Capability Webinar Updated - asq-1302.org · – Cp = – Cpk = Cp (1 – k) USL – LSL 6 * Voice...

11/3/2010

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Welcome!

The “Understanding capability and controlThe “Understanding capability and control charts” webinar will begin shortly

“Understanding capability andUnderstanding capability and control charts”Two 75-minute web sessions

Presented by Matt SavageModerated by David Shattucky

PQ Systems

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Who is PQ Systems?• Established in 1980• Established in 1980

• Founded by Dr. Michael J. Cleary

• Headquartered in Dayton, OH

• Vision statement– PQ Systems provides products and services that help

customers demonstrate proof of their qualitycustomers demonstrate proof of their quality performance.

• International offices

Agenda• Day 1• Day 1

– What is capability analysis?– Control charts and capability analysis– Control chart review– Process control vs. process capability

P f bilit t d– Purpose of a capability study– Types of capability studies– Types of capability indices– Mathematical definitions

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Agenda (cont.)Da 2• Day 2– When to use a capability study

– How capability analysis is performed

– Interpreting study results

– Capability analysis with non-normal datap y y

Using GotoWebinar

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Webinar goals

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20

1.3434.64 34.64

12.8518.243

1.394

1.3411.245

18.243

20

1.57.657

DataData17.998

08

34.64

10.45

5.08.08

1.34

36.10

6.357

2.536

.015

.04934 64

5.082.245

1.394

6.357

.015

08

10.45

50.4

2.245

.049

.08

11.24517.8

2.245

50.4 15.24

11.245

7.65717.998

34.64

15.24

4.698

34.64

7.657

.08

17.812.85

18.243

Cpk PpkCp PpCr PrCpu PpuCpu PpuCpl PplCpm

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A+ B- F

Cp = 24

Cpk = 24

Cp = 24

Cpk = 1

Cp = 24

Cpk = -1

What is capability analysis?

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Y YY

USL

Y

USLLSL

X X

LSL T USL LSL T USL

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customers

engineering others

LSL

TS

USL

engineering others

management regulatory agencies

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Y

USLLSL

Y

UCL

≠XX

LCL

Mean≠Control limits reflect actualprocess variation

Specification limits reflect allowable process variation

Referred to as Voice of the Process

Referred to as Voice of the Customer

What does this tell you about process control?What does this tell you about process capability or process performance?

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Four states of quality

Quality

In control but not capable

In control and capable

Out-of-control, but capable

Out-of-control and not capable

Out of control & not capable

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In control, but not capable

Out-of-control, but capable

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In control & capable

Relationship of control charts to capability analysis

C t l Ch tCapability analysis Control Chart

• The mean is calculated

• Standard deviation is calculated

• Standard deviation of the averages is used

• Averages are plotted (n > 1)

Capability analysis• The mean is calculated• Standard deviation is

calculated • Standard deviation of each

value is used• Individual observations are

plotted• Data must be time ordered

• Cannot tell the process capability (n > 1)

p• Data must be time ordered for

Cpk• Order is not necessary for Ppk• Cannot tell if the process is

out-of-control

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Types of control charts: Count data

• p chart A chart that expresses the result inNonconforming • p-chart – A chart that expresses the result in percentage form

• np-chart – A chart that expresses the result in count form

• c-chart – A chart that expresses the result in t f

Nonconforming or Defective

Nonconformitiesor Defects

count form

• u-chart – A chart that expresses the results in problems (defects) per xx units.

Control charts: Central tendency

• X or individuals chart

• MA or moving average

• where the average is plotted– Control limits based on the range for n < 10

– Control limits based on the standard deviation. n > 10

• Median chart where the median value is plotted

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Control charts: Variability

• Mr or moving range chart when the sample size (n) is one. When n=1, Mr=2 is most commonly usedused.

• Range chart – Used when the sample size is typically less than 10

• Sigma chart – Used when the sample size is typically 10 or more

Purpose of process control

C t l h t d tControl charts are used to:

• Assess central tendency (compare the average to the target)

• Monitor variability

• Determine if any trends exist

• To determine stability and therefore predictability

• To understand the performance of a process over time

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Purpose of capability analysis

X + σ + USL + LSL Cpk

AQLAQL

M3

Purpose

Slide 33

M3 Purpose of Capability Study:

Communicate the level of quality

Blend average & sigma with specifications and get a result. Cpk consists of Mean, Sigma, USL & LSL

Communicate the minimum level of quality (maximum amount of defects that are expected.) AQL

Create a quantifiable customer & supplier agreement

So a grade-like system can be put in place.

Compare the specifications to the expected output and then assess if the system is acceptableMattS, 9/25/2008

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Types of capability studies

Short term Long term Past Future

Types of capability indices

• Cp

• Cr

• Cpk

• Cpu

C l

• Pp

• Pr

• Ppk

• Ppu

P l• Cpl • Ppl

• Cpm

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Symbols and formulas– = Mean = Average Mean Average– X = Average of the– = Average of range values– = Average of the moving range values– USL = Upper Specification Limit (Upper

Tolerance)Tolerance)– LSL = Lower Specification Limit (Lower

Tolerance)– T = Target Specification

Symbols and formulas– Sigma-e = estimated sigma = g g

d2

– Sigma-i = standard deviation of the individual readings. Excel uses the STDEV function

=

– sigma = standard deviation

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Symbols and formulas

Pr = (6 * )

Pp = (USL – LSL)

(6 * )

Cr = (6 * )

Cp = (USL – LSL)

(6 * )

(USL – LSL) (USL – LSL)


– = estimated sigma = d2

– k = ( –Target*)

[Smaller of (USL – Target*) or (Target* – LSL)]( g )]

* Target must be the midpoint of the USL & LSL

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– Cp =

– Cpk = Cp (1 – k)

USL – LSL

6 *

Voice of the Customer

Voice of the Process=

– = (USL – )


(USL )

– = ( - LSL)

– Cpk = Smaller of ( or )

Sigma*

Sigma*

– Cpk = Smaller of ( or )

– Ppk = Smaller of ( or )

3

3

* Where sigma used in z value is estimated sigma

* Where sigma used in z value is sigma of the individual values

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Cpu = (USL )


– Cpu = (USL – )

– Cpl = ( – LSL)

(3 * )

(3 * )

– Cpk = Smaller of Cpu and Cpl

When can / should I perform a capability study?capability study?

• The process should be stable (in control)

• The process should be bell-shaped for the usual calculations

• When you want to know how the system y yperforms in relation to specification limit(s)

• When it is required

• When you want to impress your boss

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Key assumptions

• A control chart is completed• A control chart is completed

• The control chart shows no special causes

• One or both specifications exist

• The specifications are “correct”

• The distribution of the data is bell shaped• The distribution of the data is bell-shaped

If key assumptions are not met• If a control chart is not completedIf a control chart is not completed,

• Create a control chart• If the control chart shows special causes

• Incorporate or remove special causes• If one or both specifications do not exist

• Determine what the specifications are• If the specifications are not “correct”

• Use “What if …” analysis• If the distribution of the data is not bell-shaped

• Use alternative capability calculations

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Control chart refresher

What is it?• What is it?

• What does it look like?

• When is it used?

• How is it made?

-R Control Chart

What is it?Average and range chart

– Monitors how a system changes over time

– Monitors averages to show trends and shifts

Monitors range to indicate changes in– Monitors range to indicate changes in variation

– Used for variables data where the subgroup size is larger than one

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When is it used?

-R Control ChartWhen is it used?Answer “yes” to the following:

– Do you need to assess the variability in the system?

– Can the data be collected or does a collection of data already exist?of data already exist?

– Is the time order of the data preserved?

– Is the data in variables format?

– Is the data collected in subgroups greater than one?

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-R Control Chart

Exercise

-R Control Chart

How is it made?

– Assumes data has been collected• Ideally 25 or more data points.

A l b d d i– Any unusual occurrences observed during data collection should have been noted.

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How is it made?

-R Control ChartHow is it made?1. Complete the header information.

2. Record the data.

3. Calculate the average and range for each subgroup.

4. Calculate the overall averages.

5. Calculate the control limits.

6. Scale the control chart.

7. Draw the average line and control limits.

8. Plot the values on the control chart.

9. Interpret the control chart.

How is it made?


2. Record the data.








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-R Control Chart

2. Record the data.

How is it made?


2. Record the data.








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3. Calculate the average for each subgroup.-R Control Chart

Sum X1 = 45.3 + 45.0 + 44.2 + 45.0 + 44.5= 224.0

XX

n

X1

= 224.05

= 44.8

3. Calculate the range for each subgroup.-R Control Chart

lowesthighest XXR1

= 45.3 - 44.2

= 1.1 1.1

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How is it made?


2. Record the data.








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-R Control Chart

kXX

= X1 + X2 + X3 + … + Xkk

= 44.80+45.06+45.80+…+45.70

• For the average chart

25= 1133.06

25= 45.32

-R Control Chart4. Calculate the overall averages.

• For the average range

= R1 + R2 + R3 + … + Rkk

= 1.1+0.5+1.5+…+0.7

kRR

25= 27.60

25= 1.10

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How is it made?


2. Record the data.








-R Control Chart5 Calculate the control limits5. Calculate the control limits.


)*/() * (3 2 ndRXUCL X

)*/() * (3 - 2 ndRXLCL X

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5 Calculate the control limits

-R Control Chart5. Calculate the control limits.


) x (A 2 RXUCL X

) x (A - 2 RXLCL X

-R Control ChartWeighting Factors

Subgroup size (n) A2 D3 D4

2 1.880 0 3.267

3 1.023 0 2.574

4 0.729 0 2.282

5 0.577 0 2.114

6 0.483 0 2.004

7 0.419 0.076 1.924

8 0.373 0.136 1.864

9. 0.337 0.184 1.816

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5 Calculate the control limits

-R Control Chart5. Calculate the control limits.


)1.10 x (0.577 45.32 XUCL

)1.10 x (0.577 - 45.32 XLCL

-R Control Chart5 Calculate the control limits5. Calculate the control limits.

• For the range chart

= 2.114 x 1.10

RDUCLR 4 = 0 x 1.10

RDLCLR 3

= 2.33 = 0

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When to calculate control limits

• When you have time ordered data

• When you have enough data

• When you want to know if the process is (statistically) stable

When to calculate new limits

• Does the data display a distinctly different kind of behavior from the past?

• Is the reason for this change known?

• Is the new process behavior desirable?

• Is it intended and expected that the new behaviorIs it intended and expected that the new behavior will continue?

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-R Control ChartHow is it made?How is it made?1. Complete the header information.

2. Record the data.









2. Record the data.








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73


2. Record the data.








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75


2. Record the data.








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-R Control Chart9. Interpret the control chart.

• Basic rules for interpretation– Any point lying outside the control limits.

– Run of seven points:» Seven or more points in a row above or below the center

line.

» Seven or more points in a row going in one direction, up or ddown.

– Any non-random pattern, including the following typical cases:

» Too close to the average.

» Too far from the average.

» Cycles.

Control chart exercise #2

Time Reading-1 Reading-2 Reading-3 Reading-4 Reading-5Time Reading 1 Reading 2 Reading 3 Reading 4 Reading 5

9:00 AM 35 37 41 36 40

10:00 AM 40 39 42 38 43

11:00 AM 47 41 39 40 41

12:00 PM 40 36 37 39 36

1:00 PM 43 44 42 40 39

2:00 PM 38 39 41 40 37

3:00 PM 42 36 40 41 38

4:00 PM 43 44 43 39 41

5:00 PM 36 38 35 30 37

6:00 PM 36 41 40 42 39

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Time Reading-1 Reading-2 Reading-3 Reading-4 Reading-5Time Reading 1 Reading 2 Reading 3 Reading 4 Reading 5

9:00 AM 35 37 41 36 40

10:00 AM 40 39 42 38 43

11:00 AM 47 41 39 40 41

12:00 PM 40 36 37 39 36

1:00 PM 43 44 42 40 39

2:00 PM 38 39 41 40 37

3:00 PM 42 36 40 41 38

4:00 PM 43 44 43 39 41

5:00 PM 36 38 35 30 37

6:00 PM 36 41 40 42 39

Control chart exercise #2X-bar Range

N t t9:00 AM 37.8 6

10:00 AM 40.4 5

11:00 AM 41.6 8

12:00 PM 37.6 4

1:00 PM 41.6 5

2:00 PM 39 4

3:00 PM 39 4 6

Next steps:

• Calculate limits

• Complete the chart

• Interpret the chart3:00 PM 39.4 6

4:00 PM 42 5

5:00 PM 35.2 8

6:00 PM 39.6 6

Average 39.42 5.7

• Consider capability analysis

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X = 39 42X = 39.42

R = 5.7 = 5.7 / 2.326 = 2.45

UCLx = 39.42 + (0.577 x 5.7) = 42.71

LCLx = 39.42 – (0.577 x 5.7) = 36.13

Or

13.36)5*/().75 * (3 - 42.93 2 dLCL X

71.42)5*/()5.7 * (3 42.93 2 dUCL X


X = 39 42X = 39.42

R = 5.7 = 5.7 / 2.326 = 2.45

Other (wrong) sigma = 2.96

Why is using sigma–i, the actual sigma, incorrect for control limits? ___

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X-MR Control Chart

What is it?What is it?Individual - moving range chart

– Monitors how a system changes over time

– Monitor individual values to show trends and shifts

– Monitor moving range to indicate changes in variation

Used for variables data where the subgroup size is– Used for variables data where the subgroup size is equal to one

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X-MR Control Chart

When is it used?When is it used?Answer “yes” to the following:

– Do you need to assess the variability in thesystem?

– Can the data be collected or does a collection f d l d i ?of data already exist?

– Is the time order of the data preserved?

– Is the data in variables format?

– Is the data collected in subgroups of one?

X-MR Control Chart

Exercise

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X-MR Control Chart

How is it made?How is it made?

– Assumes data has been collected• Ideally 25* or more data points.

– Any unusual occurrences observed during data collection should have been noted.

* The number of samples necessary will be reviewed later.

X-MR Control Chart

How is it made?How is it made?1. Complete the header information.

2. Record the data.

3. Calculate the moving ranges.


5 Calculate the control limits5. Calculate the control limits.





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X-MR Control Chart


2. Record the data.








X-MR Control Chart

2 Record the data2. Record the data.

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X-MR Control Chart


2. Record the data.








X-MR Control Chart3. Calculate the moving ranges.

• There is no moving range for the first sample.

MR1 = | X2 – X1|= | 97 – 90 |= | 7 | = 7

MR2 = | X3 – X2 |= | 118 – 97 |= | 21 | = 21

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X-MR Control Chart


2. Record the data.








X-MR Control Chart4. Calculate the overall averages.

F th i di id l h t• For the individual chart

X =sum of data pointsno. of subgroups

=

=

103.6

90 + 97 + 118 + . . . + 11225=

=

=

2,59125

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X-MR Control Chart4. Calculate the overall averages.

F th i• For the average moving range

MR = MR1 + MR2 + MR3 + . . . + MRk–1

k –1= 7 + 21 + 10 + . . . + 23

24= 188

24= 7.8

X-MR Control Chart


2. Record the data.








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X-MR Control Chart5. Calculate the control limits.

• For the individual chart

RMEXUCLx 2

RMEXLCLx 2



Weighting Factors Subgroup size (n) E2 D3 D4

2 2.660 0 3.267

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752061038.7660.26.103

2

RMEXUCLx

752061038.7660.26.103

2

RMEXLCLx

4.12475.206.103

9.8275.206.103


• For the moving range chart

RMDUCLMR 4

RMDLCL RMDLCLMR 3

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• For the moving range chart

8.7267.34

RMDUCLMR

8.703

RMDLCLMR

5.25 0

X-MR Control Chart


2. Record the data.








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X-MR Control Chart


2. Record the data.








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X-MR Control Chart


2. Record the data.








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X-MR Control Chart

Exercise

X-MR Control Chart


2. Record the data.








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X-MR Control Chart9. Interpret the control chart.p

• Basic rules for interpretation– Any point lying outside the control limits.

– Run of seven points:

» Seven or more points in a row above or below the center line.

» Seven or more points in a row going in one direction, up or down.

– Any non-random pattern, including the following typical cases:

» Too close to the average.

» Too far from the average.

» Cycles.

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The chart must be correct

• The right chart type – bar, run, or control chart

• If using a control chart, the right type of control chart

• Use software designed to aid with your analysis

• Use enough data to support your needsg pp y

Use enough data to support your needs

• How much data is necessary for control limits on a control chart?

• “An X-mr chart can be made with as few as four original values without an undue risk of a false alarm.” Donald J. Wheeler, Ph.D.

• Data beyond a control limit? Investigate it —even with only a few samples.

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Purpose of a control chart

Common and special cause variation

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How capability analysis is performedfor bell-shaped distributionsp

• Calculate the and the standard deviation

• Sketch the distribution (optional, but helpful )

• Sketch the specification line(s)

• Calculate and• Calculate and

• Translate to an expected percent out of spec

• Calculate the Cp, Cpk and/or Pp, Ppk.

How capability analysis is performed

• Calculate the and the standard deviation• Calculate the and the standard deviation.

= Sum of Xin

= Sum of ranges n n - 1

MR =Sum of moving ranges

Estimated sigma =

d2

Actual sigma =

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How capability analysis is performed• Sketch the distribution (optional, but helpful)( p , p )

= 10.0

= 5.81

USL = 15.8

LSL = 5.73

n = 5


= 10.0

= 5.81

USL = 15.8

LSL = 5.73

n = 5

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• Calculate the and the standard deviation• Calculate the and the standard deviation.Sum of ranges

n Estimated sigma =d2

= = 5.81

Subgroup size d2 values

2 1.128

3 1.693

4 2 059

Est. sigma =

n = 5

5.812.326

4 2.059

5 2.326

6 2.534

7 2.704

8 2.847

9 2.970

10 3.078

Est. sigma = 2.4978 ~ 2.50

3 x sigma = 7.50

Mean +/- 3sigma = 17.50, 2.50


= 10.0

= 5.81

USL = 15.8

LSL = 5.73

n = 5

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= 10.0

= 5.81

USL = 15.8

LSL = 5.73

n = 5


= 10.0

= 5.81

USL = 15.8

LSL = 5.73

n = 5

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• Calculate andCalculate and

(USL – Mean)

sigma=

( Mean- LSL)

(15.8 – 10.0)

2.50=

(10 – 5.73)

= 2.32

( )

sigma=

( )

2.50= =~ 1.71

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• Calculate andCalculate and

= 2.32 0.0102 1.02%

1 71 0 0436 4 36%= 1.71 0.0436 4.36%

We expect 1.02% + 4.36% of data to be outside the specification limits.

• √ Calculate the and the standard


• √ Calculate the and the standard deviation

• √ Sketch the distribution (optional)

• √ Sketch the specification line(s)

• √ Calculate and• √ Calculate and

• √ Translate this to an expected percentage

• Calculate the Cp, Cpk and/or Pp, Ppk

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• Cp = spec spread / process spreadCp spec spread / process spread• Cp = (15.8 – 5.73) / (6 * 2.50)• Cp = 10.07 / 15 = 0.67

=

=

2.32

1.71

• Cpk = average to closest spec / (½ of the process spread)• Cpk = Smallest Z value / 3• Cpk = 1.71 / 3 = 0.57

Characteristics of the normal distribution

68.26% 95.44%

Since +/- 3 sigma = 99.73%, each tail

99.73%

g ,of the curve has the remaining 0.135% of data

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Non bell-shaped distribution

3 standarddeviations

3 standarddeviations


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Calculate the area under the curve so that the amount of data in the left tail of the distribution equals 0.135%. This point is the PL.

Calculate the area under the curve so that the amount of data in the right tail of the distribution equals 0.135%. This point is the PU.

Calculate how many standard deviation fit between the PL and the mean and between the mean and the PU. If this was bell shaped, it p ,would be 3 standard deviations to the left and 3 to the right.

This forms the basis for calculating Cp, Cr, Cpk and other statistics.

Non bell-shaped distribution•Determine the number of standard deviation that fit between the PL and the mean.

• Determine the number of standard deviation that fit between the mean and the PU.

• Use this number in place of 3.

• Cpk is the smaller of the two equations below.(USL – Mean)

_?_ x sigma=

( Mean- LSL)=

_?_ x sigmaCpk

Cpk

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Getting the most from capability analysis

Cpmˆ

LSLUSLCpm

6

1

)(ˆ

2

n

TXi

Cpm

Average is at the target value

Cp = 0.5Cpk = 0.5Cpm = 0.5

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Average is off target

Cp = 2.0Cpk = 1.0Cpm = 0.65

Average is at the target value

Cp = 1.5Cpk = 1.5Cpm = 1.5

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Excel exercise

A B

Central location: _________ Spread: _________

CCp = _________

CCpk = _________

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A B

Central location: _________ Spread: _________

CCp = _________

C k CCpk = _________

Control charts are not designed for comparisons to

Communication with capability indices

• Control charts are not designed for comparisons to a specification

• Capability analysis allows you to compare different processes

• Capability analysis brings together the process limits (+/- 3 sigma) and specification limits

The two voices: Voice of the

customerVoice of

the process

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Summary• Tips and tools to use for effective control charts p

and capability analysis

• How capability analysis and control charts can be used more effectively

• The differences between various capability indices

PQ Systems software

• David Shattuck• (800) 777-3020 x 119• [email protected]• www.pqsystems.com

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Final thoughts

The primary goal of these two web-based sessions isThe primary goal of these two web based sessions is to educate and clarify. It is not to mislead anyone.

“He uses statistics as a drunk usesa street lamp, for support rather than illumination.” – Andrew Lang

“… and go on till you come to the end; then stop."

Alice's Adventures in Wonderland Lewis CarrollAlice's Adventures in Wonderland - Lewis Carroll

Matthew Savage, [email protected]

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Capability Webinar Updated - asq-1302.org · – Cp = – Cpk = Cp (1 – k) USL – LSL 6 * Voice...

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