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Chapter 6 1Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 2Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Learning Objectivesg j
Chapter 6 3Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 4Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 5Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Subgroup Data with Unknown μ and σ
Chapter 6 6Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 7Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 8Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 9Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Phase I Application of and R Chartsx• Eqns 6.4 and 6.5 are trial control limits
– Determined from m initial samplesT i ll 20 25 b f i b 3 d 5• Typically 20-25 subgroups of size n between 3 and 5
– Any out-of-control points should be examined for assignable causes• If assignable causes are found, discard points from calculations
and revise the trial control limits• Continue examination until all points plot in control• Adopt resulting trial control limits for use• If no assignable cause is found, there are two options
1 Eliminate point as if an assignable cause were found and1. Eliminate point as if an assignable cause were found and revise limits
2. Retain point and consider limits appropriate for controlIf there are many out of control points they should be examined for
Chapter 6 10Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
– If there are many out-of-control points they should be examined for patterns that may identify underlying process problems
Example 6.1 The Hard Bake Process
Chapter 6 11Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 12Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 13Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 14Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 15Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 16Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Revision of Control Limitsand Center Lines
• Effective use of control charts requires periodicEffective use of control charts requires periodic review and revision of control limits and center lines
• Sometimes users replace the center line on the chart xpwith a target value
• When R chart is out of control, out-of-control points
x
, pare often eliminated to recompute a revised value of which is used to determine new limits and center line
R
on R chart and new limits on chartx
Chapter 6 17Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Phase II Operation of ChartsPhase II Operation of Charts
• Use of control chart for monitoring future production,Use of control chart for monitoring future production, once a set of reliable limits are established, is called phase II of control chart usage (Figure 6.4)
• A run chart showing individuals observations in each sample, called a tolerance chart or tier diagram(Figure 6.5), may reveal patterns or unusual observations in the data
Chapter 6 18Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 19Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 20Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 21Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Control vs Specification LimitsControl vs. Specification Limits• Control limits are derived
f lfrom natural process variability, or the natural tolerance limits of a process
• Specification limits are determined externally, for
l b texample by customers or designers
Th i h i l• There is no mathematical or statistical relationship between the control limits and the specification limits
Chapter 6 22Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
and the specification limits
Rational SubgroupsRational Subgroups
• charts monitor between-sample variabilityxcharts monitor between sample variability• R charts measure within-sample variability• Standard deviation estimate of σ used to construct
x
• Standard deviation estimate of σ used to construct control limits is calculated from within-sample variabilityy
• It is not correct to estimate σ using
Chapter 6 23Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Guidelines for Control Chart DesignGuidelines for Control Chart Design
• Control chart design requires specification of sample size, g q p pcontrol limit width, and sampling frequency– Exact solution requires detailed information on statistical
characteristics as well as economic factors– The problem of choosing sample size and sampling frequency is one of
allocating sampling effort• For chart, choose as small a sample size is consistent with x p
magnitude of process shift one is trying to detect. For moderate to large shifts, relatively small samples are effective. For small shifts, larger samples are needed.g p
• For small samples, R chart is relatively insensitive to changes in process standard deviation. For larger samples (n > 10 or 12), s or s2 charts are better choices.
Chapter 6 24Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
12), s or s charts are better choices.
Chapter 6 25Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 26Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 27Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 28Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
• An assumption in performance properties is that the underlying distribution of quality characteristic is normal– If underlying distribution is not normal, sampling distributions can be y g , p g
derived and exact probability limits obtained• Burr (1967) notes the usual normal theory control limits are
very robust to normality assumption• Schilling and Nelson (1976) indicate that in most cases,
samples of size 4 or 5 are sufficient to ensure reasonable robustness to normality assumption for chartx
• Sampling distribution of R is not symmetric, thus symmetric 3-sigma limits are an approximation and α-risk is not 0.0027. R chart is more sensitive to departures from normality than chart
xchart.
• Assumptions of normality and independence are not a primary concern in phase I
Chapter 6 29Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 30Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
If the shift is 1.0σ and the sample size is n = 5, then β = 0.75.
Chapter 6 31Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 32Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 33Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 34Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 35Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 36Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 37Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 38Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Development of the control limits:
Chapter 6 39Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Thius produces the control limits in equation (6.27)
This produces the control limits in equation (6 28)This produces the control limits in equation (6.28)
Chapter 6 40Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 41Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 42Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 43Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 44Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 45Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 46Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 47Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 48Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 49Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 50Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 51Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 52Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 53Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 54Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 55Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Average Run LengthsC d (1987b) h d th t• Crowder (1987b) showed that ARL0 of combined individuals and moving-range chart with conventional 3 sigma limits isconventional 3-sigma limits is generally much less than ARL0(= 370) of standard Shewhart control chartcontrol chart
• Ability of individuals chart to detect small shifts is very poordetect small shifts is very poor– Rather than narrowing the
3-sigma limits, correct approach to detecting smallapproach to detecting small shifts is a cumulative-sum or exponentially weighted moving-average control
Chapter 6 56Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
g gchart (Chapter 9)
Normality
• Borror, Montgomery, and Runger (1999) found in-control ARL is dramatically affected by nonnormal datadramatically affected by nonnormal data
• One approach for nonnormal data is to determine control limits for individuals control chart based on percentiles of correct underlying distribution
Chapter 6 57Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
– Requires at least 100 and preferably 200 observations
Chapter 6 58Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 59Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 60Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 61Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 62Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Chapter 6 63Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
Learning ObjectivesLearning Objectives
Chapter 6 64Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.
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