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Chapter 6 Variables Control Charts &5² Statistical Quality Control (D. C. Montgomery)

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Page 1: Chapter 6 Variables Control Charts

Chapter 6Variables Control Charts

許湘伶

Statistical Quality Control(D. C. Montgomery)

Page 2: Chapter 6 Variables Control Charts

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)

Page 3: Chapter 6 Variables Control Charts

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

Page 4: Chapter 6 Variables Control Charts

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

]

Page 5: Chapter 6 Variables Control Charts

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

Page 6: Chapter 6 Variables Control Charts

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)

Page 7: Chapter 6 Variables Control Charts

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

Page 8: Chapter 6 Variables Control Charts

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)

Page 9: Chapter 6 Variables Control Charts

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.

Page 10: Chapter 6 Variables Control Charts

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

Page 11: Chapter 6 Variables Control Charts

Statistical Basis of the Charts VIII

Define A2 = 3d2√

n

Page 12: Chapter 6 Variables Control Charts

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

Page 13: Chapter 6 Variables Control Charts

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

Page 14: Chapter 6 Variables Control Charts

Statistical Basis of the Charts XI

Page 15: Chapter 6 Variables Control Charts

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

Page 16: Chapter 6 Variables Control Charts

Example II

Page 17: Chapter 6 Variables Control Charts

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

●●

LCL

UCL

CL

Number of groups = 25Center = 0.325208StdDev = 0.1398143

LCL = 0UCL = 0.6876425

Number beyond limits = 0Number violating runs = 0

Page 18: Chapter 6 Variables Control Charts

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

●●

●●

LCL

UCL

CL

Number of groups = 25Center = 1.50561StdDev = 0.1398143

LCL = 1.31803UCL = 1.693191

Number beyond limits = 0Number violating runs = 0

Page 19: Chapter 6 Variables Control Charts

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")

Page 20: Chapter 6 Variables Control Charts

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

Page 21: Chapter 6 Variables Control Charts

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%

Page 22: Chapter 6 Variables Control Charts

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%

Page 23: Chapter 6 Variables Control Charts

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%

Page 24: Chapter 6 Variables Control Charts

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

Page 25: Chapter 6 Variables Control Charts

Phase II Operation I

Page 26: Chapter 6 Variables Control Charts

Phase II Operation II

Page 27: Chapter 6 Variables Control Charts

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

Page 28: Chapter 6 Variables Control Charts

Phase II Operation IV

Page 29: Chapter 6 Variables Control Charts

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.

Page 30: Chapter 6 Variables Control Charts

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

Page 31: Chapter 6 Variables Control Charts

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

Page 32: Chapter 6 Variables Control Charts

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

Page 33: Chapter 6 Variables Control Charts

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)

Page 34: Chapter 6 Variables Control Charts

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

Page 35: Chapter 6 Variables Control Charts

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)

Page 36: Chapter 6 Variables Control Charts

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

Page 37: Chapter 6 Variables Control Charts

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

Page 38: Chapter 6 Variables Control Charts

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

Page 39: Chapter 6 Variables Control Charts

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

Page 40: Chapter 6 Variables Control Charts

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

Page 41: Chapter 6 Variables Control Charts

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

Page 42: Chapter 6 Variables Control Charts

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

Page 43: Chapter 6 Variables Control Charts

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 ↑)

Page 44: Chapter 6 Variables Control Charts

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

Page 45: Chapter 6 Variables Control Charts

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

Page 46: Chapter 6 Variables Control Charts

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

Page 47: Chapter 6 Variables Control Charts

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

Page 48: Chapter 6 Variables Control Charts

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)

Page 49: Chapter 6 Variables Control Charts

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.

Page 50: Chapter 6 Variables Control Charts

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

Page 51: Chapter 6 Variables Control Charts

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

Page 52: Chapter 6 Variables Control Charts

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

Page 53: Chapter 6 Variables Control Charts

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

Page 54: Chapter 6 Variables Control Charts

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

Page 55: Chapter 6 Variables Control Charts

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

Page 56: Chapter 6 Variables Control Charts

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.

Page 57: Chapter 6 Variables Control Charts

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.

Page 58: Chapter 6 Variables Control Charts

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

Page 59: Chapter 6 Variables Control Charts

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

Page 60: Chapter 6 Variables Control Charts

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]

Page 61: Chapter 6 Variables Control Charts

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

Page 62: Chapter 6 Variables Control Charts

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)

Page 63: Chapter 6 Variables Control Charts

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

Page 64: Chapter 6 Variables Control Charts

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

Page 65: Chapter 6 Variables Control Charts

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

Page 66: Chapter 6 Variables Control 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)

Page 67: Chapter 6 Variables Control Charts

The Average Run Length for the x chart II

Page 68: Chapter 6 Variables Control Charts

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

Page 69: Chapter 6 Variables Control Charts

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

Page 70: Chapter 6 Variables Control Charts

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

Page 71: Chapter 6 Variables Control Charts

Control Charts for x and s IV

Example 6.3: 活塞環的內半徑

Page 72: Chapter 6 Variables Control Charts

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

Page 73: Chapter 6 Variables Control Charts

Control Charts for x and s VI

Page 74: Chapter 6 Variables Control Charts

Control Charts for x and s VII

Page 75: Chapter 6 Variables Control Charts

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

Page 76: Chapter 6 Variables Control Charts

The x and s Control Charts with VariableSample Size II

Page 77: Chapter 6 Variables Control Charts

The x and s Control Charts with VariableSample Size III

Page 78: Chapter 6 Variables Control Charts

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

Page 79: Chapter 6 Variables Control Charts

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

Page 80: Chapter 6 Variables Control Charts

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

Page 81: Chapter 6 Variables Control Charts

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

Page 82: Chapter 6 Variables Control Charts

The Shewhart Control Chart for IndividualMeasurement III

Page 83: Chapter 6 Variables Control Charts

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")

Page 84: Chapter 6 Variables Control Charts

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

Page 85: Chapter 6 Variables Control Charts

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.

Page 86: Chapter 6 Variables Control Charts

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")

Page 87: Chapter 6 Variables Control Charts

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

Page 88: Chapter 6 Variables Control Charts

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

Page 89: Chapter 6 Variables Control Charts

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

Page 90: Chapter 6 Variables Control Charts

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