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SEMI M56-1103 © SEMI 2003 6 R1-4.1.1 The ch aracteristics being measured are independe nt or an inde pendent com bination of their values is used. R1-4.1.2 Al l tests are perform ed before a decision to reject or pass i …

SEMI M56-1103 © SEMI 2003 5

RELATED INFORMATION 1

EXTENSIONS OF THE METHODOLOGY

NOTICE: This related information is not an official part of M56. It was derived from task force deliberations

during the development of the document. This related information was approved for publication by full letter ballot

procedures on September 3, 2003.

R1-1 Introduction

R1-1.1 In general, when a single characteristic on an

item is measured once on a single gauge and 100%

sampling is employed, the model will take the form

described in this practice.

R1-1.2 If the situation is more complex, the nature of

the model will be different. Factors that can affect the

nature of the model include the following:

• number of items examined (lot acceptance

sampling vs. 100% sampling),

• number of times an item is inspected (single vs.

multiple),

• effect of the inspection process on the item

(destructive vs. non-destructive),

• number of item characteristics examined for a

single decision (one vs. many), and

• cost functions associated with the business

decisions (fixed vs. variable).

R1-1.3 In addition, it is possible to have more than one

set of cost functions for each type of item that is

inspected by the metrology system. Because much of

this is context specific, it would be impossible to cover

all possible models. Other extensions to the model

include relaxation of the assumption of constant

variance, the introduction of variability in the bias, and

the use of guard banding. The foundation of all these

extensions, however, is the use of

and

R1-2 Extension to Multiple Gauges

R1-2.1 To extend the model to multiple gauges, one

must make the additional assumption that all measuring

gauges are measuring the same characteristic.

R1-2.1.1 In addition, define a conforming item as one

that all gauges show the measured characteristic to be

in specification.

R1-2.1.2 A non-conforming item is taken to be one in

which at least one gauge shows the measured

characteristic to be outside of specification.

R1-2.2 It is then possible to define a set of

and

error rates for each gauge. Let

, …,

be the

values and

, …,

be the

values associated with

the n different gauges.

R1-2.3 The overall

value,

, is calculated from the

equation:

∏

−

−−

)1(

απ

where:

,d)(

∫

∞−

USL

xxf

and

f (x) = PDF of the characteristic being measured.

NOTE 1: This equation is based on the probability P that the

item is conforming but that one or more gauges give a

conforming result:

= P[Item is conforming, ≥1 gauges show nonconforming]

= P[Item is conforming] −

P[Item is conforming, All gauges show conforming]

R1-2.4 The overall

value,

, is calculated from the

equation:

∏

NOTE 2: This equation is based on the probability P that the

item is nonconforming but that all gauges give a conforming

result:

= P[Item is nonconforming, All gauges show pass]

R1-3 Extension to Multiple Inspections with

the Same Metrology System

R1-3.1 Multiple inspection with the same metrology

system is a special case of inspection with multiple

gauges. If the same measurement system is used to

measure the item characteristic repeatedly, one merely

lets

and

for all i, as the

and

error rates

will not change for the same gauge.

R1-4 Extension to Decisions Based on

Multiple Characteristics

R1-4.1 It is also possible to develop a model where a

decision is based on more than one characteristic. As

with the case of multiple gauges, several assumptions

must be made.

SEMI M56-1103 © SEMI 2003 6

R1-4.1.1 The characteristics being measured are

independent or an independent combination of their

values is used.

R1-4.1.2 All tests are performed before a decision to

reject or pass is made.

R1-4.1.3 F(x) has been deconvolved from F·G.

R1-4.1.4 A conforming item is defined as one in which

all measured characteristics are shown to be in

specification.

R1-4.1.5 A non-conforming item is taken to be one in

which at least one characteristic is outside of

specification.

R1-4.2 It is then possible to define a set of

and

errors for each gauge. Let

, …,

be the

values and

, …,

be the

values associated with

the n different characteristics.

R1-4.3 The overall

value,

, is calculated from the

equation:

∏∏

−−

)1(

απ

where:

,d)(

∫

∞−

USL

xxf

and

(x) = PDF of the characteristic i.

NOTE 3: This equation is based on the probability P that all

characteristics conform, but that at least one test failed:

= P[All characteristics conform, At least one test failed]

= P[All characteristics conform] −

P[All characteristics conform, All tests passed]

R1-4.4 The overall

value,

, is calculated from the

equation:

∏∏

−−

)1(

where:

= proportion of observations within specification

for characteristic i.

NOTE 4: This equation is based on the probability P that one

or more characteristics are nonconforming but that all tests

passed:

= P[≥1 characteristic nonconforming, All tests passed]

= P[All pass] −

P[All characteristics conform, All tests passed]

RELATED INFORMATION 2

PICTORIAL REPRESENTATION OF PROCESS DISTRIBUTION AND

MEASUREMENT VARIABILITY AND BIAS

NOTICE: This related information is not an official part of SEMI M56. It was derived from task force

deliberations during the development of the document. This related information was approved for publication by

full letter ballot procedures on September 3, 2003.

Process

Characteristic

PDF [= f

(

)

]

Measurement

Variability PDF

(

)

]

Observed

Measurement

Bias

]

True Value

]

NOTE: The observed measurement is displaced from the true value, x, by an amount U with a frequency of occurrence for each

value of U given by the measurement variability PDF.

Figure R2-1

Relationship of Process Characteristic PDF, Measurement Variability PDF, True Value, Bias, and an

Observed Measurement