© 2008 prentice hall, inc.s6 – 1 operations management supplement 6 – statistical process...
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© 2008 Prentice Hall, Inc. S6 – 1
Operations ManagementOperations ManagementSupplement 6 – Supplement 6 – Statistical Process Statistical Process ControlControl
PowerPoint presentation to accompany PowerPoint presentation to accompany Heizer/Render Heizer/Render Principles of Operations Management, 7ePrinciples of Operations Management, 7eOperations Management, 9e Operations Management, 9e
© 2008 Prentice Hall, Inc. S6 – 2
Variability is inherent Variability is inherent in every processin every process Natural or common Natural or common
causescauses Special or assignable causesSpecial or assignable causes
Provides a statistical signal when Provides a statistical signal when assignable causes are presentassignable causes are present
Detect and eliminate assignable Detect and eliminate assignable causes of variationcauses of variation
Statistical Process Control Statistical Process Control (SPC)(SPC)
© 2008 Prentice Hall, Inc. S6 – 3
Natural VariationsNatural Variations Also called common causesAlso called common causes
Affect virtually all production processesAffect virtually all production processes
Expected amount of variationExpected amount of variation
Output measures follow a probability Output measures follow a probability distributiondistribution
For any distribution there is a measure For any distribution there is a measure of central tendency and dispersionof central tendency and dispersion
If the distribution of outputs falls within If the distribution of outputs falls within acceptable limits, the process is said to acceptable limits, the process is said to be “in control”be “in control”
© 2008 Prentice Hall, Inc. S6 – 4
Assignable VariationsAssignable Variations
Also called special causes of variationAlso called special causes of variation Generally this is some change in the processGenerally this is some change in the process
Variations that can be traced to a specific Variations that can be traced to a specific reasonreason
The objective is to discover when The objective is to discover when assignable causes are presentassignable causes are present Eliminate the bad causesEliminate the bad causes
Incorporate the good causesIncorporate the good causes
© 2008 Prentice Hall, Inc. S6 – 5
SamplesSamples
To measure the process, we take samples To measure the process, we take samples and analyze the sample statistics following and analyze the sample statistics following these stepsthese steps
(a)(a) Samples of the Samples of the product, say five product, say five boxes of cereal boxes of cereal taken off the filling taken off the filling machine line, vary machine line, vary from each other in from each other in weightweight
Fre
qu
ency
Fre
qu
ency
WeightWeight
##
#### ##
####
####
##
## ## #### ## ####
## ## #### ## #### ## ####
Each of these Each of these represents one represents one sample of five sample of five
boxes of cerealboxes of cereal
Figure S6.1Figure S6.1
© 2008 Prentice Hall, Inc. S6 – 6
SamplesSamples
To measure the process, we take samples To measure the process, we take samples and analyze the sample statistics following and analyze the sample statistics following these stepsthese steps
(b)(b) After enough After enough samples are samples are taken from a taken from a stable process, stable process, they form a they form a pattern called a pattern called a distributiondistribution
The solid line The solid line represents the represents the
distributiondistribution
Fre
qu
ency
Fre
qu
ency
WeightWeightFigure S6.1Figure S6.1
© 2008 Prentice Hall, Inc. S6 – 7
SamplesSamples
To measure the process, we take samples To measure the process, we take samples and analyze the sample statistics following and analyze the sample statistics following these stepsthese steps
(c)(c) There are many types of distributions, including There are many types of distributions, including the normal (bell-shaped) distribution, but the normal (bell-shaped) distribution, but distributions do differ in terms of central distributions do differ in terms of central tendency (mean), standard deviation or tendency (mean), standard deviation or variance, and shapevariance, and shape
WeightWeight
Central tendencyCentral tendency
WeightWeight
VariationVariation
WeightWeight
ShapeShape
Fre
qu
ency
Fre
qu
ency
Figure S6.1Figure S6.1
© 2008 Prentice Hall, Inc. S6 – 8
SamplesSamples
To measure the process, we take samples To measure the process, we take samples and analyze the sample statistics following and analyze the sample statistics following these stepsthese steps
(d)(d) If only natural If only natural causes of causes of variation are variation are present, the present, the output of a output of a process forms a process forms a distribution that distribution that is stable over is stable over time and is time and is predictablepredictable
WeightWeightTimeTimeF
req
uen
cyF
req
uen
cy PredictionPrediction
Figure S6.1Figure S6.1
© 2008 Prentice Hall, Inc. S6 – 9
SamplesSamples
To measure the process, we take samples To measure the process, we take samples and analyze the sample statistics following and analyze the sample statistics following these stepsthese steps
(e)(e) If assignable If assignable causes are causes are present, the present, the process output is process output is not stable over not stable over time and is not time and is not predicablepredicable
WeightWeightTimeTimeF
req
uen
cyF
req
uen
cy PredictionPrediction
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Figure S6.1Figure S6.1
© 2008 Prentice Hall, Inc. S6 – 10
Control ChartsControl Charts
Constructed from historical data, the Constructed from historical data, the purpose of control charts is to help purpose of control charts is to help distinguish between natural variations distinguish between natural variations and variations due to assignable and variations due to assignable causescauses
© 2008 Prentice Hall, Inc. S6 – 11
Process ControlProcess Control
Figure S6.2Figure S6.2
FrequencyFrequency
(weight, length, speed, etc.)(weight, length, speed, etc.)SizeSize
Lower control limitLower control limit Upper control limitUpper control limit
(a) In statistical (a) In statistical control and capable control and capable of producing within of producing within control limitscontrol limits
(b) In statistical (b) In statistical control but not control but not capable of producing capable of producing within control limitswithin control limits
(c) Out of control(c) Out of control
© 2008 Prentice Hall, Inc. S6 – 12
Types of DataTypes of Data
Characteristics that Characteristics that can take any real can take any real valuevalue
May be in whole or May be in whole or in fractional in fractional numbersnumbers
Continuous random Continuous random variablesvariables
VariablesVariables AttributesAttributes Defect-related Defect-related
characteristics characteristics
Classify products Classify products as either good or as either good or bad or count bad or count defectsdefects
Categorical or Categorical or discrete random discrete random variablesvariables
© 2008 Prentice Hall, Inc. S6 – 13
Central Limit TheoremCentral Limit Theorem
Regardless of the distribution of the Regardless of the distribution of the population, the distribution of sample means population, the distribution of sample means drawn from the population will tend to follow drawn from the population will tend to follow a normal curvea normal curve
1.1. The mean of the sampling The mean of the sampling distribution distribution ((xx)) will be the same will be the same as the population mean as the population mean
x = x =
nn
xx = =
2.2. The standard deviation of the The standard deviation of the sampling distribution sampling distribution ((xx)) will will equal the population standard equal the population standard deviation deviation (()) divided by the divided by the square root of the sample size, nsquare root of the sample size, n
© 2008 Prentice Hall, Inc. S6 – 14
Population and Sampling Population and Sampling DistributionsDistributions
Three population Three population distributionsdistributions
Beta
Normal
Uniform
Distribution of Distribution of sample meanssample means
Standard Standard deviation of deviation of the sample the sample meansmeans
= = xx = = nn
Mean of sample means = xMean of sample means = x
| | | | | | |
--33xx --22xx --11xx xx ++11xx ++22xx ++33xx
99.73%99.73% of all x of all xfall within fall within ± 3± 3xx
95.45%95.45% fall within fall within ± 2± 2xx
Figure S6.3Figure S6.3
© 2008 Prentice Hall, Inc. S6 – 15
Control Charts for VariablesControl Charts for Variables
For variables that have For variables that have continuous dimensionscontinuous dimensions Weight, speed, length, Weight, speed, length,
strength, etc.strength, etc.
x-charts are to control x-charts are to control the central tendency of the processthe central tendency of the process
R-charts are to control the dispersion of R-charts are to control the dispersion of the processthe process
These two charts must be used togetherThese two charts must be used together
© 2008 Prentice Hall, Inc. S6 – 16
Setting Chart LimitsSetting Chart Limits
For x-Charts when we know For x-Charts when we know
Upper control limit Upper control limit (UCL)(UCL) = x + z = x + zxx
Lower control limit Lower control limit (LCL)(LCL) = x - z = x - zxx
wherewhere xx ==mean of the sample means or mean of the sample means or a target value set for the processa target value set for the process
zz ==number of normal standard number of normal standard deviationsdeviations
xx ==standard deviation of the standard deviation of the sample meanssample means
==/ n/ n
==population standard population standard deviationdeviation
nn ==sample sizesample size
© 2008 Prentice Hall, Inc. S6 – 17
Setting Control LimitsSetting Control LimitsHour 1Hour 1
SampleSample Weight ofWeight ofNumberNumber Oat FlakesOat Flakes
11 1717
22 1313
33 1616
44 1818
55 1717
66 1616
77 1515
88 1717
99 1616
MeanMean 16.116.1
== 11
HourHour MeanMean HourHour MeanMean
11 16.116.1 77 15.215.2
22 16.816.8 88 16.416.4
33 15.515.5 99 16.316.3
44 16.516.5 1010 14.814.8
55 16.516.5 1111 14.214.2
66 16.416.4 1212 17.317.3n = 9n = 9
LCLLCLxx = x - z = x - zxx = = 16 - 3(1/3) = 15 ozs16 - 3(1/3) = 15 ozs
For For 99.73%99.73% control limits, z control limits, z = 3= 3
UCLUCLxx = x + z = x + zxx = 16 + 3(1/3) = 17 ozs= 16 + 3(1/3) = 17 ozs
© 2008 Prentice Hall, Inc. S6 – 18
17 = UCL17 = UCL
15 = LCL15 = LCL
16 = Mean16 = Mean
Setting Control LimitsSetting Control Limits
Control Chart Control Chart for sample of for sample of 9 boxes9 boxes
Sample numberSample number
|| || || || || || || || || || || ||11 22 33 44 55 66 77 88 99 1010 1111 1212
Variation due Variation due to assignable to assignable
causescauses
Variation due Variation due to assignable to assignable
causescauses
Variation due to Variation due to natural causesnatural causes
Out of Out of controlcontrol
Out of Out of controlcontrol
© 2008 Prentice Hall, Inc. S6 – 19
Setting Chart LimitsSetting Chart Limits
For x-Charts when we don’t know For x-Charts when we don’t know
Lower control limit Lower control limit (LCL)(LCL) = x - A = x - A22RR
Upper control limit Upper control limit (UCL)(UCL) = x + A = x + A22RR
wherewhere RR ==average range of the samplesaverage range of the samples
AA22 ==control chart factor found in control chart factor found in Table S6.1 Table S6.1
xx ==mean of the sample meansmean of the sample means
© 2008 Prentice Hall, Inc. S6 – 20
Control Chart FactorsControl Chart Factors
Table S6.1Table S6.1
Sample Size Sample Size Mean Factor Mean Factor Upper Range Upper Range Lower Lower RangeRange
n n AA22 DD44 DD3322 1.8801.880 3.2683.268 00
33 1.0231.023 2.5742.574 00
44 .729.729 2.2822.282 00
55 .577.577 2.1152.115 00
66 .483.483 2.0042.004 00
77 .419.419 1.9241.924 0.0760.076
88 .373.373 1.8641.864 0.1360.136
99 .337.337 1.8161.816 0.1840.184
1010 .308.308 1.7771.777 0.2230.223
1212 .266.266 1.7161.716 0.2840.284
© 2008 Prentice Hall, Inc. S6 – 21
Setting Control LimitsSetting Control Limits
Process average x Process average x = 12= 12 ounces ouncesAverage range R Average range R = .25= .25Sample size n Sample size n = 5= 5
© 2008 Prentice Hall, Inc. S6 – 22
Setting Control LimitsSetting Control Limits
UCLUCLxx = x + A= x + A22RR
= 12 + (.577)(.25)= 12 + (.577)(.25)= 12 + .144= 12 + .144= 12.144 = 12.144 ouncesounces
Process average x Process average x = 12= 12 ounces ouncesAverage range R Average range R = .25= .25Sample size n Sample size n = 5= 5
From From Table S6.1Table S6.1
© 2008 Prentice Hall, Inc. S6 – 23
Setting Control LimitsSetting Control Limits
UCLUCLxx = x + A= x + A22RR
= 12 + (.577)(.25)= 12 + (.577)(.25)= 12 + .144= 12 + .144= 12.144 = 12.144 ouncesounces
LCLLCLxx = x - A= x - A22RR
= 12 - .144= 12 - .144= 11.857 = 11.857 ouncesounces
Process average x Process average x = 12= 12 ounces ouncesAverage range R Average range R = .25= .25Sample size n Sample size n = 5= 5
UCL = 12.144UCL = 12.144
Mean = 12Mean = 12
LCL = 11.857LCL = 11.857
© 2008 Prentice Hall, Inc. S6 – 24
R – ChartR – Chart
Type of variables control chartType of variables control chart
Shows sample ranges over timeShows sample ranges over time Difference between smallest and Difference between smallest and
largest values in samplelargest values in sample
Monitors process variabilityMonitors process variability
Independent from process meanIndependent from process mean
© 2008 Prentice Hall, Inc. S6 – 25
Setting Chart LimitsSetting Chart Limits
For R-ChartsFor R-Charts
Lower control limit Lower control limit (LCL(LCLRR)) = D = D33RR
Upper control limit Upper control limit (UCL(UCLRR)) = D = D44RR
wherewhere
RR ==average range of the samplesaverage range of the samples
DD33 and D and D44==control chart factors from control chart factors from Table S6.1 Table S6.1
© 2008 Prentice Hall, Inc. S6 – 26
Setting Control LimitsSetting Control Limits
UCLUCLRR = D= D44RR
= (2.115)(5.3)= (2.115)(5.3)= 11.2 = 11.2 poundspounds
LCLLCLRR = D= D33RR
= (0)(5.3)= (0)(5.3)= 0 = 0 poundspounds
Average range R Average range R = 5.3 = 5.3 poundspoundsSample size n Sample size n = 5= 5From From Table S6.1Table S6.1 D D44 = 2.115, = 2.115, DD33 = 0 = 0
UCL = 11.2UCL = 11.2
Mean = 5.3Mean = 5.3
LCL = 0LCL = 0
© 2008 Prentice Hall, Inc. S6 – 27
Mean and Range ChartsMean and Range Charts
(a)(a)
These These sampling sampling distributions distributions result in the result in the charts belowcharts below
(Sampling mean is (Sampling mean is shifting upward but shifting upward but range is consistent)range is consistent)
R-chartR-chart(R-chart does not (R-chart does not detect change in detect change in mean)mean)
UCLUCL
LCLLCL
Figure S6.5Figure S6.5
x-chartx-chart(x-chart detects (x-chart detects shift in central shift in central tendency)tendency)
UCLUCL
LCLLCL
© 2008 Prentice Hall, Inc. S6 – 28
Mean and Range ChartsMean and Range Charts
R-chartR-chart(R-chart detects (R-chart detects increase in increase in dispersion)dispersion)
UCLUCL
LCLLCL
Figure S6.5Figure S6.5
(b)(b)
These These sampling sampling distributions distributions result in the result in the charts belowcharts below
(Sampling mean (Sampling mean is constant but is constant but dispersion is dispersion is increasing)increasing)
x-chartx-chart(x-chart does not (x-chart does not detect the increase detect the increase in dispersion)in dispersion)
UCLUCL
LCLLCL
© 2008 Prentice Hall, Inc. S6 – 29
Steps In Creating Control Steps In Creating Control ChartsCharts
1.1. Take samples from the population and Take samples from the population and compute the appropriate sample statisticcompute the appropriate sample statistic
2.2. Use the sample statistic to calculate control Use the sample statistic to calculate control limits and draw the control chartlimits and draw the control chart
3.3. Plot sample results on the control chart and Plot sample results on the control chart and determine the state of the process (in or out of determine the state of the process (in or out of control)control)
4.4. Investigate possible assignable causes and Investigate possible assignable causes and take any indicated actionstake any indicated actions
5.5. Continue sampling from the process and reset Continue sampling from the process and reset the control limits when necessarythe control limits when necessary
© 2008 Prentice Hall, Inc. S6 – 30
Manual and AutomatedManual and AutomatedControl ChartsControl Charts
© 2008 Prentice Hall, Inc. S6 – 31
Control Charts for AttributesControl Charts for Attributes
For variables that are categoricalFor variables that are categorical Good/bad, yes/no, Good/bad, yes/no,
acceptable/unacceptableacceptable/unacceptable
Measurement is typically counting Measurement is typically counting defectivesdefectives
Charts may measureCharts may measure Percent defective (p-chart)Percent defective (p-chart)
Number of defects (c-chart)Number of defects (c-chart)
© 2008 Prentice Hall, Inc. S6 – 32
Control Limits for p-ChartsControl Limits for p-Charts
Population will be a binomial distribution, Population will be a binomial distribution, but applying the Central Limit Theorem but applying the Central Limit Theorem
allows us to assume a normal distribution allows us to assume a normal distribution for the sample statisticsfor the sample statistics
UCLUCLpp = p + z = p + zpp^̂
LCLLCLpp = p - z = p - zpp^̂
wherewhere pp ==mean fraction defective in the samplemean fraction defective in the samplezz ==number of standard deviationsnumber of standard deviationspp ==standard deviation of the sampling distributionstandard deviation of the sampling distribution
nn ==sample sizesample size
^̂
pp(1 -(1 - p p))nn
pp = =^̂
© 2008 Prentice Hall, Inc. S6 – 33
p-Chart for Data Entryp-Chart for Data EntrySampleSample NumberNumber FractionFraction SampleSample NumberNumber FractionFractionNumberNumber of Errorsof Errors DefectiveDefective NumberNumber of Errorsof Errors DefectiveDefective
11 66 .06.06 1111 66 .06.0622 55 .05.05 1212 11 .01.0133 00 .00.00 1313 88 .08.0844 11 .01.01 1414 77 .07.0755 44 .04.04 1515 55 .05.0566 22 .02.02 1616 44 .04.0477 55 .05.05 1717 1111 .11.1188 33 .03.03 1818 33 .03.0399 33 .03.03 1919 00 .00.00
1010 22 .02.02 2020 44 .04.04
Total Total = 80= 80
(.04)(1 - .04)(.04)(1 - .04)
100100pp = = = .02= .02^̂p p = = .04= = .04
8080
(100)(20)(100)(20)
© 2008 Prentice Hall, Inc. S6 – 34
.11 .11 –
.10 .10 –
.09 .09 –
.08 .08 –
.07 .07 –
.06 .06 –
.05 .05 –
.04 .04 –
.03 .03 –
.02 .02 –
.01 .01 –
.00 .00 –
Sample numberSample number
Fra
ctio
n d
efec
tive
Fra
ctio
n d
efec
tive
| | | | | | | | | |
22 44 66 88 1010 1212 1414 1616 1818 2020
p-Chart for Data Entryp-Chart for Data Entry
UCLUCLpp = p + z = p + zpp = .04 + 3(.02) = .10= .04 + 3(.02) = .10^̂
LCLLCLpp = p - z = p - zpp = .04 - 3(.02) = 0 = .04 - 3(.02) = 0^̂
UCLUCLpp = 0.10= 0.10
LCLLCLpp = 0.00= 0.00
p p = 0.04= 0.04
© 2008 Prentice Hall, Inc. S6 – 35
.11 .11 –
.10 .10 –
.09 .09 –
.08 .08 –
.07 .07 –
.06 .06 –
.05 .05 –
.04 .04 –
.03 .03 –
.02 .02 –
.01 .01 –
.00 .00 –
Sample numberSample number
Fra
ctio
n d
efec
tive
Fra
ctio
n d
efec
tive
| | | | | | | | | |
22 44 66 88 1010 1212 1414 1616 1818 2020
UCLUCLpp = p + z = p + zpp = .04 + 3(.02) = .10= .04 + 3(.02) = .10^̂
LCLLCLpp = p - z = p - zpp = .04 - 3(.02) = 0 = .04 - 3(.02) = 0^̂
UCLUCLpp = 0.10= 0.10
LCLLCLpp = 0.00= 0.00
p p = 0.04= 0.04
p-Chart for Data Entryp-Chart for Data Entry
Possible assignable
causes present
© 2008 Prentice Hall, Inc. S6 – 36
Control Limits for c-ChartsControl Limits for c-Charts
Population will be a Poisson distribution, Population will be a Poisson distribution, but applying the Central Limit Theorem but applying the Central Limit Theorem
allows us to assume a normal distribution allows us to assume a normal distribution for the sample statisticsfor the sample statistics
wherewhere cc ==mean number defective in the samplemean number defective in the sample
UCLUCLcc = c + = c + 33 c c LCLLCLcc = c = c -- 33 c c
© 2008 Prentice Hall, Inc. S6 – 37
c-Chart for Cab Companyc-Chart for Cab Company
c c = 54= 54 complaints complaints/9/9 days days = 6 = 6 complaintscomplaints//dayday
|1
|2
|3
|4
|5
|6
|7
|8
|9
DayDay
Nu
mb
er d
efec
tive
Nu
mb
er d
efec
tive14 14 –
12 12 –
10 10 –
8 8 –
6 6 –
4 –
2 –
0 0 –
UCLUCLcc = c + = c + 33 c c
= 6 + 3 6= 6 + 3 6= 13.35= 13.35
LCLLCLcc = c - = c - 33 c c
= 6 - 3 6= 6 - 3 6= 0= 0
UCLUCLcc = 13.35= 13.35
LCLLCLcc = 0= 0
c c = 6= 6
© 2008 Prentice Hall, Inc. S6 – 38
Managerial Issues andManagerial Issues andControl ChartsControl Charts
Select points in the processes that Select points in the processes that need SPCneed SPC
Determine the appropriate charting Determine the appropriate charting techniquetechnique
Set clear policies and proceduresSet clear policies and procedures
Three major management decisions:Three major management decisions:
© 2008 Prentice Hall, Inc. S6 – 39
Which Control Chart to UseWhich Control Chart to Use
Using an x-chart and R-chart:Using an x-chart and R-chart: Observations are variablesObservations are variables
Collect Collect 20 - 2520 - 25 samples of n samples of n = 4= 4, or n , or n = = 55, or more, each from a stable process , or more, each from a stable process and compute the mean for the x-chart and compute the mean for the x-chart and range for the R-chartand range for the R-chart
Track samples of n observations eachTrack samples of n observations each
Variables DataVariables Data
© 2008 Prentice Hall, Inc. S6 – 40
Which Control Chart to UseWhich Control Chart to Use
Using the p-chart:Using the p-chart: Observations are attributes that can Observations are attributes that can
be categorized in two states be categorized in two states We deal with fraction, proportion, or We deal with fraction, proportion, or
percent defectivespercent defectives Have several samples, each with Have several samples, each with
many observationsmany observations
Attribute DataAttribute Data
© 2008 Prentice Hall, Inc. S6 – 41
Which Control Chart to UseWhich Control Chart to Use
Using a c-Chart:Using a c-Chart: Observations are attributes whose Observations are attributes whose
defects per unit of output can be defects per unit of output can be countedcounted
The number counted is a small part of The number counted is a small part of the possible occurrencesthe possible occurrences
Defects such as number of blemishes Defects such as number of blemishes on a desk, number of typos in a page on a desk, number of typos in a page of text, flaws in a bolt of clothof text, flaws in a bolt of cloth
Attribute DataAttribute Data
© 2008 Prentice Hall, Inc. S6 – 42
Patterns in Control ChartsPatterns in Control Charts
Normal behavior. Normal behavior. Process is “in control.”Process is “in control.”
Upper control limitUpper control limit
TargetTarget
Lower control limitLower control limit
Figure S6.7Figure S6.7
© 2008 Prentice Hall, Inc. S6 – 43
Upper control limitUpper control limit
TargetTarget
Lower control limitLower control limit
Patterns in Control ChartsPatterns in Control Charts
One plot out above (or One plot out above (or below). Investigate for below). Investigate for cause. Process is “out cause. Process is “out of control.”of control.”
Figure S6.7Figure S6.7
© 2008 Prentice Hall, Inc. S6 – 44
Upper control limitUpper control limit
TargetTarget
Lower control limitLower control limit
Patterns in Control ChartsPatterns in Control Charts
Trends in either Trends in either direction, 5 plots. direction, 5 plots. Investigate for cause of Investigate for cause of progressive change.progressive change.
Figure S6.7Figure S6.7
© 2008 Prentice Hall, Inc. S6 – 45
Upper control limitUpper control limit
TargetTarget
Lower control limitLower control limit
Patterns in Control ChartsPatterns in Control Charts
Two plots very near Two plots very near lower (or upper) lower (or upper) control. Investigate for control. Investigate for cause.cause.
Figure S6.7Figure S6.7
© 2008 Prentice Hall, Inc. S6 – 46
Upper control limitUpper control limit
TargetTarget
Lower control limitLower control limit
Patterns in Control ChartsPatterns in Control Charts
Run of 5 above (or Run of 5 above (or below) central line. below) central line. Investigate for cause. Investigate for cause. Figure S6.7Figure S6.7
© 2008 Prentice Hall, Inc. S6 – 47
Upper control limitUpper control limit
TargetTarget
Lower control limitLower control limit
Patterns in Control ChartsPatterns in Control Charts
Erratic behavior. Erratic behavior. Investigate.Investigate.
Figure S6.7Figure S6.7