Quality and Reliability Methods - SAS

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104 Cumulative Sum Control Charts Chapter 6 Formulas for CUSUM Charts Figure 6.9 Show Parameters Formulas for CUSUM Charts Notation The following notation is used in these formulas: μ denotes the mean of the population, also referred to as the process mean or the process level. μ 0 denotes the target mean (goal) for the population. Sometimes, the symbol X 0 is used for μ 0 . See American Society for Quality Statistics Division (2004). You can provide μ 0 as the Target on the known statistics dialog. σ denotes the population standard deviation. σ 0 denotes a known standard deviation. You can provide σ o as the sigma on the known statistics dialog or through JSL. σˆ denotes an estimate of σ. n denotes the nominal sample size for the cusum scheme. δ denotes the shift in μ to be detected, expressed as a multiple of the standard deviation. You can provide δ as the delta on the dialog or through JSL. Δ denotes the shift in μ to be detected, expressed in data units. If the sample size n is constant across subgroups, then Δ = δσ = ( δσ) ⁄ n X Note that some authors use the symbol D instead of Δ. You can provide Δ as the Shift on the dialog or through JSL. One-Sided CUSUM Charts Positive Shifts If the shift δ to be detected is positive, the CUSUM computed for the t th subgroup is S t = max(0, S t – 1 + (z t – k))
Chapter 6 Cumulative Sum Control Charts 105 Formulas for CUSUM Charts for t = 1, 2,..., n, where S 0 = 0, z t is defined as for two-sided schemes, and the parameter k, termed the reference value, is positive. The cusum S t is referred to as an upper cumulative sum. Since S t can be written as the sequence S t cumulates deviations in the subgroup means greater than k standard errors from μ 0 . If S t exceeds a positive value h (referred to as the decision interval), a shift or out-of-control condition is signaled. Negative Shifts X i – ( μ 0 + kσ ) Xi max0, S t – 1 + --------------------------------------- σ Xi If the shift to be detected is negative, the cusum computed for the t th subgroup is S t = max(0, S t – 1 – (z t + k)) for t = 1, 2,..., n, where S 0 = 0, z t is defined as for two-sided schemes, and the parameter k, termed the reference value, is positive. The cusum S t is referred to as a lower cumulative sum. Since S t can be written as X i – ( μ 0 – kσ ) Xi max0, S t – 1 – -------------------------------------- σ Xi the sequence S t cumulates the absolute value of deviations in the subgroup means less than k standard errors from μ 0 . If S t exceeds a positive value h (referred to as the decision interval), a shift or out-of-control condition is signaled. Note that S t is always positive and h is always positive, regardless of whether δ is positive or negative. For schemes designed to detect a negative shift, some authors define a reflected version of S t for which a shift is signaled when S t is less than a negative limit. Lucas and Crosier (1982) describe the properties of a fast initial response (FIR) feature for CUSUM schemes in which the initial CUSUM S 0 is set to a “head start” value. Average run length calculations given by them show that the FIR feature has little effect when the process is in control and that it leads to a faster response to an initial out-of-control condition than a standard CUSUM scheme. You can provide head start values on the dialog or through JSL. Constant Sample Sizes When the subgroup sample sizes are constant (= n), it might be preferable to compute cusums that are scaled in the same units as the data. Cusums are then computed as S t = max( 0, S t – 1 + ( X i – ( μ 0 + kσ ⁄ n) )) for δ > 0 and the equation S t = max( 0, S t – 1 – ( X i – ( μ 0 – kσ ⁄ n) )) for δ < 0. In either case, a shift is signaled if S t exceeds h' = hσ ⁄ n . Some authors use the symbol H for h'.
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Chapter 11 Pareto Plots 193 Defect
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Chapter 19 Reliability and Survival
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References Abernethy, Robert B. (19
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References 407 Carroll, R.J., Ruppe
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References 409 Haaland, P.D. (1989)
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References 411 Lawless, J.F. (1982)
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References 413 Nelson, W.B. (2003),
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References 415 Snee, R.D. and Marqu
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Index JMP Quality and Reliability M
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Index 419 Fit SEV 242 Fit Weibull 2
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Index 421 Q Quantiles table 356 R R
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Index 423 Weibull Plot 360 Westgard
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Quality and Reliability Methods - SAS

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