8/4/2015ieng 486: statistical quality & process control 1 ieng 486 - lecture 16 p, np, c, &...

04/19/23 IENG 486: Statistical Quality & Process Control 1

IENG 486 - Lecture 16

P, NP, C, & U Control Charts

(Attributes Charts)


Assignment:

Reading: Chapter 3.5 Chapter 7

Sections 7.1 – 7.2.2: pp. 288 – 304 Sections 7.3 – 7.3.2: pp. 308 - 321

Chapter 6.4: pp. 259 - 265 Chapter 9

Sections 9.1 – 9.1.5: pp. 399 - 410 Sections 9.2 – 9.2.4: pp. 419 - 425 Sections 9.3: pp. 428 - 430

Assignment: CH7 # 6; 11; 27a,b; 31; 47 Access Excel Template for P, NP, C, & U Control Charts


Process for Statistical Control Of Quality

Removing special causes of variation

Hypothesis Tests

Ishikawa’s Tools

Managing the process with control charts

Process Improvement

Process Stabilization

Confidence in “When to Act”

Reduce Variability

Identify Special Causes - Good (Incorporate)

Improving Process Capability and Performance

Characterize Stable Process Capability

Head Off Shifts in Location, Spread

Identify Special Causes - Bad (Remove)

Continually Improve the System

Statistical Quality Control and Improvement

Time

Center the ProcessLSL 0 USL


Review

Shewhart Control charts Are like a sideways hypothesis test (2-sided!) from a

Normal distribution UCL is like the right / upper critical region CL is like the central location LCL is like the left / lower critical region

When working with continuous variables, we use two charts: X-bar for testing for change in location R or s-chart for testing for change in spread

We check the charts using 4 Western Electric rules


Continuous & Discrete Distributions

Continuous Probability of a range of

outcomes is area under PDF (integration)

Discrete Probability of a range of

outcomes is area under PDF (sum of discrete outcomes)

35.0 2.5

37()

41.4(+2)

32.6(-2)

43.6(+3)

30.4(-3)

39.2 (+)

34.8 (-)

35.0 2.5

36()

4032 4230 3834


Continuous & Attribute Variables

Continuous Variables: Take on a continuum of values.

Ex.: length, diameter, thickness Modeled by the Normal Distribution

Attribute Variables: Take on discrete values

Ex.: present/absent, conforming/non-conforming Modeled by Binomial Distribution if classifying

inspection units into defectives (defective inspection unit can have multiple defects)

Modeled by Poisson Distribution if counting defects occurring within an inspection unit


Discrete Variables Classes

Defectives The presence of a non-conformity ruins the entire unit – the

unit is defective Example – fuses with disconnects

Defects The presence of one or more non-conformities may lower the

value of the unit, but does not render the entire unit defective Example – paneling with scratches


Binomial Distribution

Sequence of n trials Outcome of each trial is “success” or “failure” Probability of success = p r.v. X - number of successes in n trials

So: where

Mean: Variance:

~ ,X Bin n p

1n xxn

P X x p px

!

! !

n n

x x n x

E X np 2 1V X np p

04/19/23IENG 486: Statistical Quality &

Process Control 9

Binomial Distribution Example

A lot of size 30 contains three defective fuses.

What is the probability that a sample of five fuses selected at random contains exactly one defective fuse?

What is the probability that it contains one or more defectives?

]1[ XP 4)9)(.1)(.5( 328.

]0[1]1[ XPXP050

30

31

30

3

0

51

5)9)(.1)(1(1

5905.1 4095.

151

30

31

30

3

1

5


Poisson Distribution

Let X be the number of times that a certain event occurs per unit of length, area, volume, or time

So:

where x = 0, 1, 2, …

Mean: Variance:

~X Pois

!

xeP X x

x

E X 2 V X


Poisson Distribution Example

A sheet of 4’x8’ paneling (= 4608 in2) has 22 scratches.

What is the expected number of scratches if checking only one square inch (randomly selected)?

What is the probability of finding at least two scratches in 25 in2?

]1[]0[1 XPXP

4608

221 λ 00477.

25

1

125

i

λλ )(25 1λ )00477(.25 119.

]2[ XP

!1

)119(.

!0

)119(.1

1119.0119. ee

1

)119(.888.

1

)1(888.1 )106.888(.1 007.


Moving from Hypothesis Testing to Control Charts

Attribute control charts are also like a sideways hypothesis test

Detects a shift in the process Heads-off costly errors by detecting trends –

if constant control limits are used

0

2

2

0

2

2

2-Sided Hypothesis Test Shewhart Control ChartSideways Hypothesis Test

CL

LCL

UCL

Sample Number


P-Charts

Sample Control Limits: Approximate 3σ limits are

found from trial samples:

Standard Control Limits: Approximate 3σ limits

continue from standard:

Tracks proportion defective in a sample of insp. unitsCan have a constant number of inspection units in the sample

n

)p1(p3pLCL

pCL

n

)p1(p3pUCL

n

)p1(p3pLCL

pCLn

)p1(p3pUCL


P-Charts (continued)

Mean Sample Size Limits: Approximate 3σ limits are

found from sample mean:

Variable Width Limits: Approximate 3σ limits vary

with individual sample size:

More commonly has variable number of inspection unitsCan’t use run rules with variable control limits

n

pppLCL

pCL

n

pppUCL

)1(3

)1(3

i

i

n

)p1(p3pLCL

pCL

n

)p1(p3pUCL


NP-Charts





Tracks number of defectives in a sample of insp. unitsMust have a constant number of inspection units in each sample

Use of run rules is allowed if LCL > 0 - adds power !

)p1(pn3pnLCL

pnCL

)p1(pn3pnUCL

)p1(np3npLCL

npCL

)p1(np3npUCL


C-Charts





Tracks number of defects in a logical inspection unitMust have a constant size inspection unit containing the defects

Use of run rules is allowed if LCL > 0 - adds power !

negativeisLCLif0orc3cLCL

cCL

c3cUCL

negativeisLCLif0orc3cLCL

cCL

c3cUCL


U-Charts

Mean Sample Size Limits: Approximate 3σ limits are

found from sample mean:

Variable Width Limits: Approximate 3σ limits vary

with individual sample size:

Number of defects occurring in variably sized inspection unit (Ex. Solder defects per 100 joints - 350 joints in board = 3.5 insp. units)

Can’t use run rules with variable control limits, watch clustering!

n

u3uLCL

uCL

n

u3uUCL

i

i

n

u3uLCL

uCL

n

u3uUCL


Summary of Control Charts

Continuous Variable Charts Smaller changes detected faster Require smaller sample sizes Can be applied to attributes data as

well (by CLT)*

Attribute Charts Can cover several

defects with one chart Less costly inspection

Use of the control chart decision tree…


Use p-Chart

No, varies

Yes, constant

Use np-Chart

Individual Defects

Poisson Distribution

Use c-Chart

Use u-Chart

No, varies

Discrete

Attribute

What is the inspection

basis?

Is the size of the inspection

unit fixed?

Yes, constant

Is the size of the inspection sample fixed?

Continuous

Variable

Range

Standard Deviation

Which spread method

preferred?

Use X-bar and R-Chart

Use X-bar and S-Chart

Kind of inspection variable?

Defective Units

(possibly with multiple defects)Binomial Distribution

Control Chart Decision Tree


Attribute Chart Applications

Attribute control charts apply to “service” applications, too!

Number of incorrect invoices per customer Proportion of incorrect orders taken in a day Number of return service calls to resolve problem


Questions & Issues

8/4/2015ieng 486: statistical quality & process control 1 ieng 486 - lecture 16 p, np, c, &...

Documents

statistical control

process lsl

inspection unit slide

stable process capability

usl slide

binomial distribution

discrete values

poisson distribution