IENG 486: Statistical Quality & Process Control 11/14/2018 IENG 486 - Lecture 13 Statistical Basis for X-Bar & R Control Charts 11/14/2018 IENG 486: Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen & R. C. Wurl

IENG 486: Statistical Quality & Process Control Teminology Causes of Variation: Assignable Causes Keep the process from operating predictably Things that we can do something about Common Causes Random, inherent variation in the process Meaning of Control: In Specification Meets customer constraints on product In Statistical Control No Assignable Causes of variation present in the process 11/14/2018 IENG 486: Statistical Quality & Process Control

IENG 486: Statistical Quality & Process Control 11/14/2018 Assignment: Reading: CH5: 5.3 (already read 5.1-5.2 & 5.4) Start on CH6: all except 6.3.2 & 6.4 Homework 4: Textbook Problems CH5: 9, 11, 13, 23, & 24 11/14/2018 IENG 486: Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen & R. C. Wurl

Charts Based on Standard Values, x Chart IENG 486: Statistical Quality & Process Control 11/14/2018 Charts Based on Standard Values, x Chart If values for m and s are known (i.e., do not need to estimate from data) Quantity A is tabulated in Appendix VI 11/14/2018 IENG 486: Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen & R. C. Wurl

R - Chart Based on Standard Values IENG 486: Statistical Quality & Process Control 11/14/2018 R - Chart Based on Standard Values If values for R and s are known r.v. W = R / s – relative range The parameters of the distribution of W are a function of n From the relative range we can compute the mean of R 11/14/2018 IENG 486: Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen & R. C. Wurl

The Standard Deviation of R IENG 486: Statistical Quality & Process Control 11/14/2018 The Standard Deviation of R The standard deviation of R is given as: (Text does not derive this) therefore where D1 and D2 are constants tabulated in Appendix VI Caution: Be careful when using standard values make sure these values are representative of the actual process 11/14/2018 IENG 486: Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen & R. C. Wurl

Computing Trial Control Limits (from sample data) for x Chart IENG 486: Statistical Quality & Process Control 11/14/2018 Computing Trial Control Limits (from sample data) for x Chart General model for x chart Substituting estimates for μx and σx and using 3-sigma limits: Where A2 comes from Appendix VI and depends on n 11/14/2018 IENG 486: Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen & R. C. Wurl

Computing Trial Control Limits for R - Chart IENG 486: Statistical Quality & Process Control 11/14/2018 Computing Trial Control Limits for R - Chart x and R charts come as a pair General model for R chart Substituting estimates for mR and sR and using 3-sigma limits 11/14/2018 IENG 486: Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen & R. C. Wurl

Computing Trial Control Limits for R - Chart (continued) IENG 486: Statistical Quality & Process Control 11/14/2018 Computing Trial Control Limits for R - Chart (continued) where and D3 and D4 are tabulated in Appendix VI and depend on n NOTE: R chart is quite sensitive to departures from normality 11/14/2018 IENG 486: Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen & R. C. Wurl

x & R Trial Control Chart Limits: Guidelines for Sampling IENG 486: Statistical Quality & Process Control 11/14/2018 x & R Trial Control Chart Limits: Guidelines for Sampling Sample should be of size 3 to 8 (sizes 4 – 6 are more common) Sample must be homogeneous same time (consecutive units) same raw materials same operator same machine Time may pass between samples but not within samples 11/14/2018 IENG 486: Statistical Quality & Process Control (c) 2002 - 2009, D.H. Jensen & R. C. Wurl

Steps for Trial Control Limits Start with 20 to 25 samples Use all data to calculate initial control limits Plot each sample in time-order on chart. Check for out of control sample points If one (or more) found, then: Investigate the process; Remove the special cause; and Remove the special cause point and recalculate control limits. If can’t find special cause - drop point & recalculate anyway 11/14/2018 IENG 486: Statistical Quality & Process Control

Control Chart Sensitizing Rules Western Electric Rules: One point plots outside the three-sigma limits; Two out of three consecutive points plot beyond two-sigma warning limits on the same side of the center line; Four out of five consecutive points plot beyond one-sigma warning limits on the same side of the center line; or Eight consecutive points plot on one side of the center line. If chart shows lack of control, investigate for special cause 11/14/2018 IENG 486: Statistical Quality & Process Control

Control Chart Examples LCL UCL x LCL UCL x LCL UCL x LCL UCL x Rule 1 Rule 2 Rule 3 Rule 4 11/14/2018 IENG 486: Statistical Quality & Process Control

Control Chart Sensitizing Rules Additional Sensitizing Rules: One or more points very near a control limit. Six points in a row steadily increasing or decreasing. Eight points in a row on both sides of the center line, but none in-between the one-sigma warning limits on both sides of the center line. Fourteen points in a row alternating above and below the center line. Fifteen points in a row anywhere between the one-sigma warning limits (including either side of the center line). Any unusual or non-random pattern to the plotted points. 11/14/2018 IENG 486: Statistical Quality & Process Control