quality improvement

Quality Quality ImprovementImprovement

Quality Quality ImprovementImprovement

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Besterfield, Quality Improvement, 9eBesterfield, Quality Improvement, 9e

PowerPoint presentation to accompanyPowerPoint presentation to accompany

Besterfield, Quality Improvement, 9eBesterfield, Quality Improvement, 9e

Chapter 6- Control Chapter 6- Control Charts for VariablesCharts for VariablesChapter 6- Control Chapter 6- Control

Charts for VariablesCharts for Variables

Quality Improvement, 9eDale H. Besterfield

© 2013, 2008 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved

2

OutlineOutline

The Control Chart Techniques State of Introduction Control Specifications Process Capability Different Control Charts



3

Learning ObjectivesLearning Objectives

When you have completed this chapter you should:

Know the three categories of variation and their sources.

Understand the concept of the control chart method.

Know the purpose of variable control charts. Know how to select the quality characteristics,

the rational subgroup and the method of taking samples



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Be able to calculate the central value, trial control limits and the revised control limits for Xbar and

R chart. Be able to explain what is meant by a process in

control and the various out-of-control patterns. Know the difference between individual

measurements and averages; control limits and specifications.



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Know the different situations between the process spread and specifications and what can be done to correct the undesirable situation.

Be able to calculate process capability.



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The variation concept is a law of nature in that no two natural items are the same.

The variation may be quite large and easily noticeable

The variation may be very small. It may appear that items are identical; however, precision instruments will show difference

The ability to measure variation is necessary before it can be controlled

variationvariation



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There are three categories of variation in piece part production:

1. Within-piece variation: Surface

2. Piece-to-piece variation: Among pieces produced at the same time

3. Time-to-time variation: Difference in product produced at different times of the day

VariationVariation



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Materials

ToolsTools

OperatorsOperators MethodsMethods MeasurementMeasurement InstrumentsInstruments

HumanHumanInspectionInspectionPerformancePerformance

EnvironmentEnvironmentMachinesMachines

INPUTSINPUTS PROCESSPROCESS OUTPUTSOUTPUTS

VariationVariation

Sources of Variation in production processes:



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Sources of variation are:1. Equipment:

1. Toolwear2. Machine vibration3. Electrical fluctuations etc.

2. Material1. Tensile strength2. Ductility3. Thickness4. Porosity etc.

VariationVariation



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Sources of variation are:3. Environment

1. Temperature2. Light3. Radiation4. Humidity etc.

4. Operator1. Personal problem2. Physical problem etc.

VariationVariation



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Variable datax-bar and R-chartsx-bar and s-chartsCharts for individuals (x-charts)

Attribute dataFor “defectives” (p-chart, np-chart)For “defects” (c-chart, u-chart)

Control ChartsControl Charts



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ControlCharts

RChart

VariablesCharts

AttributesCharts

XChart

PChart

CChart

Continuous Numerical Data

Categorical or Discrete Numerical Data




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The control chart for variables is a means of visualizing the variations that occur in the central tendency and the mean of a set of observations. It shows whether or not a process is in a stable state.

Control Charts for VariablesControl Charts for Variables



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Figure 5-1 Example of a control chart



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Figure 6-1 Example of a method of reporting inspection results



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The objectives of the variable control charts are:

1. For quality improvement

2. To determine the process capability

3. For decisions regarding product specifications

4. For current decisions on the production process

5. For current decisions on recently produced items

Variable Control ChartsVariable Control Charts



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Procedure for establishing a pair of control charts for the average Xbar and the range R:

1. Select the quality characteristic

2. Choose the rational subgroup

3. Collect the data

4. Determine the trial center line and control limits

5. Establish the revised central line and control limits

6. Achieve the objective

Control Chart TechniquesControl Chart Techniques



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The Quality characteristic must be measurable. It can expressed in terms of the seven basic units:

1.Length2.Mass3.Time4.Electrical current5.Temperature6.Subatance7.Luminosity

Quality CharacteristicQuality Characteristic



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A rational subgroup is one in which the variation within a group is due only to chance causes.

Within-subgroup variation is used to determine the control limits.

Variation between subgroups is used to evaluate long-term stability.

Rational SubgroupRational Subgroup



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There are two schemes for selecting the subgroup samples:

1. Select subgroup samples from product or service produced at one instant of time or as close to that instant as possible

2. Select from product or service produced over a period of time that is representative of all the products or services




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The first scheme will have a minimum variation within a subgroup.The second scheme will have a minimum variation among subgroups.The first scheme is the most commonly used since it provides a particular time reference for determining assignable causes.The second scheme provides better overall results and will provide a more accurate picture of the quality.




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As the subgroup size increases, the control limits become closer to the central value, which make the control chart more sensitive to small variations in the process average

As the subgroup size increases, the inspection cost per subgroup increases

When destructive testing is used and the item is expensive, a small subgroup size is required

Subgroup SizeSubgroup Size



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From a statistical basis a distribution of subgroup averages are nearly normal for groups of 4 or more even when samples are taken from a non-normal distribution

When a subgroup size of 10 or more is used, the s chart should be used instead of the R chart. .

See Table 6-1 for (total) sample sizes

Subgroup SizeSubgroup Size



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Data collection can be accomplished using the type of figure shown in Figure 6-2.

It can also be collected using the method in Table 6-2.

It is necessary to collect a minimum of 25 subgroups of data.

A run chart can be used to analyze the data in the development stage of a product or prior to a state of statistical control

Data CollectionData Collection



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Run ChartRun Chart

Figure 6-4 Run Chart for data of Table 6-2



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Trial Central LinesTrial Central Lines

Central Lines are obtained using:

1 1

g g

i ii i

i

i

X RX and R

g g

where

X average of subgroup averages

X average of the ith subgroup

g number of subgroups

R average of subgroup ranges

R range of the ith subgroup



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Trial Control LimitsTrial Control Limits

Trial control limits are established at ±3 standard deviatons from the central value

3 3

3 3

R RX X

R RX X

X

R

UCL X UCL R

LCL X LCL R

where

UCL=upper control limit

LCL=lower control limit

population standard deviation of the subgroup averages

population standard deviation of the range



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In practice calculations are simplified by using the following equations where A2,D3

and D4 are factors that vary with the

subgroupsize and are found in Table B of the Appendix.

2 4

2 3

RX

RX

UCL X A R UCL D R

LCL X A R LCL D R



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Figure 6-5 Xbar and R chart for preliminary data with trial control limits



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Revised Central LinesRevised Central Lines

d dnew new

d d

d

d

d

X X R RX and R

g g g g

where

X discarded subgroup averages

g number of discarded subgroups

R discarded subgroup ranges



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Standard ValuesStandard Values

00 0 0

2

new newR

X X R R andd

0 0 2 0

0 0 1 0

RX

RX

UCL X A UCL D

LCL X A LCL D



33Figure 6-6 Trial control limits and revised control limits for Xbar and R charts



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Achieve the ObjectiveAchieve the Objective

Figure 5-7 Continuing use of control charts, showing improved quality



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Revised Central LinesRevised Central Lines

d dnew new

d d

d

d

d

X X R RX and R

g g g g

where

X discarded subgroup averages

g number of discarded subgroups

R discarded subgroup ranges

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Sample Standard Deviation Sample Standard Deviation Control ChartControl Chart

For subgroup sizes >=10, an s chart is more accurate than an R Chart.Trial control limits are given by:

1 1

3 4

3 3

g gi ii i

sX

sX

s Xs X

g g

UCL X A s UCL B s

LCL X A s LCL B s



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Revised Limits for s chartRevised Limits for s chart

0

00 0

4

0 0 6 0

0 0 5 0

4 5 6, , ,

dnew

d

dnew

d

sX

sX

d

X XX X

g g

s s ss s

g g c

UCL X A UCL B

LCL X A LCL B

where

s discarded subgroup averages

c A B B factors found in Table B



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Process in Control When special causes have been

eliminated from the process to the extent that the points plotted on the control chart remain within the control limits, the process is in a state of control

When a process is in control, there occurs a natural pattern of variation

State of ControlState of Control



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Figure 6-9 Natural pattern of variation of a control chart



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Types of errors: Type I, occurs when looking for a special

cause of variation when in reality a common cause is present

Type II, occurs when assuming that a common cause of variation is present when in reality there is a special cause




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When the process is in control:

1. Individual units of the product or service will be more uniform

2. Since the product is more uniform, fewer samples are needed to judge the quality

3. The process capability or spread of the process is easily attained from 6ơ

4. Trouble can be anticipated before it occurs




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When the process is in control:

5. The % of product that falls within any pair of values is more predictable

6. It allows the consumer to use the producer’s data

7. It is an indication that the operator is performing satisfactorily


Common Common CausesCauses

Special Special CausesCauses

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Figure 6-11 Frequency Distribution of subgroup averages with control limits



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When a point (subgroup value) falls outside its control limits, the process is out of control.

Out of control means a change in the process due to a special or assignable cause.A process can also be considered out of control even when the points fall inside the 3ơ limits




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It is not natural for seven or more consecutive points to be above or below the central line.

Also when 10 out of 11 points or 12 out of 14 points are located on one side of the central line, it is unnatural.

Six points in a row are steadily increasing or decreasing indicate an out of control situation




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Patterns in Control ChartsPatterns in Control Charts

Figure 6-12 Some unnatural runs-process out of control




Simplified rule: Divide space into two equal zones of 1.5σ.

Out of control occurs when two consecutive points are beyond 1.5σ.

See Figure 6-13

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Patterns in Control ChartsPatterns in Control Charts

Figure 6-13 Simplified rule for out-of-control pattern



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1. Change or jump in level.

2. Trend or steady change in level

3. Recurring cycles

4. Two populations (also called mixture)

5. Mistakes

Out-of-Control ConditionOut-of-Control Condition



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Out-of-Control PatternsOut-of-Control Patterns

Change or jump inlevel Trend or steady change in level

Recurring cycles Two populations



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SpecificationsSpecifications

Figure 5-18 Comparison of individual values compared to averages



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Calculations of the average for both the individual values and for the subgroup avergaes are the same. However the sample standard deviation is different.

SpecificationsSpecifications

X

X

nwhere

population standard deviation of subgroup averages

population standard deviation of individual values

n=subgroup size

If we assume normality, then the population standard deviation

can be

4

sestimated from

c



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If the population from which samples are taken is not normal, the distribution of sample averages will tend toward normality provided that the sample size, n, is at least 4. This tendency gets better and better as the sample size gets larger. The standardized normal can be used for the distribution averages with the modification.

Central Limit TheoremCentral Limit Theorem

X

X XZ

n



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Figure 6-19 Illustration of central limit theorem



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Figure 6-20 Dice illustration of central limit theorem



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The control limits are established as a function of the average

Specifications are the permissible variation in the size of the part and are, therefore, for individual values

The specifications or tolerance limits are established by design engineers to meet a particular function

Control Limits & Control Limits & SpecificationsSpecifications



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Figure 6-21 Relationship of limits, specifications, and distributions

Control Limits & Control Limits & SpecificationsSpecifications



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The process spread will be referred to as the process capability and is equal to 6σ

The difference between specifications is called the tolerance

When the tolerance is established by the design engineer without regard to the spread of the process, undesirable situations can result

Process Capability & Process Capability & ToleranceTolerance



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Three situations are possible: Case I: When the process capability is

less than the tolerance 6σ<USL-LSL Case II: When the process capability is

equal to the tolerance 6σ=USL-LSL Case III: When the process capability is

greater than the tolerance 6σ >USL-LSL




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Case I: When the process capability is less than the tolerance 6σ<USL-LSL


Figure 6-24 Case I 6σ<USL-LSL



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Case II: When the process capability is equal to the tolerance 6σ=USL-LSL


Figure 6-24 Case I 6σ=USL-LSL



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Case III: When the process capability is less than the tolerance 6σ>USL-LSL


Figure 6-24 Case I 6σ>USL-LSL



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The range over which the natural variation of a process occurs as determined by the system of common or random causes

Measured by the proportion of output that can be produced within design specifications

Process CapabilityProcess Capability



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This following method of calculating the process capability assumes that the process is stable or in statistical control:

Take 25 (g) subgroups of size 4 for a total of 100 measurements

Calculate the range, R, for each subgroup

Calculate the average range, RBar= ΣR/g

Calculate the estimate of the population standard deviation

Process capability will equal 6σ0


0

2

R

d



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The process capability can also be obtained by using the standard deviation:

Take 25 (g) subgroups of size 4 for a total of 100 measurements Calculate the sample standard deviation, s, for each subgroup Calculate the average sample standard deviation, sbar = Σs/g Calculate the estimate of the population

standard deviation Process capability will equal 6σo


0

4

s

c



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Process capability and tolerance are combined to form the capability index.

Capability IndexCapability Index

0

0

6

6

p

p

USL LSLC

where C capabilityindex

USL LSL tolerance

process capability



68

The capability index does not measure process performance in terms of the nominal or target value. This measure is accomplished by Cpk.


0

{( ) ( )

3

6

pk

p

Min USL X or X LSLC

where C capabilityindex

USL LSL tolerance

process capability



69

Cp = USL - LSL 6 ơo

(USL- ¯X), (¯X-LSL)} Cpk = min{

The Capability Index does not measureprocess performance in terms of the nominal or target




70

1. The Cp value does not change as the process center changes

2. Cp=Cpk when the process is centered3. Cpk is always equal to or less than Cp4. A Cpk = 1 indicates that the process is

producing product that conforms to specifications

5. A Cpk < 1 indicates that the process is producing product that does not conform to specifications




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6. A Cp < 1 indicates that the process is not capable

7. A Cpk=0 indicates the average is equal to one of the specification limits

8. A negative Cpk value indicates that the average is outside the specifications




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Cpk = negative number

Cpk = zero

Cpk = between 0 and 1

Cpk = 1

Cpk > 1

CCpkpk Measures Measures



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Charts for Better Operator Understanding:

1. Placing individual values on the chart: This technique plots both the individual values and the subgroup average. Not recommended since it does not provide much information.

2. Chart for subgroup sums: This technique plots the subgroup sum, ΣX, rather than the group average, Xbar.

Different Control ChartsDifferent Control Charts

( )

( )

X X

X X

UCL n UCL

UCL n LCL



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FIGURE 6-27 Chart forIndividual Values & Subgroup Averages

FIGURE 6-28 Subgroup Sum Chart



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Charts for Variable Subgroup Size:

Used when the sample size is not the same Different control limits for each subgroup As n increases, limits become narrower As n decreases, limits become wider apart Difficult to interpret and explain To be avoided




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FIGURE 6-29 Chart for Variable Subgroup Size



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Chart for Trends:

Used when the plotted points have an upward or downward trend that can be attributed to an unnatural pattern of variation or a natural pattern such as tool wear.

The central line is on a slope, therefore its equation must be determined.




79

Chart for TrendsChart for Trends

Figure 6-32 Chart for Trend

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Used when we cannot have multiple observations per time period

ValueValue XbarXbar RR

4444

4646

5454 48.0048.00 1010

3838 46.0046.00 1616

4949 47.0047.00 1616

4646 44.3344.33 1111

4545 46.6746.67 44

3131 40.6740.67 1515

5555 43.6743.67 2424

3737 41.0041.00 2424

4242 44.6744.67 1818

4343 40.6740.67 66

4747 44.0044.00 55

5151 47.0047.00 88

XX

n

RR

n

NOTE: n here is equal to 12, NOT 14

Chart for Moving Average Chart for Moving Average and and

Moving RangeMoving Range

An example

81

Extreme readings have a greater effect than in conventional charts. An extreme value is used several times in the calculations, the number of times depends on the averaging period.

Chart for Moving Average Chart for Moving Average and Moving Rangeand Moving Range



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This is a simplified variable control chart. Minimizes calculations Easier to understand Can be easily maintained by operators Recommended to use a subgroup of 3,

then all data is used.

Chart for Median and RangeChart for Median and Range



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5

5

6

5

MD Md Md

MD Md Md

R Md

R Md

UCL Md A R

LCL Md A R

UCL D R

LCL D R

For Table for A5, D5 and D6 see page 230For Table for A5, D5 and D6 see page 230





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FIGURE 6-31 Control Charts for Median and Range

FIGURE 6-31 Control Charts for Median and Range



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Used when only one measurement is taken on quality characteristic

Too expensiveTime consumingDestructiveVery few items

Chart for Individual valuesChart for Individual values



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2.660

2.660

3.267

(0)

x

x

R

R

X RX R

g g

UCL X R

LCL X R

UCL R

LCL R

To use those equations, you have to use a moving range with n=2To use those equations, you have to use a moving range with n=2

Chart for Individual ValuesChart for Individual Values



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0 0

0 0

0 0

0

0

3

3

3.686

(0)

new new

x

x

R

R

X X R R

UCL X

LCL X

UCL R

LCL


Revised Limits:




88

FIGURE 6-32 Control Charts for Individual Values and Moving Range

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Charts with Non-Acceptance Charts with Non-Acceptance LimitsLimits

Non-Acceptance limits have the same Relationship to averages as specificationshave to individual values. Control Limits tell what the process is capable of doing, and reject limits tell when the product is conforming to specifications.

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Charts with Non-Acceptance Charts with Non-Acceptance LimitsLimits

Figure 6-35 Relationship of non-acceptance limits, control limitsand specifications.



Exponential Weighted Exponential Weighted AverageAverage

Gives greatest weight to most recent values The EWMA is defined by the euqation Vt = lXt + 11 - l2Vt-1 where V t = the EWMA of the most recent

plotted point V t− 1 = the EWMA of the previous plotted

point l = the weight given to the subgroup average

or individual value Xt = the subgroup average or individual value

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Gives greatest weight to most recent values The EWMA is defined by the euqation Vt = λ Xbart + (1 – λ) Vt-1

where Vt = most recent plotted point Vt−1 = previous plotted point λ= weight given to subgroup

average or individual value Xbar = the subgroup average or

individual value

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UCL = Xdbar + A2Rbar(((SqRt(λ/(2 – λ)))

LCL = Xdbar - A2Rbar(((SqRt(λ/(2 – λ)))

93



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Exponential Weighted

Average



Computer ProgramComputer Program

Computer Program file names are:Xbar and RMd and RX and MREWMAProcess Capability

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Documents

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common cause of variation