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Description of how to build control charts for variables

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Page 1: 07 Control Charts for Variables

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Control Charts for Variables

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Control Charts for Variables

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Introduction

• Variable - a single quality characteristic that can be measured on a numerical scale.

• When working with variables, we should monitor both the mean value of the characteristic and the variability associated with the characteristic.

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Control Charts for and R

Notation for variables control charts• n - size of the sample (sometimes called a

subgroup) chosen at a point in time• m - number of samples selected• = average of the observations in the ith sample

(where i = 1, 2, ..., m)• = grand average or “average of the averages

(this value is used as the center line of the control chart)

x

ix

x

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Notation and values• Ri = range of the values in the ith sample

Ri = xmax - xmin

• = average range for all m samples• µ is the true process mean• σ is the true process standard deviation

R

Control Charts for and Rx

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Statistical Basis of the Charts• Assume the quality characteristic of interest is normally

distributed with mean µ, and standard deviation, σ.• If x1, x2, …, xn is a sample of size n, then the average of

this sample is

• is normally distributed with mean, µ, and standard deviation,

nxxxx n+++

=L21

xnx /σσ =

Control Charts for and Rx

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Statistical Basis of the Charts• The probability is 1 - α that any sample mean will fall

between

• The above can be used as upper and lower control limits on a control chart for sample means, if the process parameters are known.

nZZ

nZZ

x

x

σµσµ

σµσµ

αα

αα

2/2/

2/2/

and

−=−

+=+

Control Charts for and Rx

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Estimating the Process Standard Deviation• The process standard deviation can be estimated

using a function of the sample average range.

• This is an unbiased estimator of σ2

ˆdR

Control Charts for and Rx

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Control Limits for the chart

• A2 is found in Appendix VI for various values of n.

x

RAxLCLx

RAxUCL

2

2

LineCenter−=

=

+=

Control Charts for and Rx

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Control Limits for the R chart

• D3 and D4 are found in Appendix VI for various values of n.

RDLCLR

RDUCL

3

4

LineCenter=

=

=

Control Charts for and Rx

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Trial Control Limits• The control limits obtained from equations (5-4) and (5-5)

should be treated as trial control limits.• If this process is in control for the m samples collected,

then the system was in control in the past.• If all points plot inside the control limits and no systematic

behavior is identified, then the process was in control in the past, and the trial control limits are suitable for controlling current or future production.

Control Charts for and Rx

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Trial control limits and the out-of-control process• If points plot out of control, then the control limits

must be revised.• Before revising, identify out of control points and

look for assignable causes.– If assignable causes can be found, then discard the

point(s) and recalculate the control limits.– If no assignable causes can be found then 1) either

discard the point(s) as if an assignable cause had been found or 2) retain the point(s) considering the trial control limits as appropriate for current control.

Control Charts for and Rx

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Estimating Process Capability

• The x-bar and R charts give information about the capability of the process relative to its specification limits.

• Assumes a stable process.• We can estimate the fraction of nonconforming items for

any process where specification limits are involved.• Assume the process is normally distributed, and x is

normally distributed, the fraction nonconforming can be found by solving:

P(x < LSL) + P(x > USL)

Control Charts for and Rx

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Process-Capability Ratios (Cp)• Used to express process capability.• For processes with both upper and lower control limits,

Use an estimate of σ if it is unknown.

• If Cp > 1, then a low # of nonconforming items will be produced.

• If Cp = 1, (assume norm. dist) then we are producing about 0.27% nonconforming.

• If Cp < 1, then a large number of nonconforming items are being produced.

σ6LSLUSLCp

−=

Control Charts for and Rx

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Control Charts for and RProcess-Capability Ratios (Cp)• The percentage of the specification band that the process

uses up is denoted by

**The Cp statistic assumes that the process mean is centered at the midpoint of the specification band – it measures potential capability.

x

%1001ˆ

=

CPCP

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Control Limits, Specification Limits, and Natural Tolerance Limits

• Control limits are functions of the natural variability of the process

• Natural tolerance limits represent the natural variability of the process (usually set at 3-sigma from the mean)

• Specification limits are determined by developers/designers.

Control Charts for and Rx

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Control Limits, Specification Limits, and Natural Tolerance Limits

• There is no mathematical relationship between control limits and specification limits.

• Do not plot specification limits on the charts– Causes confusion between control and capability– If individual observations are plotted, then specification

limits may be plotted on the chart.

Control Charts for and Rx

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Rational Subgroups

• X-bar chart monitors the between sample variability

• R chart monitors the within sample variability.

Control Charts for and Rx

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Guidelines for the Design of the Control Chart• Specify sample size, control limit width, and

frequency of sampling• if the main purpose of the X-bar chart is to detect

moderate to large process shifts, then small sample sizes are sufficient (n = 4, 5, or 6)

• if the main purpose of the X-bar chart is to detect small process shifts, larger sample sizes are needed (as much as 15 to 25)…which is often impractical…alternative types of control charts are available for this situation…see Chapter 8

Control Charts for and Rx

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Guidelines for the Design of the Control Chart

• If increasing the sample size is not an option, then sensitizing procedures (such as warning limits) can be used to detect small shifts…but this can result in increased false alarms.

• R chart is insensitive to shifts in process standard deviation.(the range method becomes less effective as the sample size increases) may want to use S or S2 chart.

• The OC curve can be helpful in determining an appropriate sample size.

Control Charts for and Rx

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Guidelines for the Design of the Control ChartAllocating Sampling Effort• Choose a larger sample size and sample less

frequently? or, Choose a smaller sample size and sample more frequently?

• The method to use will depend on the situation. In general, small frequent samples are more desirable.

Control Charts for and Rx

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Changing Sample Size on the and R Charts• In some situations, it may be of interest to know the effect

of changing the sample size on the X-bar and R charts. Needed information:

• = average range for the old sample size• = average range for the new sample size• nold = old sample size• nnew = new sample size• d2(old) = factor d2 for the old sample size• d2(new) = factor d2 for the new sample size

x

oldRnewR

Control Charts for and Rx

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Changing Sample Size on the and R ChartsControl Limits

old2

22

old2

22

)old()new(

)old()new(

chart

RddAxLCL

RddAxUCL

x

−=

+=

=

==

=

old2

23

old2

2new

old2

24

)old()new(,0max

)old()new(

)old()new(

chart

RddDUCL

RddRCL

RddDUCL

R

Control Charts for and Rxx

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Charts Based on Standard Values

• If the process mean and variance are known or can be specified, then control limits can be developed using these values:

• Constants are tabulated in Appendix VI

chart−X chart−R

σµµ

σµ

ALCLCL

AUCL

−==

+=

σσσ

1

2

2

DLCLdCLDUCL

===

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Interpretation of and R Charts

• Patterns of the plotted points will provide useful diagnostic information on the process, and this information can be used to make process modifications that reduce variability.– Cyclic Patterns – Mixture– Shift in process level– Trend – Stratification

x

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The Effects of Nonnormality and R

• In general, the chart is insensitive (robust) to small departures from normality.

• The R chart is more sensitive to nonnormality than the chart

• For 3-sigma limits, the probability of committing a type I error is 0.00461on the R-chart. (Recall that for , the probability is only 0.0027).

xx

x

x

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Operating Characteristic Function

• How well the and R charts can detect process shifts is described by operating characteristic (OC) curves.

• Consider a process whose mean has shifted from an in-control value by k standard deviations. If the next sample after the shift plots in-control, then you will not detect the shift in the mean.

x

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• The probability of not detecting a shift in the process mean on the first sample is

L= multiple of standard error in the control limitsk = shift in process mean (# of standard deviations).

)()( nkLnkL −−Φ−−Φ=β

Operating Characteristic Function

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• The operating characteristic curves are plots of the value β against k for various sample sizes.

Operating Characteristic Function

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• If β is the probability of not detecting the shift on the next sample, then 1 - β is the probability of correctly detecting the shift on the next sample.

Operating Characteristic Function

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Construction and Operation of and S Charts

• First, S2 is an “unbiased” estimator of σ2

• Second, S is NOT an unbiased estimator of σ• S is an unbiased estimator of c4 σ

where c4 is a constant• The standard deviation of S is

x

241 c−σ

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• If a standard σ is given the control limits for the S chart are:

• B5, B6, and c4 are found in the Appendix for various values of n.

σσσ

5

4

6

BLCLcCLBUCL

===

Construction and Operation of and S Chartsx

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No Standard Given• If σ is unknown, we can use an average sample standard deviation, ∑

=

=m

iiS

mS

1

1

SBLCLSCL

SBUCL

3

4

=

=

=

Construction and Operation of and S Chartsx

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Chart when Using S

The upper and lower control limits for the chart are given as

where A3 is found in the Appendix

x

SAxLCLxCL

SAxUCL

3

3

−=

=

+=

Construction and Operation of and S Chartsx

x

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Estimating Process Standard Deviation• The process standard deviation, σ can be

estimated by

4

ˆcS

Construction and Operation of and S Chartsx

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and S Control Chartswith Variable Sample Size

• The and S charts can be adjusted to account for samples of various sizes.

• A “weighted” average is used in the calculations of the statistics.

m = the number of samples selected. ni = size of the ith sample

x

x

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• The grand average can be estimated as:

• The average sample standard deviation is:

=

== m

ii

m

iii

n

xnx

1

1

21

1

1

2)1(

−=

=

=m

ii

m

iii

mn

SnS

and S Control Chartswith Variable Sample Size

x

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• Control Limits

• If the sample sizes are not equivalent for each sample, then – there can be control limits for each point (control limits

may differ for each point plotted)

SAxLCLxCL

SAxUCL

3

3

−=

=

+=

SBLCLSCL

SBUCL

3

4

=

=

=

and S Control Chartswith Variable Sample Size

x

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S2 Control Chart

• There may be situations where the process variance itself is monitored. An S2 chart is

where and are points found from the chi-square distribution.

21),2/(1

2

2

21,2/

2

1

1

−−

−=

=−

=

n

n

nSLCL

SCLnSUCL

α

α

χ

χ

21,2/ −nαχ

21),2/(1 −− nαχ

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Shewhart Control Chartfor Individual Measurements

• What if you could not get a sample size greater than 1 (n =1)? Examples include– Automated inspection and measurement technology is

used, and every unit manufactured is analyzed.– The production rate is very slow, and it is inconvenient

to allow samples sizes of N > 1 to accumulate before analysis

– Repeat measurements on the process differ only because of laboratory or analysis error, as in many chemical processes.

• The X and MR charts are useful for samples of sizes n = 1.

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Moving Range Chart• The moving range (MR) is defined as the

absolute difference between two successive observations:

MRi = |xi - xi-1|which will indicate possible shifts or changes in the process from one observation to the next.

Shewhart Control Chartfor Individual Measurements

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X and Moving Range Charts• The X chart is the plot of the individual observations. The

control limits are

where

2

2

3

3

dMRxLCL

xCLd

MRxUCL

−=

=

+=

m

MRMR

m

ii∑

== 1

Shewhart Control Chartfor Individual Measurements

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X and Moving Range Charts• The control limits on the moving range

chart are:

0

4

==

=

LCLMRCL

MRDUCL

Shewhart Control Chartfor Individual Measurements

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ExampleTen successive heats of a steel alloy are tested for hardness. The resulting data are

Heat Hardness Heat Hardness1 52 6 522 51 7 503 54 8 514 55 9 585 50 10 51

Shewhart Control Chartfor Individual Measurements

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Example

109876543210

62

52

42Observation

Indiv

iduals

X=52.40

3.0SL=60.97

-3.0SL=43.83

10

5

0Movi

ng R

ange

R=3.222

3.0SL=10.53

-3.0SL=0.000

I and MR Chart for hardness

Shewhart Control Chartfor Individual Measurements

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Interpretation of the Charts• X Charts can be interpreted similar to charts. MR charts

cannot be interpreted the same as or R charts.• Since the MR chart plots data that are “correlated” with

one another, then looking for patterns on the chart does not make sense.

• MR chart cannot really supply useful information about process variability.

• More emphasis should be placed on interpretation of the Xchart.

xx

Shewhart Control Chartfor Individual Measurements

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• The normality assumption is often taken for granted.

• When using the individuals chart, the normality assumption is very important to chart performance.

P-Value: 0.063A-Squared: 0.648

Anderson-Darling Normality Test

N: 10StDev: 2.54733Average: 52.4

585756555453525150

.999

.99

.95

.80

.50

.20

.05

.01

.001Pr

obab

ility

hardness

Normal Probability Plot

Shewhart Control Chartfor Individual Measurements