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SMUEMIS 7364

NTUTO-570-N

Measures of Process Capability Process Capability Ratios

Updated: 2/4/04

Statistical Quality ControlDr. Jerrell T. Stracener, SAE Fellow

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Process Capability

Refers to the uniformity of the process.

Variability in the process is a measure of the uniformity of the output.

- Instantaneous variability is the natural orinherent variability at a specified time

- Variability over time

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Process Capability

A critical performance measure that addressesprocess results relative to process/productspecifications.

A capable process is one for which the processoutputs meet or exceed expectation.

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Process capability indices are used to measure theprocess variability due to common causes presentin the process

• The Cp indexInherent or potential measure of capability

specification spread process spread

• The CpK indexRealized or actual measure of capability

• Other indicesCpM, CpMK

Cp =

Process Capability Measures or Indices

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Measures of Process Capability

Customary to use the six sigma spread in thedistribution of the product quality characteristic

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Key Points

The proportion of the process output that will falloutside the natural tolerance limits.

• Is 0.27% (or 2700 nonconforming parts per million)if the distribution is normal

• May differ considerably from 0.27% if thedistribution is not normal

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Measure of Potential Process Capability, Cp

• Cp measures potential or inherent capability of theprocess, given that the process is stable

• Cp is defined as

, for two-sided specifications

and , for lower specifications only

, for upper specifications only

6σ

LSLUSLCp

3σ

LSLCpL

3σ

USLCpU

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Interpretation of Cp

is the percentage of the specification band used up by the process

100%C

1P

p

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Measure of Potential Process Capability, CpK

• CpK measures realized process capability relative toactual production, given a stable process

• CpK is defined as

3σ

USL,

3σ

LSLminCpK

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Interpretation of CpK

< 1, then conclude that the process is stable

If CpK = 1, then conclude that theprocess is marginally capable

> 1, then conclude that theprocess is capable

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Recommended Minimum Values of the ProcessCapability Ratio

Two-sided One-sidedspecifications specifications

Existing process 1.33 1.25

New processes 1.50 1.45

Safety, strength,or critical parameterexisting process 1.50 1.45

Safety, strength, or critical parameternew process 1.67 1.60

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Process Fallout (in defective ppm)

PCR One-sided specs Two-sided specs0.25 226628 4532550.50 66807 1336140.60 35931 718610.70 17865 357290.80 8198 163950.90 3467 69341.00 1350 27001.10 1484 9671.20 159 3181.30 48 961.40 14 271.50 4 71.60 1 21.70 0.17 0.341.80 0.03 0.062.00 0.0009 0.0018

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Estimation of Process Capability Ratios

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Estimation of Cp

A point estimate of Cp is:

where

^

^

p

σ6

LSLUSLC

n

1nsσ

^

n

ii xx

1

2

1-n

1s

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Estimation of Cp - example

If the specification limits are

LSL = 73.95 and USL = 74.05 and 0099.0^

^

^

p

σ6

LSLUSLC

0.00996

95.3705.47

68.1

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Estimation of CP - example

and the process uses

of the specification band.

%10068.1

1

P

%5.59

^

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Example

To illustrate the use of the one sided process capability ratios, suppose that the lower specificationlimit on bursting strength is 200 psi. We will use x = 264 and S = 32 as estimates of and , respectively, and the resulting estimate of the onesided lower process capability ratio is

^

^

pL

σ3

LSLC

323

002264

67.096

64

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Example

The fraction of defective bottles produced by thisprocess is estimated by finding the area to the leftof Z = (LSL - )/ = (200 - 264)/32 = -2 underthe standard normal distribution. The estimatedfallout is about 2.28% defective, or about 22,800nonconforming bottles per million. Note that if thenormal distribution were an inappropriate model for strength, then this last calculation would have to be performed using the appropriate probabilitydistribution.

^ ^

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Estimation of Cp

A (1 - ) 100% confidence interval for Cp is

where=

=

where are the lower /2 and upper/2 percentage points of the chi-squared distributionwith n - 1 degrees of freedom.

UL PP CC ,

UpC

LpC

116

21,2/1

^21,2/1

nC

nS

LSLUSL np

n

116

21,2/

^21,2/

nC

nS

LSLUSL np

n

212

2121 and ,nα/,nα/ χχ

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The Chi-Squared Distribution

Definition - A random variable X is said to have the Chi-Squared distribution with parameter , called degrees of freedom, if the probability density function of X is:

, for x > 0

, elsewhere

2/x1

22/

ex2/2

1 )x(f

0

where is a positive integer.

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The Chi-Squared Model

Remarks:

The Chi-Squared distribution plays a vital role in statistical inference. It has considerable application in bothmethodology and theory. It is an important component ofstatistical hypothesis testing and estimation.

The Chi-Squared distribution is a special case of theGamma distribution, i.e., when = /2 and = 2.

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Properties of the Chi-Squared Model

• Mean or Expected Value

2

• Standard Deviation

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Properties of the Chi-Squared Model

It is customary to let 2 represent the value above

which we find an area of p. This is illustrated by the shaded region below.

For tabulated values of the Chi-Squared distribution see the

Chi-Squared table, which gives values of 2 for various values

of p and . The areas, p, are the column headings; the degrees

of freedom, , are given in the left column, and the table entries

are the 2 values.

x

f(x)

2,

)(1 2,F

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Estimation of CpK

An approximate (1 - ) 100% confidence interval for CpK is

where=

=

UL PKPK CC ,

UpKC

LpKC

12

1

9

11 2^2/

^

nCn

ZC

pK

pK

12

1

9

11 2^2/

^

nCn

ZC

pK

pK

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Example

A sample of size n = 20 is taken from a stable processis used to estimate CpK, with the result that

= 1.33. An approximate 95% confidenceinterval on CpK is

=

=

LpKC

12

1

9

11 2^2/

^

nCn

ZC

pK

pK

pKC^

192

1

33.1209

196.1133.1 2

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Example

=

=

0.99 CpK 1.67

This is an extremely wide confidence interval. Based on the sample data, the ratio CpK could be less than one (a very bad situation), or it could be as large as 1.67 (a very good situation). Thus, we have learned very little about the actual process capability, because CpK is very imprecisely estimated. The reason for this is that a very small sample (n = 20) has been used.

UpKC

12

1

9

11 2^2/

^

nCn

ZC

pK

pK

192

1

33.1209

196.1133.1 2

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Testing Hypotheses About PCR’s

A practice that is becoming increasingly common in industry is to require a supplier to demonstrate process capability as part of the contractualagreement. Thus, it is frequently necessary todemonstrate that the process capability ratio Cp meets or exceeds some particular target value - say, Cp0. This problem may be formulated as a hypothesis testing problem.

H0: Cp = Cp0 (or the process is not capable)H1: Cp > Cp0 (or the process is capable)

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Testing Hypotheses About PCR’s

= = 0.10 = = 0.05

Sample Cp(high)/ Cp(high)/Size Cp(low) C/Cp(low) Cp(low) C/Cp(low)10 1.88 1.27 2.26 1.3720 1.53 1.20 1.73 1.2630 1.41 1.16 1.55 1.2140 1.34 1.14 1.46 1.1850 1.30 1.13 1.40 1.1660 1.27 1.11 1.36 1.1570 1.25 1.10 1.33 1.1480 1.23 1.10 1.30 1.1390 1.21 1.10 1.28 1.12100 1.20 1.09 1.26 1.11

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Testing Hypotheses About PCR’s

We would like to reject H0, thereby demonstratingthat the process is capable. We can formulate thestatistical test in terms of Cp, so that we will rejectH0, if Cp exceeds a critical value C.

^^

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Example

A customer has told his supplier that, in order toqualify for business with his company, the suppliermust demonstrate that his process capability exceeds Cp = 1.33. Thus his supplier is interestedin establishing a procedure to test the hypothesis.

H0: Cp = 1.33H1: Cp > 1.33

The supplier wants to be sure that if the processcapability is below 1.33 there will be a high probability of detecting this (say, 0.90), whereas ifthe process capability exceeds 1.66 there will be ahigh probability of judging the process capable

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Example

again (say, 0.90). This would imply that Cp(low) =1.33, Cp(high) = 1.66 and = = 0.10. To findthe sample size and critical value C from the table,compute

and enter the table value where = = 0.10. Thisyields n = 70 and

1.251.33

1.66

(low)C

(high)C

p

p

1.10(low)C

C

p

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Example

from which we calculate

Thus, to demonstrate capability, the supplier must take a sample of n = 70 parts and the sample process ratio Cp must exceed C = 1.46.

.101(low)CC p

.10133.1

46.1

^

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Testing Hypotheses

In many situations the reason for gathering and analyzing data is to provide a basis for deciding on a course of action. Let us assume that either of two courses of action is possible: A1 or A2, and that we would be clear whether one or the other is the better action, if only we knew the nature of a certain population - that is, if we knew the probability distribution of a certain random variable.

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Testing Hypotheses

The whole population or the distribution of probability is usually unattainable, therefore, we are forced to settle for such information as can be gleaned from a sample and to make our choice between the two actions on the basis of the sample.

1. Obtain random sample of size n2. Apply decision rule to data

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Testing Hypotheses

Statistical Hypothesis - is a statement about a probability distribution and is usually a statement about the values of one or more parameters of the distribution. For example, a company maywant to test the hypothesis that the true average lifetime of a certain type of TV is at least 500 hours, i.e., that 500.

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Testing Hypotheses

The hypothesis to be tested is called the null hypothesis and is denoted by H0. To construct a criterion for testing a given null hypothesis, analternate hypothesis must be formulated, denoted by H1.

Remark: To test the validity of a statistical hypothesis the test is conducted, and according to the test plan the hypothesis is rejected if the results are improbable under the hypothesis. If not, the hypothesis is accepted. The test leads to one of two possible actions: accept H0 or reject H0 (accept H1)

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Testing Hypotheses

Test Statistic - The statistic upon which a test of a statistical hypothesis is based.

Critical Region - The range of values of a test statistic which, for a given test, requires the rejection of H0.

Remark: Acceptance or rejection of a statistical hypothesis does not prove nor disprove the hypothesis! Whenever a statistical hypothesis is accepted or rejected on the basis of a sample, there is always the risk of making a wrong decision. The uncertainty with which a decision is made is measured in terms of probability.

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Testing Hypotheses

There are two possible decision errors associated with testing a statistical hypothesis:

A Type I error is made when a true hypothesis is rejected.

A Type II error is made when a false hypothesis is accepted.

True SituationDecision H0 true H0 falseAccept H0 correct Type II errorReject H0 Type I error correct(Accept H1)

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Uses of Results from a Process Capability Analysis

1. Predicting how well the process will hold thetolerances.

2. Assisting product developers/designers in selecting or modifying a process.

3. Assisting in establishing an interval between sampling for process monitoring.

4. Specifying performance requirements for newequipment

5. Selecting between competing vendors.6. Planning the sequence of production processes

when there is an interactive effect on processes or tolerances.

7. Reducing the variability in a manufacturing process.