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Unformatted text preview: 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 Control Charting Counts In this module, we discuss the Shewhart control charts for socalled "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 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 Probability Basis for Control Limits In this kind of circumstance, the notation p = X n = the sample fraction nonconforming is standard, and Standards Given Control Limits Stat 231 facts about the binomial distribution are that X = np and X = q np ( 1 & p ) so that (since p = & 1 n X ) p = p and 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 p LCL p = p & 3 r p ( 1 & p ) n and UCL p = p + 3 r p ( 1 & p ) n Vardeman and Morris (Iowa State University) IE 361 Module 13 5 / 23 p Chart Example Example 131 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 p . Sample 1 2 3 4 5 6 7 X 15 11 18 9 13 11 10 p .0375 .0275 .0450 .0225 .0325 .0275 .0250 Sample 8 9 10 11 12 13 14 X 19 24 7 9 13 17 7 p .0475 .0600 .0175 .0225 .0325 .0425 .0175 Sample 15 16 17 18 19 20 X 10 19 11 8 8 7 p .0250 .0475 .0275 .0200 .0200 .0175 Vardeman and Morris (Iowa State University) IE 361 Module 13 6 / 23 p Chart Example Example 131 continued (Standards Given Control Limits) Standards given control limits for p here are LCL p = p & 3 r p ( 1 & p ) n = . 03 & 3 r . 03 ( 1 & . 03 ) 400 = . 0044 and UCL p = . 03 + 3 r . 03 ( 1 & . 03 ) 400 = . 0556 Vardeman and Morris (Iowa State University) IE 361 Module 13 7 / 23 p Chart Example...
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This document was uploaded on 02/11/2012.
 Fall '09

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