probabilities with which a single point is more than k standard deviations from the target mean In Figure 118, while only one point is beyond the three sigma limit, several are beyond the two sigma limits These points are indicated by the symbol on the control chart

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Some Additional Tests for Control Charts Table 116 k 10 15 20 25 30 Probability 0317311 0133614 0045500 0012419 0002700

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Test 2 k points in a row on the same side of the centerline It is common to use k = 9 The probability that nine points in a row are on the same side of the centerline is 2 (1/2)9 = 000390625 Table 117 shows this probability for k = 7, 8, , 11

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Table 117 k 7 8 9 10 11 Probability 001563 000781 000391 000195 000098

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This test fails at samples 13, 14, 16, 17, 18, 28, 60, 61,and 91 Test 5 At least k out of k + 1 points in a row more than two sigmas from the centerline The quantity k is commonly chosen as 2 Since the probability that one point is more than two sigmas above the centerline is 00227501 and since the number of observations outside these limits is a binomial random variable, the probability that at least two out of three observations are more than two sigmas from the centerline and either above or below the centerline is

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3 (00227501)x (09771499)3 x = 0003035 x

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Table 118 gives values of this probability for other values of k This event occurred at observation 44 in Figure 118

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Table 118 k 2 3 4 Probability 0003035 0000092 0000003

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Test 6 At least k out of k + 1 points in a row more than one sigma from the centerline The value of k is commonly chosen as 4 Since the probability that one point is more than one sigma above or below the centerline is 0158655 and since the number of observations outside these limits is a binomial random variable, the probability that at least four out of ve observations are more than two sigmas from the centerline is

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