attribute control charts 2 attribute control chart learning objectives defective vs defect binomial...

Attribute Control Attribute Control ChartsCharts

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Attribute Control Chart

Learning ObjectivesLearning Objectives

Defective vs Defect Binomial and Poisson Distribution p Chart np Chart c Chart u Chart Tests for Instability

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Shewhart Control Charts - OverviewShewhart Control Charts - Overview

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Defective and DefectDefective and Defect

DefectiveA unit of product that does not meet

customer’s requirement or specification. Also known as a non-conforming unit.

ExampleA base casting that fails porosity specification

is a defective.A disc clamp that does not meet the

parallelism specification is a defective.

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DefectA flaw or a single quality characteristic that does

not meet customer’s requirement or specification.Also known as a non-conformity.There can be one or more defects in a defective.

ExampleA dent on a VCM pole that fails customer’s

specification is a defect.A stain on a cover that fails customer’s

specification is a defect.

Defective and DefectDefective and Defect

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Shewhart Control Charts for Shewhart Control Charts for Attribute DataAttribute Data

There are 4 types of Attribute Control Charts:

np

c

p

u

ConstantLot Size

VariableLot Size

Defects(Poisson Distribution)

Defectives(Binomial Distribution)

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Mean defective rate

Mean defect rate

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Types of Data and DistributionsTypes of Data and Distributions Discrete Data (Attribute)

Binomial Poisson

Continuous Data (Variable) Normal Exponential Weibull Lognormal t 2

F

Discrete Distributions

Continuous Distributions

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Types of DistributionsTypes of Distributions

Normal Distribution Exponential Distribution

Uniform Distribution Binomial Distribution

Discrete Distributions

Continuous Distributions

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Binomial Distribution Useful for attribute data (or binary data) Result from inspection criteria which are binary in

nature, e.g. pass/fail, go/nogo, accept/reject, etc. Data generated from counting of defectives.

Discrete DistributionsDiscrete Distributions

Binomial Distribution

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 1 2 3 4 5 6 7 8

Number of Rejects (X)

Pro

ba

bil

ity

of

Fin

din

g X

Re

ject

s

n,0,1,2,xp1px

nxP xnx

Plot is known as Probability Mass Function

of X

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Binomial Distribution If a process typically gives 10% reject rate (p =

0.10), what is the chance of finding 0, 1, 2 or 3 defectives within a sample of 20 units (n = 20)?



0.122

0.2700.285

0.190

0.090

0.032

0.009 0.002 0.0000.00

0.05

0.10

0.15

0.20

0.25

0.30

0 1 2 3 4 5 6 7 8

No. of Defectives (x)

Pro

ba

bili

ty o

f F

ind

ing

x D

efe

cti

ve

s n,0,1,2,xp1p

x

nxP xnx

Commonly used in Acceptance Sampling

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Binomial DistributionBinomial Distribution

Commonly used in Acceptance Sampling, where p is the probability of success (defective rate), n is the number of trials (sample size), and x is the number of successes (defectives found).

n,,2,1,0xp1px

nxP xnx

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Binomial DistributionBinomial DistributionProperties: each trial has only 2 possible outcomes - success or

failure probability of success p remains constant throughout

the n trials the trials are statistically independent the mean and variance of a Binomial Distribution are

pnnp

n

p1np2

p

p1pn2

np

pp

and

and

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The location, dispersion and shape of a binomial distribution are affected by the sample size (n) and defective rate (p).



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James James BernoulliBernoulli



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Poisson Distribution Useful for discrete data involving error rate,

defect rate (dpu, dpmo), particle count rate, etc. Data generated from counting of defects.


0,1,2,xx!

exP

x

λλ

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Poisson Distribution If a process typically gives 4.0 defect rate ( = 4

dpu), what is the chance of finding 0, 1, 2 or 3 defects per unit?


Commonly used as an approximation of the binomial distribution when: p < 0.1 (10%) n is large

0,1,2,xx!

exP

x

λλ

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Poisson DistributionPoisson DistributionThis distribution have been found to be relevant for applications involving error rates, particle count, chemical concentration, etc, where is the mean number of events (or defect rate) within a

given unit of time or space.

,2,1,0x!x

exP

x

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Poisson DistributionPoisson DistributionProperties: number of outcomes in a time interval (or space

region) is independent of the outcomes in another time interval (or space region)

probability of an occurrence within a very short time interval (or space region) is proportional to the time interval (or space region)

probability of more than 1 outcome occurring within a short time interval (or space region) is negligible

the mean and variance for a Poisson Distribution are

2 and

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The location, dispersion and shape of a Poisson distribution are affected by the mean ().

Poisson Distribution


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Simeon D Simeon D PoissonPoisson

Poisson Distribution


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Summary of ApproximationSummary of ApproximationBinomial

p < 0.1 The smaller p andlarger n the better

15 The larger the better

np > 5, > 10p = 0.5, < 0.5

Poisson

Normal

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p Chartp Chart

Fraction Non-ConformingReject Rate / Defective Rate

Percent Fallout

20100

0.5

0.4

0.3

0.2

0.1

0.0

Sample Number

Pro

port

ion

p Chart

1

P=0.2140

3.0SL=0.3880

-3.0SL=0.04000

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p Chartp Chart

Fraction non-conforming ()Ratio of number of defectives (or non-conforming items) in a population to the number of items in that population.

Sample fraction non-conforming (p)Ratio of number of defectives (d) in a sample to the sample size (n), i.e.

i

ii n

dp Is “p” a sample

statistic?

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The underlying principles of the p chart are based on the binomial distribution.

This means that if a process has a typical fraction non-conforming, p, the mean and variance of the distribution for p’s are computed from the binomial equation, giving:

p Chartp Chart

k21

k21

......n n n

x.....xxpp

k

n n

n

p1pσ

k

1i

i

2p

k = number of subgroup, should be between 20 to 25 before constructing control limits.

Xk = number of defective unit in subgroup k which has a total sample size of nk units

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The p chart also assumes a symmetrical bell-shape distribution, with symmetrical control limits on each side of the center line.

This implies that the binomial distribution is approximately close to the shape of the normal distribution, which can happen under certain conditions of p and n: p 1/2 and n > 10 implying np > 5

For other values of p, the general guideline is to have np > 10 to get a satisfactory approximation of the normal to the binomial.

p Chartp Chart

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p Chartp Chart

Following Shewhart’s principle, the Center Line and Control Limits of a p chart are:

n

p1p3pLCL

pLineCentern

p1p3pUCL

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If the sample size is not constant, then the Control Limits of a p chart may be computed by either method:a) Variable Control Limits

where ni is the actual sample size of each sampling ib) Control Limits Based on Average Sample Size

where n is the average (or typical) sample size of all the samples

n

p1p3p

LimitsControl

in

p1p3p

LimitsControl

p Chartp Chart

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When to Use Control Limits Based on Average Sample Size instead of Variable Control Limits

Smallest subgroup size, nmin, is at least 30% of the largest subgroup size, nmax.

Future sample sizes will not differ greatly from those previously observed.

When using Control Limits Based on Average Sample Size, the exact control limits of a point should be determined and examined relative to that value if:

There is an unusually large variation in the size of a particular sample

There is a point which is near the control limits.

p Chart - Average Sample Sizep Chart - Average Sample Size

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Example 1: p ChartExample 1: p ChartS/N SampledRejects 1 50 12 2 50 15 3 50 8 4 50 10 5 50 4 6 50 7 7 50 16 8 50 9 9 50 1410 50 1011 50 512 50 613 50 1714 50 1215 50 2216 50 817 50 1018 50 519 50 1320 50 11

Frozen orange juice concentrate is packed in 6-oz cardboard cans. A metal bottom panel is attached to the cardboard body. The cans are inspected for possible leak. 20 samplings of different sampling size were obtained.

Verify if the process is in control. The data are found in AttributeSPC.MTW.

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MiniTab: Stat Control Charts P

Example 1: p ChartExample 1: p Chart

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2010Subgroup 0

0.5

0.4

0.3

0.2

0.1

0.0

Pro

por

tion

P Chart for Example 1 (Variable Width Control Limits)

P=0.2173

UCL=0.3906

LCL=0.04406

2010Subgroup 0

0.5

0.4

0.3

0.2

0.1

0.0

Pro

por

tion

P Chart for Example 1 (Based on Average Sample Size)

1

P=0.2173

UCL=0.3923

LCL=0.04233

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Example 1: p ChartExample 1: p ChartMinitab allows different set of control charts to be plotted on one chartMiniTab: Stat Control Charts P

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20100

0.5

0.4

0.3

0.2

0.1

0.0

Sample Number

Pro

port

ion

P Chart for Rejects by Date

1

P=0.23

UCL=0.4085

LCL=0.05146

1-Dec-01 8-Dec-01 15-Dec-0120100

0.4

0.3

0.2

0.1

0.0

Sample Number

Pro

port

ion

P Chart for defective by Material

P=0.1525

UCL=0.3230

LCL=0

1 2

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Establish Trial Control LimitsEstablish Trial Control LimitsWhen to use it? New process, modified process, no historical data available

to calculate pHow to do it? Calculate p based on the preliminary 20 to 25 subgroups. Calculate the trial control limits using the formula

mentioned in slide 21 or 22. Sample values of p from the preliminary subgroups to be

plotted against the trial control limits. Any points exceed the trial control limits should be

investigated. If assignable causes for these points are discovered, they

should be discarded and new trial control limits to be determined.

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np Chartnp Chart

If the sample size is constant, it is possible to base a control chart on the number nonconforming (np), rather than the fraction nonconforming (p).

The Center Line and Control Limits of an np chart are:

p1pn3pn

pn

p1pn3pn

LCL

LineCenter

UCL

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Example 2: np ChartExample 2: np ChartS/N SampledRejects 1 50 12 2 50 15 3 50 8 4 50 10 5 50 4 6 50 7 7 50 16 8 50 9 9 50 1410 50 1011 50 512 50 613 50 1714 50 1215 50 2216 50 817 50 1018 50 519 50 1320 50 11

Frozen orange juice concentrate is packed in 6-oz cardboard cans. A metal bottom panel is attached to the cardboard body. The cans are inspected for possible leak. 20 samplings of 50 cans/sampling were obtained.

Verify if the process is in control. The data are found in AttributeSPC.MTW.

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Example 2: np ChartExample 2: np Chart

MiniTab: Stat Control Charts NP

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Example 2: np ChartExample 2: np Chart

20100

20

10

0

Sample Number

Sam

ple

Co

unt

np Chart for Example 2

1

NP=10.8

UCL=19.53

LCL=2.070

2010Subgroup 0

0.5

0.4

0.3

0.2

0.1

0.0

Pro

po

rtio

n

p Chart for Example 2

1

P=0.216

UCL=0.3906

LCL=0.04141

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p Chart vs np Chartp Chart vs np Chart

For ease of recording, the np chart is preferred.

The p chart offers the following advantages:

accommodation for variable sample size

provides information about process capability

X =X

n

Distribution of Sampling Averages

XX

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p

5,

p

p19Maximumn

Sample Size for p and np ChartsSample Size for p and np Charts

Sample Size is determined based on the 2 criteria:

1. Assumption to approximate Binomial Distribution to a Normal Distribution

2. To ensure that the LCL is greater than zero.

For p 0.5

For p = other values

p

10,

p

p19Maximumn

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c Chartc ChartDefects per Unit (DPU)Error Rate / Defect RateDefects per Opportunity

20100

20

10

0

Sample Number

Sam

ple

Cou

nt

c Chart

C=9.650

3.0SL=18.97

-3.0SL=0.3307

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c Chartc Chart

Each specific point at which a specification is not satisfied results in a defect or nonconformity.

The c chart is a control chart for the total number of defects

in an inspection unit based on the normal distribution as an

approximation for the Poisson distribution, which can happen when: c or 15

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c Chartc Chart

Inspection Unit The area of opportunity for the occurrence of

nonconformities.

– e.g. a HSA, a media, a PCBA

This is an entity chosen for convenience of record-keeping.

It may constitute more than 1 unit of product.

– e.g. a HSA, both surfaces of a media, 10 pieces of PCBA

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c Chartc Chart

If the number of nonconformities (defects) per inspection unit is denoted by c, then:

The Center Line and Control Limits of a c chart are:

cm

cc

2

c

m

1i ic

c3cLCL

cLineCenter

c3cUCL

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u Chartu Chart

In cases where the number of inspection units is not constant, the u chart may be used instead, with:

If the average number of defects per inspection unit is denoted by u, then

uu

aaacccuu k

k

2......

21

21

i

ii a

cu Where ci is the count of the

number of defects in number of inspection units, ai

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u Chartu Chart

The Center Line and Control Limits of a u chart are:

u3uLCL

uLineCenter

u3uUCL

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Example 3: c and u ChartsExample 3: c and u ChartsS/N Units Defects 1 5 10 2 5 12 3 5 8 4 5 14 5 5 10 6 5 16 7 5 11 8 5 7 9 5 1010 5 1511 5 912 5 513 5 714 5 1115 5 1216 5 617 5 818 5 1019 5 720 5 5

A personal computer manufacturer plans to establish a control chart for nonconformities at the final assembly line. The number of nonconformities in 20 samples of 5 PCs are shown here.

Verify if the process is in-control.

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Example 3: c and u ChartsExample 3: c and u Charts

MiniTab’s Stat Control Charts C

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MiniTab’s Stat Control Charts U

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20100

4

3

2

1

0

Sample Number

Sam

ple

Cou

nt

u Chart for Example 3

U=1.930

3.0SL=3.794

-3.0SL=0.06613

20100

20

10

0

Sample Number

Sam

ple

Cou

nt

c Chart for Example 3

C=9.650

3.0SL=18.97

-3.0SL=0.3307

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u (or c) Chart vs p (np) Chartu (or c) Chart vs p (np) Chart

The u (or c) chart offers the following advantages:

More informative as the type of nonconformity is noted.

Facilitates Pareto analysis.

Facilitates Cause & Effect Analysis.

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c - Chart Measures the total number of defects in a subgroup

The subgroup size can be 1 unit of product if we expect to have a relatively large number of defects/unit

Requires a constant subgroup size u - Chart

Measures the number of defects/unit of product (dpu) The subgroup size can be constant or variable

p - Chart Measures the proportion of defective units in a subgroup The subgroup size can be constant or variable

np - Chart Measures the number of defective items in a subgroup Requires a constant subgroup size

Selecting the Appropriate ChartSelecting the Appropriate Chart

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Exercise #1Exercise #1 Strength of 5 test pieces sampled every hour(Xbar-R)

Number of defectives in 100 parts(np)

Number of solder defects in a printed circuit board assembly(C)

Diameter of 40 units of products sampled every day(Xbar-S)

Percent defective of a lot produced in every 30-min period(p)

Surface defects of surface area of varying sizes(u)

In a maintenance group dealing with repair work, the number of maintenance requests that require a second call to complete the repair every week

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Test for InstabilityTest for Instability

Suitable for all charts

Suitable only for X-Chart

_

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Tests for InstabilityTests for Instability

CAUTION :CAUTION : Do not apply “tests” blindly

Not every “test” is relevant for all charts

Excessive number of “tests” Increased -error

Nature of application

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Variables vs Attributes ChartsVariables vs Attributes Charts

Attributes Control Charts facilitate monitoring of more than 1 quality characteristics.

Variables Control Charts provide leading indicators of trouble; Attributes Control Charts react after the process has actually produced bad parts.

For a specified level of protection against process drift, Variables Control Charts require a smaller sample size.

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End of TopicWhat Question

Do You Have

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Reading ReferenceReading Reference

Introduction to Statistical Quality Control,

Douglas C. Montgomery, John Wiley & Sons,

ISBN 0-471-30353-4

attribute control charts 2 attribute control chart learning objectives defective vs defect binomial...

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