IENG 486: Statistical Quality & Process Control 12/4/2018 IENG 486 - Lecture 14 X-bar & s Charts: Trial Limits & Standard Limits, Control Chart Operating Characteristics 12/4/2018 IENG 486: Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen

IENG 486: Statistical Quality & Process Control 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 12/4/2018 IENG 486: Statistical Quality & Process Control

R - (Range) Control Chart X-Bar (Means) Control Chart X-Bar & R-Charts The X-Bar Chart checks variability in location between samples The R-Chart checks for changes in sample variation R - (Range) Control Chart LCL UCL Sample Number R X-Bar (Means) Control Chart x 12/4/2018 IENG 486: Statistical Quality & Process Control

IENG 486: Statistical Quality & Process Control X-Bar & Sigma-Charts Used when sample size is greater than 10 X-Bar Control Limits: Approximate 3 limits are found from S & table Sigma-Chart Control Limits: Approximate, asymmetric 3 limits from S & table 12/4/2018 IENG 486: Statistical Quality & Process Control

IENG 486: Statistical Quality & Process Control X-Bar & Sigma-Charts Limits can also be generated from historical data: X-Bar Control Limits: Approximate 3 limits are found from known 0 & table Sigma-Chart Control Limits: Approximate, asymmetric 3 limits from 0 & table 12/4/2018 IENG 486: Statistical Quality & Process Control

Operating Characteristic (OC) Curve IENG 486: Statistical Quality & Process Control 12/4/2018 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 from m0 (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 12/4/2018 IENG 486: Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen

Ex. Probability of NOT Detecting Shift IENG 486: Statistical Quality & Process Control 12/4/2018 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. 12/4/2018 IENG 486: Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen

IENG 486: Statistical Quality & Process Control 12/4/2018 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 12/4/2018 IENG 486: Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen

IENG 486: Statistical Quality & Process Control 12/4/2018 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) 12/4/2018 IENG 486: Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen

Probability of Detecting Shift for Subsequent Samples IENG 486: Statistical Quality & Process Control 12/4/2018 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) 12/4/2018 IENG 486: Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen

Average Run Length (ARL) IENG 486: Statistical Quality & Process Control 12/4/2018 Average Run Length (ARL) Expected number of samples taken before shift is detected is called the Average Run Length (ARL) 12/4/2018 IENG 486: Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen

Performance of Any Shewhart Control Chart IENG 486: Statistical Quality & Process Control 12/4/2018 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) 12/4/2018 IENG 486: Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen

IENG 486: Statistical Quality & Process Control 12/4/2018 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 12/4/2018 IENG 486: Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen

IENG 486: Statistical Quality & Process Control 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 12/4/2018 IENG 486: Statistical Quality & Process Control