Module 4C - IE 361 Module 4 Metrology Applications of Some...

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IE 361 Module 4 Metrology Applications of Some Intermediate Statistical Methods for Separating Components of Variation Reading: Section 2.2 Statistical Quality Assurance for Engineers (Section 2.3 of Revised SQAME ) Prof. Steve Vardeman and Prof. Max Morris Iowa State University Vardeman and Morris (Iowa State University) IE 361 Module 4 1 / 22
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A Relatively Simple First Method for Separating Process and Measurement Variation In Module 2 we observed that 1 repeated measurement of a single measurand with a single device allows one to estimate σ device , and 2 single measurements made on multiple measurands from a stable process using a linear device allow one to estimate σ y = q σ 2 x + σ 2 device and remarked that these facts might allow one to somehow °nd a way to estimate σ x (a process standard deviation) alone. Our °rst goal in this module is to provide one simple method of doing this. The next °gure illustrates a data collection plan that combines the elements 1. and 2. above. Vardeman and Morris (Iowa State University) IE 361 Module 4 2 / 22
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One Method of Separating Process and Measurement Variation Figure: One Possible Data Collection Plan for Estimating a Process Standard Deviation, σ x Vardeman and Morris (Iowa State University) IE 361 Module 4 3 / 22
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One Method of Separating Process and Measurement Variation Here we will use the notation y for the single measurements on n items from the process and the notation y 0 for the m repeat measurements on a single measurand. The sample standard deviation of the y ±s, s y , is a natural empirical approximation for σ y = q σ 2 x + σ 2 device and the sample standard deviation of the y 0 ±s, s , is a natural empirical approximation for σ device . That suggests that one estimate the process standard deviation with c σ x = q max ° 0 , s 2 y ° s 2 ± (1) as indicated in display (2.3), page 20 of SQAME . (The maximum of 0 and s 2 y ° s 2 under the root is there simply to ensure that one is not trying to take the square root of a negative number in the rare case that s exceeds s y .) Vardeman and Morris (Iowa State University) IE 361 Module 4 4 / 22
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One Method of Separating Process and Measurement Variation c σ x is not only a sensible single number estimate of σ x , but can also be used to make approximate con°dence limits for the process standard deviation. The so-called Satterthwaite approximation suggests that one use c σ x s ˆ ν χ 2 upper and c σ x s ˆ ν χ 2 lower as limits for σ x where appropriate "approximate degrees of freedom" are ˆ ν = c σ x 4 s 4 y n ° 1 + s 4 m ° 1 Vardeman and Morris (Iowa State University) IE 361 Module 4 5 / 22
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One Method of Separating Process and Measurement Variation Example 4-1 In Module 2, we considered m = 5 measurements made by a single analyst
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