chapter 6 variables control charts
TRANSCRIPT
Chapter 6Variables Control Charts
許湘伶
Statistical Quality Control(D. C. Montgomery)
Introduction I
I Variable: a numerical measurementI A single measurable quality characteristic, such as a
dimension(尺寸), weight, or volume, is called a variable.
deal with a quality characteristics, necessary to monitor both
1. the mean value of the quality characteristic: x control chart2. variability:
I a control chart for the standard deviation: s control chartI a control chart for the range: R control chart (widely used)
Introduction II
I Separate x and R charts are maintained for each qualitycharacteristic of interest.
I Important:maintain(使繼續) control over both the process mean andprocess variability
Statistical Basis of the Charts IAssumptions
I a quality characteristics is normally distributed with µand σ
I Size n: xi ∼ N (µ, σ2), i = 1, . . . ,n
I Average:
x =∑n
i=1 xi
n ∼ N (µ, σ2x), σx = σ
x
I The probability is 1− α that any sample mean will fallbetween (µ, σ: known)
[µ− Zα/2σx , µ+ Zα/2σx
]=[µ− Zα/2
σ√n, µ+ Zα/2
σ√n
]
Statistical Basis of the Charts II
I If the underlying distribution is nonnormal: the centrallimit theorem
I We usually will not know µ and σI 怎麼做?
I Estimated from preliminary samples or subgroups takenwhen the process is thought to be in control
Statistical Basis of the Charts IIIThe best estimator of µ:
I m samples: m = 20 ∼ 25I each sample containing n observations: n = 4, 5, 6
I The grand average:
¯x =∑m
i=1 xim (the center line on the x chart)
Statistical Basis of the Charts IVthe estimator of σ:1. the standard deviation2. the ranges of the m samples (the range method)
I Range of a sample of size n:
x1, . . . , xn ⇒ R = xmax − xmin
I The average range:
R =∑m
i=1 Rim
I In Chap. 4: relative range W : W = Rσ
Statistical Basis of the Charts VProperties of relative range
I The parameters of the distribution of W are afunction of sample size n
I E(W ) = d2
I An estimator of σ:
σ = Rd2
(d2: Appendix Table VI)
I R: the average range of the m preliminary samples
⇒ σ = Rd2
( an unbiased estimator of σ)
Statistical Basis of the Charts VI補充: the distribution of the sample range
I if xii.i.d∼ F(x), i = 1, . . . ,n
⇒ R = xmax − xmin = x(n) − x(1)
I If the samples are taken from N (0, 1)
⇒ fR(r) = n(n+1)∫ ∞−∞
[Φ(x + r)− Φ(x)]n−2φ(x)φ(x+r)dx, r > 0
I If the samples are taken from N (0, σ2)
⇒W = Rσ∼ fR(r)
I The moments of the range R can be derived form the p.d.f.
Statistical Basis of the Charts VIIthe x control chart:
I[µ− Zα/2
σ√n , µ+ Zα/2
σ√n
]I Zα/2 = 3I σ = R
d2
I The parameters of the x chart:
UCL = ¯x + 3d2√
n R
Center Line = ¯x
LCL = ¯x + 3d2√
n R
Statistical Basis of the Charts VIII
Define A2 = 3d2√
n
Statistical Basis of the Charts IX
R chart:I The center line: RI An estimate of σR: (Under normal distribution assumption)
R = Wσ ⇒ σR = d3σ
⇒ σR = d3Rd2
where d3= the s.d. of W
Statistical Basis of the Charts XI The parameters of the R chart:
UCL = R + 3d3Rd2
Center Line = R
LCL = R − 3d3Rd2
I Assume D3 = 1− 3d3d2, D4 = 1 + 3d3
d2
Statistical Basis of the Charts XI
Example I
Example 6.1I Hard-bake processI 25 samples, each of size 5 wafersI It is best to begin with the R chart.I R =
∑Ri
25 = 0.32521 ¯x = 1.5056I n = 5⇒ Appendix Table VI D3 = 0, D4 = 2.114
R chart: LCL = RD3 = 0, UCL = RD4 = 0.68749
I Appendix Tale VI A2 = 0.577
x chart: LCL = ¯x−A2R = 1.31795, UCL = ¯x+A2R = 1.69325
Example II
Example IIIR Chart
for predata
Group
Gro
up s
umm
ary
stat
istic
s
1 2 3 4 5 6 7 8 9 11 13 15 17 19 21 23 25
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
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LCL
UCL
CL
Number of groups = 25Center = 0.325208StdDev = 0.1398143
LCL = 0UCL = 0.6876425
Number beyond limits = 0Number violating runs = 0
Example IVxbar Chartfor predata
Group
Gro
up s
umm
ary
stat
istic
s
1 2 3 4 5 6 7 8 9 11 13 15 17 19 21 23 25
1.4
1.5
1.6
1.7
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LCL
UCL
CL
Number of groups = 25Center = 1.50561StdDev = 0.1398143
LCL = 1.31803UCL = 1.693191
Number beyond limits = 0Number violating runs = 0
Example V
I The process is in control at the state levels and adopt thetrial control limits for use in phase II, where monitoring offuture production is of interest
library(qcc)ex6_1=read.table("ex6_1.csv",header=T,sep=",")
predata=ex6_1[1:25,]barx=mean(rowMeans(predata))barR=mean(apply(predata,1,function(x) max(x)-min(x)))qcc(predata,type="R")qcc(predata,type="xbar")
Estimating Process Capability II Estimate the mean flow width of the resist: ¯x = 1.5056
micronsI The process s.d.: σ = R
d2= 0.1398 microns
I The specification limits: 1.50± 0.50 micronsI The control chart data may be used to describe the
capability of the process to produce wafers relative to thesespecifications.
p =P{x < 1.00}+ P{x > 2.00}
=Φ(1.00− 1.50560.1398 ) + 1− Φ(2.00− 1.5056
0.1398 ) = 0.00035
I about 0.035% (350 parts per million) of wafers producedwill be outside of the specifications
Estimating Process Capability II
ex61_xbar=qcc(predata,type="xbar")process.capability(ex61_xbar,spec.limits=c(1,2))
Process Capability Analysisfor predata
1.0 1.2 1.4 1.6 1.8 2.0
LSL USLTarget
Number of obs = 125Center = 1.50561StdDev = 0.1398143
Target = 1.5LSL = 1USL = 2
Cp = 1.19Cp_l = 1.21Cp_u = 1.18Cp_k = 1.18Cpm = 1.19
Exp<LSL 0.015%Exp>USL 0.02%Obs<LSL 0%Obs>USL 0%
Estimating Process Capability III
Process capability ratio (Cp; PCR)I a quality characteristic with both upper and lower
specification limits:
Cp = USL − LSL6σ
I Another method: the percentage uses up about p% of thespecification band
P =(
1Cp
)100%
Estimating Process Capability IVI hard-bake process:
Cp = 2.00− 1.006(0.1398) = 1.192> 1
⇒ the “natural” tolerance limits in the process are insidethe lower and upper specification limits
P =(
1Cp
)100% = 83.89%
Revision of Control Limits and Center Lines II require periodic revision of the control limits and center
linesI every weekI every monthI every 20, 50, or 100 samples
I Replace the CL of the x chart with a target value (¯x0)I If the R chart exhibits control, this can be helpful in
shifting the process average to the desired value. (by a fairlysimple adjustment of a manipulatable(可操縱的) variable)
I If the mean is not easily influenced by a simple processadjustment ⇒ a complex and unknown function of severalprocess variables and a target value ¯x may not be helpful
I If R chart is out of control ⇒ eliminate the out-of-controlpoints, recompute a revised value of R
Phase II Operation I
Phase II Operation II
Phase II Operation III
qcc(ex6_1[1:25,], type="xbar", newdata=ex6_1[26:45,])qcc(ex6_1[1:25,], type="R", newdata=ex6_1[26:45,])
I Examining control chart data:helpful to construct a run chart of the individualobservation in each sample
I tier chart or tolerance diagram: box plots is usually asimple way to construct the tier diagram
Phase II Operation IV
CL, SL, NTL I
I There is no connection or relationship between{the control limits on the x and R chartsthe specification limits on the process
I Control limits: driven by the natural variability of theprocess (natural tolerance limits(NTL) of the process)
I UNTL, LNTL: 3σ above and below the process meanI Specification limits: determined externally(在外面); may be
set by management, the manufacturing engineers, thecustomers etc.
CL, SL, NTL II
CL and SLThere is no mathematical or statistical relationship be-tween the control limits and specification limits
I Control chart ⇒ usecontrol limits
I tolerance chart(individualobservations) ⇒ helpful toplot the specification limits
Rational Subgroups I
x chart:I monitors the average quality level in the processI Samples should be selected: maximized the chances for
shifts in the process average to occur between samplesI between-sample variability: variability in the process
over time
R chart:I measures the variability within a sampleI within-sample variability: the instantaneous(即時的)
process variability at a given time
Rational Subgroups III Carefully examining how the control limits for the x and R
charts are determined from past dataI The estimate of the process s.d. σ used in constructing the
control limits is calculated from the variability within eachsample ⇒ reflects only within-sample variability
�����
������
��XXXXXXXXXXXXXs =
√∑mi=1
∑nj=1(xij − ¯x)2
mn − 1 to estimate σ
I σ will be overestimated
I combines both between-sample and within-samplevariability
Guidelines for the Design of the ControlChart I
x and R charts:1. sample size(樣本大小)2. control limit width(管制界線寬度)3. frequency of sampling(抽樣頻率)
Complete solution to know: (經濟考量)I the cost of sampling(抽樣成本)I the costs of investigating and possibly correcting the
process in response to out-of-control signal(調查和矯正失控製程成本)
I the costs associated with producing a product that doesnot meet specifications(製品不合格成本)
Guidelines for the Design of the ControlChart II
Some general guidelines that will aid in control chart designI x chart: detect{
large shifts (2σ or large) ⇒ n = 4, 5, 6small shifts ⇒ n = 15 ∼ 25(large sample size)
I smaller samples ⇒ less risk of a process shift occurringwhile a sample is taken
Guidelines for the Design of the ControlChart III
I R chart:insensitive to shift in the process s.d. for small samples
I n = 5⇒ about a 40% chance to detecting the shift σ → 2σI large sample size (n > 10 or 12): more effective
use a control chart for s or s2 (����XXXXR chart)
Guidelines for the Design of the ControlChart IV
Allocating sampling(抽樣配置) problem:choosing
1. the sample size
2. the frequency of sampling
I have only a limited number of resources to allocate to theinspection process
I available strategies:small, frequent samples: n=5/every half hour
⇒ favored by the current industrylarger samples less frequently: n = 20/every two hours
Guidelines for the Design of the ControlChart V
The rate of production:I influences the choice of sample size & sampling frequency
I Ex: 50,000 units per hour(high rates of production)I 在高速生產的過程,在同一時間收集n = 5 或 n = 20不會造成太大的差異
I 若檢驗成本不高,high-speed production processes通常會監測較大的樣本數
Control Limits:I Usually, 3σI type I errors are very expensive to investigate ⇒ as wide as
3.5σI out-of-control signals are quickly and easily investigated ⇒
2.5 or 2.75σ
Changing sample size on the x and R charts I
I Assume: n is constant from sample to sample
How about n is not constant?
I the center line on the R chart is changed ⇒ x and s chartsI making a permanent (固定性的) change, i.e., nold → nnew
Changing sample size on the x and R charts II
Notations:
Rold = average range for the old sample sizeRnew = average range for the new sample sizenold = old sample size
nnew = new sample sized2(old) = factors d2 for the old sample size
d2(new) = factors d2 for the new sample size
Changing sample size on the x and R charts IIIx chart
UCL = ¯x + A2(new)
[d2(new)d2(old)
]Rold
UCL = ¯x − A2(new)
[d2(new)d2(old)
]Rold
R chart
UCL = D4(new)
[d2(new)d2(old)
]Rold
CL = Rnew =[d2(new)
d2(old)
]Rold
UCL = D3(new)
[d2(new)d2(old)
]Rold
Changing sample size on the x and R charts IV
Example 6.2I the hard-bake process in Example 6.1
I nold = 5 good control−→ reduce nnew = 3I The new control charts:
Type n R d2 A2
Old 5 0.32521 2.326New 3 0.2367 1.693 1.023
Rnew =[
d2(new)d2(old)
]Rold = 0.2367
Changing sample size on the x and R charts VThe new control limits on the x chart:
UCL = ¯x + A2(new)
[d2(new)d2(old)
]Rold = 1.7478
UCL = ¯x − A2(new)
[d2(new)d2(old)
]Rold = 1.2634
The new parameters for the R chart:
UCL = D4(new)
[d2(new)d2(old)
]Rold = 0.6093
CL = Rnew = 0.2367
UCL = max{0, D3(new)
[d2(new)d2(old)
]Rold} = 0
Changing sample size on the x and R charts VI
n ↓⇒1. the width of the
control limits onx chart ↑ (∵ σ√
n )2. the center line ↓
and the uppercontrol limits ↓(∵ d2 ↑ when n ↑)
Probability Limits on the x and R charts I
Name of control limits: α = 0.002⇒ |Z0.001| = 3.09I Western Europe: 0.001(= α/2) probability limits (one
direction)I United States: three-sigma limits; a multiple of the
standard deviation of the statistic (k × σ);
x d→ normally distributed⇒ x chart: k = Zα/2 = 3.09 when α = 0.002
Probability Limits on the x and R charts II
R chart: using the percentage points of the distribution of therelative range W = R/σ
I the subgroup size: nI W = R
σ ⇒√Var(R) = σ
√Var(W )
P(σW0.001(n) ≤ R ≤ σW0.999(n)) = 1− α = 0.998
I (Wα/2(n),W1−α/2(n)) = (W0.001,W0.999(n))I Estimate σ by R/d2
Probability Limits on the x and R charts III
I The 0.001 and 0.999 limits for R:
(W0.001(n)(R/d2),W0.999(n)(R/d2))
⇒UCL = W0.999(n)(R/d2) = D0.999RUCL = W0.001(n)(R/d2) = D0.001R⇒when 3 ≤ n ≤ 6, produce LCL ≥ 0
Charts Based on Standard Values I
I Possible to specify standard values for the process meanand standard deviation:
Standards: µ and σ
The x chart based on standard values
UCL = µ+ 3 σ√n = µ+ Aσ
Center line = µ
LCL = µ− 3 σ√n = µ−Aσ
Charts Based on Standard Values II
The R chart based on standard valuesI σ = R/d2
I d2: the mean of the distribution of the relative range(E(R
σ ) = d2)
σR = d3σ (where d3 =√
Var(W ))
UCL = d2σ + 3d3σ = D2σ (D2 = d2 + 3d3)Center line = d2σ
LCL = d2σ − 3d3σ = D1σ (D1 = d2 − 3d3)
Charts Based on Standard Values IIII Care: when standard values of µ and σ are givenI May be these standards are not really applicable(適當的) to
the processI Standard value of σ seem to give more trouble than
standard value of µ.I If the process is really in control at some other mean and
standard deviation, then the analyst may spendconsiderable effort looking for assignable causes that do notexist.
I In processes where the mean of the quality characteristic iscontrolled by adjustments to the machine, standard ortarget values of µ are sometimes helpful in achievingmanagement goals with respect to process performance.
Interpretation of x and R I
I Interpreting patterns on the x chart: must determinewhether or not the R chart is in control
I First eliminate the R chart assignable causesI Never attempt to interpret the x chart when the R chart
indicates an out-of-control condition
Interpretation of x and R IICase 1: Cyclic patterns
I x chart-Systematic environmental changes: 溫度、操作員疲勞、人員輪班或機器輪流、電壓變動
I R chart: 維修排程、人員疲勞、工具磨損I Ex: the on-off cycle of a compressor(壓縮機) in the filling
machine-systematic variability
Interpretation of x and R IIICase 2: Mixture pattern
I 特徵: with relative few points near the center lineI generated by two overlapping distributions generating the
processI overcontrol: adjustments too oftenI Parallel machines: output product from several sources
Interpretation of x and R IVCase 3: Shift in process level
I New workersI Changes in methodsI raw materials or machinesI a change in the skill
Interpretation of x and R V
Case 4: Trend in process level
I gradual wearing(逐步的磨損)
I 工具或重要製成元件的惡化
I 人員疲勞
I 季節影響: 溫度I Monitoring and analyzing
processes with trends:regression control chart
Interpretation of x and R VICase 5: Stratification
I lack of natural variabilityI incorrect calculation of control limitsI come from two different distribution: R will be incorrectly
inflated(膨脹) causing the limits on the x chart to be toowide
Interpretation of x and R VII
I In interpreting patterns on the x and R charts:consider the two chart jointly
I If the underlying distribution is normal,x and R charts: statistically independent
I If there is correlation between x and R values:the underlying distribution is skewedThose analyses may be in error if specifications have beendetermined assume normality.
The Effect of Non-normality on x and Rcharts I
I Assumption: the underlying distribution of the qualitycharacteristic is normal
I Interested in knowing the effect of departures fromnormality on x and R charts
I robustness??I Literature: Burr (1967)
The usual normal theory control limit constants are veryrobust to the normality assumption and can be employedunless the population is extremely non-normal.
The Effect of Non-normality on x and Rcharts II
I Studies: the Uniform, right triangular, gamma, and twobimodal distributions
I In most cases, samples of size 4 or 5 are sufficient to ensurereasonable robustness to the normality assumption
I Worst cases:small values or r in Gamma distribution, ex: r = 1/2 or 1
I actual α-risk≤0.014 if n ≥ 4I Normal distribution: 0.0027
The Effect of Non-normality on x and Rcharts III
I The sampling distribution of R is not symmetricI Symmetric 3σ control limits are only an approximation
I the actual α-risk on R chart: 0.00461 if n = 4 not 0.0027
I the R chart is more sensitive to departure from normalitythan the x chart
The OC Function II The ability of the x and R charts to detect shift in process
quality
OC curve for an x control chartI σ: assumed known and constantI mean shifts: µ0
(in-control)=⇒ µ1 = µ0 + kσ
I The probability of not detecting the shift on the firstsubsequent sample: (β-risk)
β = P {LCL ≤ x ≤ UCL|µ = µ1 = µ0 + kσ}
(x ∼ N (µ, σ2/n), Contol limits: µ0 ± Lσ/√
n)
= Φ[
UCL − (µ0 + kσ)σ/√
n
]− Φ
[LCL − (µ0 + kσ)
σ/√
n
]= Φ
[L − k
√n]− Φ
[−L − k
√n]
The OC Function II
I L = 3, k = 2,n = 5⇒ β =Φ[3−2
√5]−Φ[−3−2
√5] ∼=
0.0708⇒ the probability thatsuch a shift will be detectson the first subsequencesample: 1− β = 0.9292
I k = 1,n = 5⇒ β = 0.75
The OC Function III
I β: the probability of not detecting the shift on the firstsubsequent sample
I 1− β: the probability that such a shift will be detects onthe first subsequence sample
I The probability that the shift is detected on the secondsample:
β(1− β) = 0.75(0.25) = 0.19
I The probability that the shift is detected on the rthsubsequence sample:
βr−1(1− β)
The OC Function IV
I average run length: The expected number of samples takenbefore the shift is detected: (the expectation of thegeometric distribution)
ARL =∞∑
r=1rβr−1(1− β) = 1
1− β
I Ex: n = 5, k = 1⇒ ARL = 10.25 = 4
I Small sample sizes often result in a relatively large β-risk
The OC Function VOC curve for the R chart
I The distribution of the relative range W = R/σI σ0: in-control value of the standard deviationI shift to a new value: σ1 > σ0
I The probability of not detecting a shift on the first samplefollowing the shift
β = P{
LCL ≤ R ≤ UCL|σ1}
(Contol limits: d2σ0 ± Lσ0d3)
= P((d2 − 3d3)σ0 ≤ R ≤ (d2 + 3d3)σ0|σ1)
= P(λ−1(d2 − 3d3) ≤ Rσ1≤ λ−1(d2 + 3d3)|σ1)
where λ = σ1σ0
The OC Function VI
I λ = σ1σ2
= 2,n = 5⇒ β ≥0.6⇒ have only about a 40%chance of detecting theshift on each subsequentsample
I R chart is insensitive tosmall or moderate shiftsfor n = 4− 6
Recommendation: use at least 20 to 25 preliminary subgroupsin establishing x and R charts
The Average Run Length for the x chart I
ARL = 1P(one point plots out of control)
In-control: ARL0 = 1α
Out-of-control: ARL1 = 11− β
Average time to signal(ATS): the average time to signal
ATS = ARL× h (h = intervals of sampling time)
I : the expected number of individual units sampled:
I = ARL× n (n = the sample size)
The Average Run Length for the x chart II
Control Charts for x and s I
x and s charts:
1. the sample size n is moderately large: n ≥ 10 or 122. the sample size n is variable
I the unbiased estimator of σ2:
sample variance: s2 =∑n
i=1(xi − x)n − 1
I s is not an unbiased estimator of σ:
E(s) = c4σ,√
Var(s) = σ√1− c2
4(H.W.)
I c4 =√
2n−1
Γ(n/2)Γ((n−1)/2) : a constant that depends on n
Control Charts for x and s IIs chart: the standard value σ is given
UCL = c4σ + 3σ√1− c2
4 = B6σ
Center line = c4σ
LCL = c4σ − 3σ√1− c2
4 = B5σ
s chart: σ is unknownEstimator of σ: s/c4 where s = 1
m∑m
i=1 si
UCL = s + 3 sc4
√1− c2
4 = B4s
Center line = s
LCL = s − 3 sc4
√1− c2
4 = B3s
Control Charts for x and s IIIx chartEstimator of σ: s/c4 where s = 1
m∑m
i=1 si
UCL = ¯x + 3 sc4√
n= ¯x + A3s
Center line = ¯x
LCL = ¯x − 3 sc4√
n= ¯x −A3s
I if using s =√∑n
i=1(xi−x)2
n ⇒ the definition of c4,B3,B4,A3
are altered(改變)I Traditionally: preferred the R chart to the s chart
the simplicity of calculating R from each sample
Control Charts for x and s IV
Example 6.3: 活塞環的內半徑
Control Charts for x and s V
Example 6.3: the piston ring inside diameter measure-ments(活塞環的內半徑)
I m = 25, n = 5⇒ ¯x = 74.001, s = 0.0094x chart:
UCL = ¯x + A3s = 74.014Center line = ¯x = 74.001
LCL = ¯x −A3s = 73.988
s chart:
UCL = B4s = 0.0196Center line = s = 0.0094
LCL = B3s = 0
Estimation of σ: σ = sc4
= 0.00940.9400 = 0.01
Control Charts for x and s VI
Control Charts for x and s VII
The x and s Control Charts with VariableSample Size I
I easy to apply in cases where the sample sizes are variableI ni : the number of observations in the ith sampleI the center line of x and s control charts:
¯x =∑m
i=1 ni xi∑mi=1 ni
s =[∑m
i=1(ni − 1)s2i∑m
i=1 ni −m
]1/2
I A3,B3,B4: depend on the sample size used in eachindividual subgroup
x chart:
UCL = ¯x + A3sCenter line = ¯x
LCL = ¯x −A3s
s chart:
UCL = B4sCenter line = s
LCL = B3s
The x and s Control Charts with VariableSample Size II
The x and s Control Charts with VariableSample Size III
The x and s Control Charts with VariableSample Size IV
Alternative:1. using an average sample size n
I ni are not very different or in a presentation to managementI the average sample size may not be an integer
2. a modal(most common) sample size: 最常出現的sample sizeni來估計σ的值。
I 有17個ni = 5⇒ average all the si for which ni = 5
s = 0.171517 = 0.01010⇒ σ = s
c4= 0.01010
0.94000 = 0.01
The s2 control Chart I
s2 chart
UCL = s2
n − 1χ2α/2,n−1
Center line = s2
LCL = s2
n − 1χ21−α/2,n−1
I (n−1)s2
σ2 ∼ χ2n−1
P(σ2χ2
1−α/2,n−1n − 1 ≤ s2 ≤
σ2χ2α/2,n−1
n − 1
)= 1− α
The Shewhart Control Chart for IndividualMeasurement I
n = 1:I Automated(自動化) inspection and measurement
technology is usedI Data: available relatively slowly; inconvenient to allow
sample sizes of n > 1I Repeat measurements on the process differ only because of
laboratory or analysis errorI Multiple measurements are taken on the same unit of
productI differ very little⇒ s.d. too small; Ex; 一捲紙塗料的厚度I Individual measurements: transactional, business and
service process, no basis for rational subgrouping
The Shewhart Control Chart for IndividualMeasurement II
Control chart for individual units: Moving range control chartI moving range of two successive observations:
MRi = |xi − xi−1|
I n = 2⇒ D3 = 0, D4 =3.267
I UCL = D4MR = 25.45I UCL = D4MR = 0
The Shewhart Control Chart for IndividualMeasurement III
The Shewhart Control Chart for IndividualMeasurement IV
library(qcc)ex6_5 = c(310,288,297,298,307,303,294,297,308,306,294,
299,297,299,314,295,293,306,301,304)qcc(ex6_5, type = "xbar.one", plot = TRUE)Ex6_5_r = matrix(cbind(ex6_5[1:length(ex6_5)-1],ex6_5[2:length(ex6_5)]), ncol=2)qcc(Ex6_5_r , type="R", plot = TRUE,title="R chart for Ex6_5")
The Shewhart Control Chart for IndividualMeasurement V
I The interpretation of the individuals control chart issimilar to that of the ordinary x control chart.
I Sometimes a point will plot outside the control limits onboth the individual chart and the moving range chart.
I a large value of x will also lead to a large value of themoving range
I most likely indicates that the mean is out of controlI not both the mean and the variance of the process are out
of control
The Shewhart Control Chart for IndividualMeasurement VI
Phase II Operation and Interpretation of the individualcharts
I The individualmeasurements on the xchart are assumed to beuncorrelated, and anyapparent pattern on thischart should be carefullyinvestigated.
The Shewhart Control Chart for IndividualMeasurement VII
ex6_5_new =c(305,282,305,296,314,295,287,301,298,311,310,292,305,299,304,310,304,305,333,328)
qcc(ex6_5, type = "xbar.one", plot = TRUE, newdata=ex6_5_new)Ex6_5_r_new = matrix(cbind(ex6_5_new[1:length(ex6_5_new)-1],ex6_5_new[2:length(ex6_5_new)]), ncol=2)Ex6_5_r_all =rbind(Ex6_5_r,Ex6_5_r_new)qcc(Ex6_5_r_all[1:19,],type="R",newdata=Ex6_5_r_all[20:38,],title="R chart for Ex6_5")
The Shewhart Control Chart for IndividualMeasurement VIII
I Some authorities:I recommended not constructing and plotting the MR chart.I The MR chart cannot really provide useful information
about a shift in process variability.
I the careful in interpretation and relies primarily on theindividual chart
The Shewhart Control Chart for IndividualMeasurement IX
Crowder (1987b)I The ARL0 of the combined procedure will generally be
much less than the ARL0 of a standard Shewhart controlchart when the process is in control.(type I error α ↑)
I results closer to the Shewhart ARL0 if we use
UCL = DMR,where 4 ≤ D ≤ 5
I The ability of the individuals control chart to detect smallshifts is very poor.
Size of Shift β ARL1
1σ 0.9772 43.962σ 0.7413 6.303σ 0.5000 2.00
The Shewhart Control Chart for IndividualMeasurement X
Dangerous:I ((((
((((hhhhhhhhnarrower limitsI ARL0 ↓ but the occurrence of false alarms ↑I detecting small shifts in phase II with individual values:
Chapter 9: the cumulative sum control chart or theEWMA control chart
The Shewhart Control Chart for IndividualMeasurement XI
Individual control chart: ARL0 is dramatically affected bynon-normal data
I moderate departure from normality ⇒ the control limitsmay be inappropriate for phase II process monitoring
Methods:1. determine the control limits based on the percentiles of the
correct underlying distribution2. transform the original variable to a new variable that is
approximately normally distributedImportant to check the normality assumption: the normalprobability plot