Advanced Topics
`in
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`Statistical
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`Process
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`Control
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`The Power of Shewhart’s Charts
`
`Donald ]. Wheeler, PhD.
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`SPC Press
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`Knoxville, Tennessee
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`Mylan v. MonoSol
`Mylan V. MonoSol
`IPR2017-00200
`IPR2017-00200
`MonoSol Ex. 2029
`MonoSol EX. 2029
`
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`Copyright © 1995 SPC Press, Inc.
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`All Rights Reserved
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`Do Not Reproduce
`the material in this book
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`by any means whatsoever
`without written permission
`from SPC Press, Inc.
`
`SPC Press, Inc.
`5908 Toole Drive, Suite C
`Knoxville, Tennessee 37919
`(615) 584—5005
`Fax (615) 588—9440
`
`ISBN 0~945320—45—0
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`xiv + 470 pages
`234 figures
`51 reference tables
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`1234567890
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`if
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`Chapter Five
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`Three-Sigma Limits
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`the method of attack is to establish limits of variability ..., such that,
`“As indicated
`when [a value] is found outside these limits, looking for an assignable cause is worth
`while.”
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`"We usually choose a symmetric range characterized by limits #9 it 0'9
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`"If more than one statistic is used, then the limits on all the statistics shOuld be chosen
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`so that the probability of looking for trouble when any one of the chosen statistics
`falls outside its own limits is economic.”
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`“Experience indicates that t = 3 seems to be an acceptable economic value.”
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`“Hence the method for establishing allowable limits of variation in a statistic de«
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`pends upon theory to furnish the expected value and the standard deviation of the
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`statistic and upon empirical evidence to justify the choice of limits
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`[ expected value ]
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`i-
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`t [ standard deviation ]
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`If an observed point
`“Construct control charts with limits 6 i- 3 39 for each statistic.
`falls outside [these] limits, take this fact as an indication of trouble or lack of control.”
`WA. Shewhart 25
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`One of the foundations of Shewhart's control charts is the use of control limits which are
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`set at a distance of three standard deviations on either side of the appropriate central line.
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`Such limits are commonly referred to as "three-Sigma” limits. Dr. Shewhart carefully
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`“5 Economic Control of Quality ofManufactured Product pp. 147-148, 277, 276, 277, and 304.
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`Advanced Topics in Statistical Process Control
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`explained the rationale behind this choice in Economic Control of Quality of Manufactured
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`Product. As shown by the quotations, this choice was neither arbitrary nor accidental. It was
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`a deliberate choice, made becausa threesigma limits provided the needed sensitivity without
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`causing an unacceptable number of false alarms.
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`In short, threewsigma limits were chosen
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`because they provided an economic balance between the consequences of the two mistakes
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`one can make when interpreting data. This choice has been thoroughly validated in practice.
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`The purpose of this chapter is to provide some insight to why and how threewsigma limits
`work.
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`5.1 Why Three-Sigma Limits?
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`Three-sigma limits are not probability limits. While we Will resort to some theory to
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`demonstrate some of the properties of three-sigma limits, it is important to remember that
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`there are other considerations which were used by Shewhart in selecting this criterion. As
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`indicated by the quotations at the beginning of this chapter, the strongest justification of
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`three-sigma limits is the empirical evidence that three-sigma limits work well in practice—
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`that they provide effective action limits when applied to real world data. Thus, the following
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`arguments cannot further justify the use of three-sigma limits, but they can reveal one of the
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`reasons Why they work so well.
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`While it is not a rigorous probabilistic argument, the Empirical Rule provides a useful
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`way of characterizing data using a measure of location and a measure of dispersion.
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`THE EMPIRICAL RULE: Given a homogeneous set of data:
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`Part One: Roughly 60% to 75% of the data will be located within a distance of
`one standard deviation on either side of the mean.
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`Part Two: Usually 90% to 98% of the data will be located within a distance of
`two standard deviations on either side of the mean.
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`Part Three: Approximately 99% to 100% of the data will be located within a
`distance of three standard deviations on either side of the mean.
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`In order to display the robustness of the Empirical Rule six different probability models
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`are used. All are constructed so as to have MEAN(X) = 0 and SD(X) = 1.0. Therefore, the
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`interval defined by Part One of the Empirical Rule will go from ~—l.0 to 1.0, the interval
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`defined by Part Two will range from —2.0 to 2.0, while the interval defined by Part Three will
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`range from ~30 to 3.0.
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`The three parts of the Empirical Rule are illustrated in Figures 5.1, 5.2, and 5.3.
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`5 / Three~Sigme Limits
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`Cl. 738
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`Figure 5.1: Part One of the Empirical Rule
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`Part One of the Empirical Rule is the weakest part. Only four of the six distributions
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`shown in Figure 5.1 satisfy Part One. Nevertheless, Part One is still a useful guide for
`describing Where the bulk of the distribution (or the data) will be.
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`Advanced Topics in Statistical Process Contra!
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`3
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`Figure 5.2: Part Two of the Empirical Rule
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`Part Two is stronger than Part One. Oniy one of the six distributions in Figure 5.2 does
`not satisfy Part Two.
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`1'18
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`5/ Three-Sigma Limits
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`Figure 5.3: Part Three of the Empirical Rule
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`Part Three is the strongest part of the Empirical Rule. With regard to probability models,
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`Part Three suggests that no matter how skewed, no matter how "heavy-tailed,” virtually all
`of the distribution will fall within 3 standard deviations of the mean.
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