IENG 486 Lecture 13 Statistical Basis for XBar

IENG 486 - Lecture 13 Statistical Basis for X-Bar & R Control Charts 9/15/2020 IENG 486: Statistical Quality & Process Control 1

Teminology w Causes of Variation: n Assignable Causes l l n 9/15/2020 n Keep the process from operating predictably Things that we can do something about Common Causes l w Meaning of Control: Random, inherent variation in the process In Specification l n Meets customer constraints on product In Statistical Control l No Assignable Causes of variation present in the process IENG 486: Statistical Quality & Process Control 2

Assignment: w Reading: n n CH 5: 5. 3 (already read 5. 1 -5. 2 & 5. 4) Start on CH 6: all except 6. 3. 2 & 6. 4 w Homework 4: n Textbook Problems CH 5: 9, 11, 13, 23, & 24 9/15/2020 IENG 486: Statistical Quality & Process Control 3

Charts Based on Standard Values, x Chart w If values for m and s are known (i. e. , do not need to estimate from data) w Quantity A is tabulated in Appendix VI 9/15/2020 IENG 486: Statistical Quality & Process Control 4

R - Chart Based on Standard Values w If values for R and s are known w r. v. W = R / s – relative range w The parameters of the distribution of W are a function of n w From the relative range we can compute the mean of R 9/15/2020 IENG 486: Statistical Quality & Process Control 5

The Standard Deviation of R w The standard deviation of R is given as: (Text does not derive this) w therefore w where n D 1 and D 2 are constants tabulated in Appendix VI w Caution: Be careful when using standard values n make sure these values are representative of the actual process 9/15/2020 IENG 486: Statistical Quality & Process Control 6

Computing Trial Control Limits (from sample data) for x Chart w General model for x chart w Substituting estimates for μx and σx and using 3 -sigma limits: w Where A 2 comes from Appendix VI and depends on n 9/15/2020 IENG 486: Statistical Quality & Process Control 7

Computing Trial Control Limits for R - Chart w x and R charts come as a pair w General model for R chart w Substituting estimates for m. R and s. R and using 3 -sigma limits 9/15/2020 IENG 486: Statistical Quality & Process Control 8

Computing Trial Control Limits for R - Chart (continued) w where and w D 3 and D 4 are tabulated in Appendix VI and depend on n w NOTE: R chart is quite sensitive to departures from normality 9/15/2020 IENG 486: Statistical Quality & Process Control 9

x & R Trial Control Chart Limits: Guidelines for Sampling w Sample should be of size 3 to 8 (sizes 4 – 6 are more common) w Sample must be homogeneous n same time (consecutive units) n same raw materials n same operator n same machine w Time may pass between samples but not within samples 9/15/2020 IENG 486: Statistical Quality & Process Control 10

Steps for Trial Control Limits w w 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 n If one (or more) found, then: 1. 2. 3. n 9/15/2020 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 IENG 486: Statistical Quality & Process Control 11

Control Chart Sensitizing Rules w Western Electric Rules: 1. 2. 3. 4. w One point plots outside three-sigma limits; Two out of three consecutive points plot beyond twosigma 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 9/15/2020 IENG 486: Statistical Quality & Process Control 12

Control Chart Examples UC L x x LC L Rule 1 9/15/2020 Rule 2 Rule 3 IENG 486: Statistical Quality & Process Control Rule 4 13

Control Chart Sensitizing Rules w 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. 8. Fourteen points in a row alternating above and below the center line. 9. Fifteen points in a row anywhere between the one-sigma warning limits (including either side of the center line). 10. Any unusual or non-random pattern to the plotted points. 5. 6. 7. 9/15/2020 IENG 486: Statistical Quality & Process Control 14