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Statistical Quality & Process Control: X-Bar & R Charts Overview

Learn about X-Bar and R Charts, their use in checking variability in location between samples, and operating characteristics. Understand control limits for X-Bar and Sigma-Charts, Probability of Detecting Shifts, and Average Run Length. Explore the Performance of Shewhart Control Charts and ARL Curve for x Charts.

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Statistical Quality & Process Control: X-Bar & R Charts Overview

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  1. IENG 486 - Lecture 14 X-bar & s Charts: Trial Limits & Standard Limits, Control Chart Operating Characteristics IENG 486: Statistical Quality & Process Control

  2. Assignment • Reading: CH 8 • 8.1 – 8.3.2 • 8.3.4 • 8.7.1 • Homework: • CH 6 Textbook Problems: • 1; 6 a, b, d only – use Spreadsheet Template (Mat’ls pg) • 11; 20 a only; 24; 30 IENG 486: Statistical Quality & Process Control

  3. The X-Bar Chart checks variability in locationbetween samples The R-Chart checks for changes in sample variation UCL UCL LCL LCL Sample Number Sample Number X-Bar (Means) Control Chart R - (Range) Control Chart x R X-Bar & R-Charts IENG 486: Statistical Quality & Process Control

  4. X-Bar Control Limits: Approximate 3 limits are found from S & table Sigma-Chart Control Limits: Approximate, asymmetric 3 limits from S & table X-Bar & Sigma-Charts • Used when sample size is greater than 10 IENG 486: Statistical Quality & Process Control

  5. X-Bar Control Limits: Approximate 3 limits are found from known 0 & table Sigma-Chart Control Limits: Approximate, asymmetric 3 limits from 0 & table X-Bar & Sigma-Charts • Limits can also be generated from historical data: IENG 486: Statistical Quality & Process Control

  6. Operating Characteristic (OC) Curve • Ability of the x and R charts to detect shifts (sensitivity) is described by OC curves • For x chart; say we know s • Mean shifts fromm0 (in-control value) to m1 = m0 +ks (out-of-control value) • The probability of NOT detecting the shift on the first sample after shift is IENG 486: Statistical Quality & Process Control

  7. Ex. Probability of NOT Detecting Shift • A 3-sigma x chart is used to monitor a normally distributed quality characteristic. The process std dev is 1.2 and the sample size is 5. The process mean is in-control at 22. • Find the probability that a shift to 24.4 is not detected on the first sample after the shift. IENG 486: Statistical Quality & Process Control

  8. OC Curve for x Chart • Plot of b vs. shift size (in std dev units) for various sample sizes n • x chart not effective for small shift sizes, i.e., k 1.5s • Performance gets better for larger n and L or larger shifts IENG 486: Statistical Quality & Process Control

  9. OC curve for R Chart • Uses distribution of relative range r.v., i.e., • Suppose • s0 - in-control std dev • s1 - out-of-control std dev • OC curve for R chart plots b vs. ratio of in-control to out-of-control standard deviation for various sample sizes • That is, plot β vs. l = s1/s0 • R chart not very effective for detecting shifts for small sample sizes (see Fig. 5-14 in text) IENG 486: Statistical Quality & Process Control

  10. Probability of Detecting Shift for Subsequent Samples • After the shift has occurred: • P(NOT detecting shift ON 1st sample) • P(DETECTING shift ON 1st sample) • P(DETECTING shift ON 2nd sample) • P(DETECTING shift ON rth sample) • P(DETECTING shift BY 2nd sample) • P(DETECTING shift BY rth sample) IENG 486: Statistical Quality & Process Control

  11. Average Run Length (ARL) • Expected number of samples taken before shift is detected is called the Average Run Length (ARL) IENG 486: Statistical Quality & Process Control

  12. Performance of Any Shewhart Control Chart • In-Control ARL: • Average number of points plotted on control chart before a false alarm occurs(ideally, should be large) • Out-of-Control ARL: • Average number of points, after the process goes out-of-control, before the control chart detects it(ideally, should be small) IENG 486: Statistical Quality & Process Control

  13. ARL Curve for x Chart • Plot of ARL1 vs. shift size (in sd units) for various sample sizes n: • Average Time to Signal, (ATS): • Number of time periods that occur until signal is generated on control chart • h - time interval between samples IENG 486: Statistical Quality & Process Control

  14. Next Assignment • Reading: CH 7, CH 6 • 8.1 – 8.3.2, • 8.3.4, • 8.7.1 • Get Tables 8.2 & 8.3 for your engineering notebook • Homework: • CH 8 Textbook Problems: • 9, 10, 25 IENG 486: Statistical Quality & Process Control

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