control charts for variables 1 introduction variable - a single quality characteristic that can be...

Control Charts for Variables 1

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.


Control Charts for and R (1)

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


Control Charts for and R(2)

Notation and values

• Ri = range of the values in the ith sample

Ri = xmax - xmin

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

x

R



Statistical Basis of the Charts

• Assume the quality characteristic of interest is normally distributed with mean , and standard deviation, .

• If x1, x2, …, xn is a sample of size n, then he average of this sample is

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

x

n

xxxx n21

x

n/x



Statistical Basis of the Charts• The probability is 1 - 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.

x

nZZ

andn

ZZ

2/x2/

2/x2/



Control Limits for the chart

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

x

x

RAxLCL

xLineCenter

RAxUCL

2

2



Control Limits for the R chart

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

x

UCL D R

Center Line R

LCL D R

4

3



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

x

R

d2

Relative range ： W=R/σ

E(W)=E(R/σ)=d2


Control Charts for and R (8)x

σR=d3σ

R=Wσ

Var(W)=d32

σR=d3(R/d2)^


Control Charts for and R (9)Trial Control Limits

• The control limits obtained from equations 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.

x


Control Charts for and R(10)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. If future samples still indicate control, then the unexplained points can probably be safely dropped.

x



• Before revising, identify out of control points and look for assignable causes.– When many of the initial samples plot out of

control against the trial limits, (as few data will remain which we can recompute reliable control limits.) it is better to concentrate on the pattern formed by these points.

x



• When setting up and R control charts, it is best to begin with the R chart. Because the control limits on the chart depend on the process variability, unless process variability is in control, these limits will not have much meaning.

x

x

x


Example 5-1


Estimating Process Capability

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


The fraction of nonconforming of Example 5-1

• Assumes a stable process.• 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)


製程能力 (process capability)

• 意義– 對於穩定之製程所持有之特定成果，能夠合

理達成之能力界限。• 製程能力的表示法

– 製程準確度 (Ca 值 )– 製程精密度 (Cp 值 ) 製程能力比 (process

capability ratio ， PCR)– 製程能力指數 (Cpk 值 )


製程準確度 Ca(capability of accuracy)

Ca=製程平均值 - 規格中心值

規格公差 /2

=X-μT/2

=

T ：規格公差 = 規格上限 - 規格下限


製程準確度之評價方法• A 級

– 小於等於 12.5% 理想的狀態，故維持現狀。• B 級

– 大於 12.5% ，小於等於 25% 儘可能調整改進至 A級。

• C 級– 大於 25% ，小於等於 50% 應立即檢討並加以改善

• D 級– 大於 50% ，小於 100% 採取緊急措施，並全面檢

討，必要時應考停止生產。


製程精密度 Cp(process capability ratio ， PCR)

雙邊規格：

Cp=規格公差

6 個估計標準差=

T6σ

單邊規格：

Cp=規格上限 - 製程平均值


SU-X3σ

=

Cp=製程平均值 - 規格下限


X-SL3σ

或 =


製程精密度評價方式• A+ 級

– 大於等於 1.67 可考慮放寛產品變異，尋求管理的簡單化或成本降低方法。

• A 級– 小於 1.67 ，大於等於 1.33 理想的狀態，故維持現狀。

• B 級– 小於 1.33 ，大於等於 1.00 確實進行製程管制，儘可能改善為

A 級。• C 級

– 小於 1.00 ，大於等 0.67 產生不良品，產品須全數選別，並管理改善製程。

• D 級– 小於 0.67 採取緊急對策，進行品質改善，探求原因並重新檢

討規格。


製程能力指數 Cpk

Cpk=(1-|Ca|)Cp

當 Ca=0 時 Cpk=Cp

= =Cpk=min{ SU-X

3σ ，X-SL

3σ }


製程能力指數評價方式• A 級

– 大於等於 1.33 製程能力足夠• B 級

– 小於 1.33 ，大於等於 1.00 能力尚可應再努力

• C 級– 小於 1.00 應加以改善


Process Capability Ratios (Cp)(assume the process is centered at the

midpoint of the specification band)

If Cp > 1, then a low number 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.


The percentage of the specification band

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

%100C

1P̂

p


Revision of Control Limits and Center Line

• The effective use of control chart will require periodic revision of the control limits and center lines.– Every week, every month, or every 25, 50, or 100

samples.

– When revising control limits, it is highly desirable to use at least 25 samples or subgroups in computing control limits.

• If the R chart exhibits control, the center line of the Chart will be replaced with a target value. This can be helpful in shifting the process average to the desired value.

X


Phase II Operation of Charts

• Use of control chart for monitoring future production, once a set of reliable limits are established, is called phase II of control chart usage

• A run chart showing individuals observations in each sample, called a tolerance chart or tier diagram, may reveal patterns or unusual observations in the data


Control Limits, Specification Limits, and Natural Tolerance Limits(1)

• 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.



• 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.


Rational Subgroups

• X bar chart monitors the between sample variability (variability in the process over time).– Sample should be selected in such a way that

maximizes the chances for shifts in the process average to occur between samples.

• R chart monitors the within sample variability (the instantaneous process variability at a given time).– Samples should be selected so that variability within

samples measures only chance or random causes.


Guidelines for the Design of the Control Chart(1)

• 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



• 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 for small samples. The range method becomes less effective as the sample size increases, for large n, may want to use S or S2 chart.

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



Allocating 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.

• For economic considerations, if the cost associated with producing defective items is high, smaller, more frequent samples are better than larger, less frequent ones.



• If the rate of production is high, then more frequent sampling is better.

• If false alarms or type I errors are very expensive to investigate, then it may be best to use wider control limits than three-sigma.

• If the process is such that out-of-control signals are quickly and easily investigated with a minimum of lost time and cost, then narrower control limits are appropriate.


Changing Sample Size on the and R Charts1

• 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:– Variable sample size.– A permanent change in the sample size

because of cost or because the process has exhibited good stability and fewer resources are being allocated for process monitoring.

x



• = 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

oldR

newR



Control Limits

x

old2

22

old2

22

R)old(d

)new(dAxLCL

R)old(d

)new(dAxUCL

chartx

old2

23

old2

2new

old2

24

R)old(d

)new(dD,0maxUCL

R)old(d

)new(dRCL

R)old(d

)new(dDUCL

chartR

A2 、 D3 、 D4 are selected for the new sample size


Example 5-2


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

ALCL

CL

AUCL

chartX

1

2

2

DLCL

dCL

DUCL

chartR


Interpretation of and R Charts • In interpreting patterns on the X bar chart,

we must first determine whether or not the R chart is in control.

• 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


The Effects of Nonnormality

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

• In most cases, samples of size four or five are sufficient to ensure reasonable robustness to the normality assumption.

x


The Effects of Nonnormality R1

• 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).

x

x


The Operating Characteristic Function(1)• 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. The probability of this occurring is called the -risk.

x


The Operating Characteristic Function(2)

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

L= multiple of standard error in the control limits

k = shift in process mean (#of standard deviations).

)nkL()nkL(


The Operating Characteristic Function(3)• The operating characteristic curves are plots of

the value against k for various sample sizes.


The Operating Characteristic Function(4)

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


Average run length ， ARL

In-control

ARL0=1/α

Out-of-control

ARL1=1/(1-)


Average time to signal ， ATS and

Expected number of individual units sampled ， I

ATS=ARLh

I=nARL


Construction and Operation of and S Charts(1)

• First, S2 is an “unbiased” estimator of 2

• Second, S is NOT an unbiased estimator of

• S is an unbiased estimator of c4

where c4 is a constant

• The standard deviation of S is

x

24c1



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

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

x

5

4

6

BLCL

cCL

BUCL



No Standard σGiven

• If is unknown, we can use an average

sample standard deviation,

x

m

1iiS

m

1S

SBLCL

SCL

SBUCL

3

4



Chart when Using S

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

where A3 is found in the Appendix

x

x

UCL x A S

CL x

LCL x A S

3

3

x



Estimating Process Standard Deviation

• The process standard deviation, can be estimated by

x

S

c4


Example 5-3


The and S Control Charts with Variable Sample Size(1)

• 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


The and S Control Charts with Variable Sample Size(2)

• The grand average can be estimated as:

• The average sample standard deviation is:

x

m

i ii 1

m

ii 1

n xx

n

2/1

1

1

21

m

ii

m

iii

mn

SnS


The and S Control Charts with Variable Sample Size

• 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)

x

SAxLCL

xCL

SAxUCL

3

3

SBLCL

SCL

SBUCL

3

4


The 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.

21n),2/(1

2

2

21n,2/

2

1n

SLCL

SCL

1n

SUCL

21n,2/ 2

1n),2/(1


The Shewhart Control Chart for Individual Measurements(1)• 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. There is no basis for rational subgrouping.

– 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.


個別值與移動全距管制圖 (2)

• 使用理用– 產品需經很久的時間才能製造完成– 分析或測定一件產品的品質較為麻煩且費時– 產品係非常貴重的物品– 產品品質頗為均勻 ( 自動化生產 )– 屬破壞性試驗– 在於管制製造條件如溫度、壓力、濕度等


The Shewhart Control Chart for Individual Measurements(3)

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.



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

observations. The control limits are

where

2

2

d

MR3xLCL

xCL

d

MR3xUCL

m

MRMR

m

1ii



X and Moving Range Charts

• The control limits on the moving range chart are:

0LCL

MRCL

MRDUCL 4


Example 5-5



Example Ten successive heats of a steel alloy are tested

for hardness. The resulting data areHeat Hardness Heat Hardness 1 52 6 52 2 51 7 50 3 54 8 51 4 55 9 58

5 50 10 51



Example

109876543210

62

52

42

Observation

Indi

vid

uals

X=52.40

3.0SL=60.97

-3.0SL=43.83

10

5

0Mov

ing

Ran

ge

R=3.222

3.0SL=10.53

-3.0SL=0.000

I and MR Chart for hardness



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 X chart.

xx



Average Run Lengths

The ability of the individuals control chart to detect small shifts is very poor.

Size of Shift β ARL1

1σ 0.9772 43.96

2σ 0.8413 6.03

3σ 0.5000 2.00



• 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

.001P

roba

bilit

y

hardness

Normal Probability Plot

control charts for variables 1 introduction variable - a single quality characteristic that can be...

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