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Module 13C - IE 361 Module 13 Control Charts for...

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IE 361 Module 13 Control Charts for Counts ("Attributes Data") Reading: Section 3.3 of Statistical Quality Assurance Methods for Engineers Prof. Steve Vardeman and Prof. Max Morris Iowa State University Vardeman and Morris (Iowa State University) IE 361 Module 13 1 / 23
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Control Charting Counts In this module, we discuss the Shewhart control charts for so-called "fraction nonconforming" and "mean nonconformities per unit" contexts. These are the Shewhart p and np charts and the Shewhart u (and c ) charts. These tools are easy enough to explain and use, but are typically really NOT very e/ective in modern applications where acceptable nonconformity rates are often so small as to be stated in "parts per million." Vardeman and Morris (Iowa State University) IE 361 Module 13 2 / 23
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Control Charts for Fraction Nonconforming The scenario under which a p chart or ( np chart) is potentially appropriate is one where periodically groups of n items (or outcomes) from a process are looked at and X = the number of outcomes among the n that are "nonconforming" is observed. This is illustrated in the °gure below, where dark balls represent nonconforming outcomes. Figure: Cartoon Representing n Process Outcomes, X of Which are Nonconforming Vardeman and Morris (Iowa State University) IE 361 Module 13 3 / 23
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Probability Basis for Control Limits In this kind of circumstance, the notation ˆ p = X n = the sample fraction nonconforming is standard, and control charts for ˆ p are called p charts control charts for X ( = n ˆ p ) are called np charts If the process producing items/outcomes is physically stable, a reasonable probability model for X (met in Stat 231) is the binomial ( n , p ) distribution, where p = the current probability that any particular outcome is nonconforming (the mental °ction here is that the particular n outcomes observed are a random sample of a huge pool of outcomes, a fraction p of which are nonconforming). Vardeman and Morris (Iowa State University) IE 361 Module 13 4 / 23
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Standards Given Control Limits Stat 231 facts about the binomial distribution are that μ X = np and σ X = q np ( 1 ° p ) so that (since ˆ p = ° 1 n ± X ) μ ˆ p = p and σ ˆ p = r p ( 1 ° p ) n These facts in turn lead to standards given ( np chart) control limits for X LCL X = np ° 3 q np ( 1 ° p ) and UCL X = np + 3 q np ( 1 ° p ) and standards given ( p chart) control limits for ˆ p LCL ˆ p = p ° 3 r p ( 1 ° p ) n and UCL ˆ p = p + 3 r p ( 1 ° p ) n Vardeman and Morris (Iowa State University) IE 361 Module 13 5 / 23
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p Chart Example Example 13-1 Below are some arti°cially generated (using p = . 03) n = 400 binomial data (except for sample 9, where a larger value of p was used) and the corresponding values of ˆ p . Sample 1 2 3 4 5 6 7 X 15 11 18 9 13 11 10 ˆ p .0375 .0275 .0450 .0225 .0325 .0275 .0250 Sample 8 9 10 11 12 13 14 X 19 24 7 9 13 17 7 ˆ p .0475 .0600 .0175 .0225 .0325 .0425 .0175 Sample 15 16 17 18 19 20 X 10 19 11 8 8 7 ˆ p .0250 .0475 .0275 .0200 .0200 .0175 Vardeman and Morris (Iowa State University) IE 361 Module 13 6 / 23
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p Chart Example Example 13-1 continued (Standards Given Control Limits)
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