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SMUEMIS 7364

NTUTO-570-N

More Control Charts Material Updated: 3/24/04

Statistical Quality ControlDr. Jerrell T. Stracener, SAE Fellow

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Operating Characteristic (OC) Function for the x - Chart

• The OC curve describes the ability of the x-chart to detect shifts in process quality.

• For an x-chart with known & constant mean shifts from in-control value, 0 to another value 1, where

1 = 0 + K

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Operating Characteristic (OC) Function for the x – Chart (continued)

μβμOC

σ

KσμLσnμΦ

σ

KσμLσnμΦ

nσ

Kσμ-LCLΦ

nσ

Kσμ-UCLΦ

Kσμμμ|UCLXLCLP

μ)|sample subsequent

first on theshift detectingP(not

00

00

00

01

n

n

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Operating Characteristic (OC) Function for the x – Chart (continued)

where

andL is usually 3, the three-sigma limits

dzey 2

zy 2

2π

1Φ

,nkLΦnkLΦ

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Example

If n=5 & L=3, determine & plot the OC function vs K, where 1= 0 + K.

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Example - Solution

KβKOC 53Φ53Φ

ΦΦ

kk

nkLnkL

k b

-3.0 0.000104397-2.5 0.00479646-2.0 0.070492119-1.5 0.361631295-1.0 0.777546112-0.5 0.9700606330.0 0.9973000660.5 0.9700606331.0 0.7775461121.5 0.3616312952.0 0.0704921192.5 0.004796463.0 0.000104397

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-4.0 -2.0 0.0 2.0 4.0 6.0

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OC Function of the Fraction Nonconforming Control Chart

pβpOC

nLCLFnUCLF

p1pd

np1p

d

n

p|nLCLDPp|nUCLDP

p|LCLp̂Pp|UCLp̂P

p)|control lstatisticain is process a

that hypothesis thegP(acceptin

nLCL

0d

dndnUCL

0d

dnd

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OC Function of the Fraction Nonconforming Control Chart

Where

[nUCL] denotes the largest integer nUCLand <nLCL> denotes the smallest integer nLCL

Note: The OC curve provides a measure of the sensitivity of the control chart – i.e., its ability to detect a shift in the process fraction nonconforming from the nominal value p to some other value p.

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Example

For a fraction nonconforming control chart with parameters

n = 50,LCL = 0.0303,

andUCL = 0.3697,

Determine & plot the OC curve.

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Example - Solution

pβpOC p|52.1DPp|49.81DP

p|0.030350DPp|0.369750DP

p P(D<=18|p) P(D<=1|p) P(D<=18|p) - P(D<=1|p)0.01 1.0000 0.9106 0.08940.03 1.0000 0.5553 0.44470.05 1.0000 0.2794 0.72060.10 1.0000 0.0338 0.96620.15 0.9999 0.0291 0.97080.20 0.9975 0.0002 0.99730.25 0.9713 0.0001 0.97120.30 0.8594 0.0000 0.85940.35 0.6216 0.0000 0.62160.40 0.3356 0.0000 0.33560.45 0.1273 0.0000 0.12730.50 0.0325 0.0000 0.03250.55 0.0053 0.0000 0.0053

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Example - Solution

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

p

OC(p)

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OC Function for c-charts and u-charts

• For the c-chart cβcOC

LCLFUCLF

!!

c|LCLXPc|UCLXP

c)|control lstatisticain is process a

that hypothesis thegP(acceptin

LCL

0d

UCL

0d

x

ce

x

ce xcxc

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OC Function for c-charts and u-charts

• For the u-chart

uβuOC

nLCL xwhere

!x

u|nLCLXPu|nUCLXP

u)|control lstatisticain is process a

that hypothesis thegP(acceptin

nUCL

0d

x

nu nue

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Example

Determine & plot the OC function for a u-chart with parameter.

LCL = 6.48,and

UCL = 32.22.

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Example - Solution

uβuOC

nUCL

0d !x

u|UCLxUCLP

u|UCLcPu|UCLcP

u|UCLxPu|UCLxP

xnu nue

nn

nn

u P(D<=33|c) P(D<=6|c) P(D<=33|c) - P(D<=6|c)0.01 1.000 0.999 0.0010.03 1.000 0.996 0.0040.05 1.000 0.762 0.2380.10 1.000 0.450 0.5500.15 1.000 0.220 0.7800.20 0.999 0.008 0.9910.25 0.997 0.000 0.9970.30 0.950 0.000 0.9500.35 0.744 0.000 0.7440.40 0.546 0.000 0.5460.45 0.410 0.000 0.4100.50 0.151 0.000 0.1510.55 0.038 0.000 0.038

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-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Example - Solution

u

OC(u)

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Average Run Length for x-Charts

• Performance of Control Charts can be characterized by their run length distribution.

• Run Length (RL) of a control chart is defined to be the number of samples until the process characteristic exceeds the control limits for the first time.

• Run Length, RL, is a random variable and therefore has a probability distribution

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Average Run Length for x-Charts

Let p = P(x falls outside control limits)

Then

P(RL = 1) = P(x1 falls outside CL)=pP(RL = 2) = P(x1 falls inside CL & x2 falls outside of CL)

= (1-p)pP(RL = 3) = P(x1, x2 fall inside CL & x3 falls outside of CL)

= (1-p)(1-p)p

P(RL = i) = P(x1, x2, … xi-1 fall inside CL & xi falls outside of

CL)= (1-p)i-1p

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Average Run Length for x-Charts

Therefore, the probability mass function of RL is

The mean or expected value of RL is

kRLP 1,2,...Kfor pp1 1K

RLEμ

1a

1a

32

321

1K

1K

p1ap

...p-14p-13p-121p

...p-14pp-13pp-12pp

pp1K

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Average Run Length for x-Charts

• The Average Run Length, ARL, indicates the number of samples needed, on the average before x will exceed the control chart limits.

p

1

p-1-1

1p 2

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Probability of Out-Of-Control Signal and ARL

• Process in control with mean 0

• p = 1 – P(LCL x UCL) = 0.0027

• ARL

i.e., one the average we would expect 1 out-of-control signal out of 370 samples.

,3700.0027

1

p

1

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Probability of Out-Of-Control Signal and ARL

• Process in control with mean 10+with constant

• What happens if the process goes out of control?

• How long does it take until the control charts detects the shift?

• Probability of detecting shift

n

xn

3μ3μP1δp 00

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Probability of Out-Of-Control Signal and ARL

nδ3Pnδ3P

nδ3nδ3P1

n

σ

δσμn

σ3μ

Z

n

σ

δσμn

σ3μ

P10000

ZZ

Z

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Example

For example, if n = 5, and = 1,

and

2225.0

7775.00000.01

764.01236.5

53P53P

ZZ 1p

495.40.2225

1

1p

1

ARL