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SEMI M56-1103 © SEMI 2003 4 7.4 Calculate α and β as follows (Note 4). Note that the symbols in the equations for α and β have the following meanings: f ( x ) = PDF of process characteristic x , USL = u pper specificatio…

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SEMI M56-1103 © SEMI 2003 3
5.4.6 cumulative distribution function (CDF) — a
mathematical formula that describes the probability a
measurable event occurs at or below a specific value.
5.4.7 distribution — a characterization of the
probability of realization for a measurable event over
the range of values that the measurements may assume.
5.4.8 joint probability — a probability density or
cumulative distribution function comprised of two or
more random variables.
5.4.9 lower specification limit (LSL) — value of a char-
acteristic below which a product is said to be non-
conforming.
5.4.10 measurand — particular attribute of a phe-
nomenon, body, or substance subject to measurement
[VIM].
5.4.11 measurement variability — differences associ-
ated with making multiple measurements on a given
measurand under specific conditions.
NOTE 2: Measurement variability (its distribution) is
illustrated in Related Information 2 (Figure R2-1). Common
(general) estimators of measurement variability are the
variance, standard deviation, and variance components.
Specific estimators include repeatability and reproducibility.
5.4.12 probability density function (PDF) — a
mathematical formula that specifies the relationship
between values that a random variable may assume and
their likelihood of occurrence. It is the first derivative
of the CDF.
5.4.13 random variable — a measurable event
occurring such that any value from its distribution is
equally likely to take place.
5.4.14 standard deviation,
σ
— the positive square
root of the variance.
NOTE 3: The standard deviation of a population may be
estimated from experimentally obtained data by the sample
standard deviation (s):
∑
=
−
−
=
n
i
i
xx
n
s
1
2
)(
1
1
where:
n = number of data values,
x
i
= value of the i
th
data point, and
x = mean of the data distribution.
5.4.15 upper specification limit (USL) — value of a
characteristic, above which a product is said to be non-
conforming.
5.4.16 variance — a statistical estimator that quantifies
spread around the mean of a PDF.
6 Summary of Practice
6.1 The process distribution for the characteristic of
interest is estimated or determined.
6.2 The bias and standard deviation for the
characteristic of interest are determined for each mea-
surement instrument to be evaluated or compared.
6.3 The quantities
α
and
β
are calculated from the
appropriate formula depending on the nature of the
specification (LSL only, USL only, or both).
6.4 Costs, based on the applicable business model, are
assigned to each of the four possible measurement out-
comes.
6.5 The cost components arising from measurement
variability are calculated for each measurement
instrument being evaluated or compared.
6.6 The costs due to measurement variability are
compared to establish the most cost effective
measurement solution for the application.
7 Procedure
7.1 Estimate the PDF for the process characteristic to
be studied. This can be done using empirical data that
represents the process. Although actual data may be
used to create a discrete PDF, it is sometimes
convenient to use the data to parametrically fit a PDF
model (e.g., log normal).
7.2 Unless already known, establish the bias and
standard deviation for each measurement gauge to be
compared in accordance with SEMI E89.
7.3 In all cases take the measurement influence into
account so that it does not broaden the PDF. This may
be done by taking repeated measurements at each point
in the measurement range and calculating the mean, or
by deconvolving the process characteristic PDF,
f(x),
and the measurement variability CDF, G(u), generally
assumed to be a Gaussian (or normal) distribution with
arithmetic mean equal to the bias, so that:
x
x
uuG
M
u
M
d
2
)(
exp
2
1
)()(
2
2
−
−=Φ=
∫
∞−
σ
δ
πσ
(1)
where:
δ
= bias,
σ
M
= standard deviation of the measurement
distribution.
NOTE 4: The quantity σ
M
includes the effects of the change
in bias over the time interval in which σ
M
has been
established.
SEMI M56-1103 © SEMI 2003 4
7.4 Calculate
α
and
β
as follows (Note 4). Note that
the symbols in the equations for
α
and
β
have the
following meanings:
f (x) = PDF of process characteristic x,
USL = upper specification limit,
LSL = lower specification limit, and
Φ(u) = Gaussian CDF (see Equation (1) in Section
7.3).
7.4.1 For a characteristic with only a USL, use the
following equations to calculate
α
and
β
:
∫
∞−
+−
Φ−=
USL
M
xxf
xUSL
d)(1
σ
δ
α
∫
∞
+−
Φ=
USL
M
xxf
xUSL
d)(
σ
δ
β
7.4.2 For a characteristic with only an LSL, use the
following equations to calculate
α
and
β
:
∫
∞
+−
Φ=
LSL
M
xxf
xLSL
d)(
σ
δ
α
∫
∞−
+−
Φ−=
LSL
M
xxf
xLSL
d)(1
σ
δ
β
7.4.3 For a characteristic with both upper and lower
specifications limits, use the following equations to
calculate
α
and
β
:
xxf
xLSL
xxf
xUSL
USL
LSL
M
USL
LSL
M
d)(
d)(1
∫
∫
+−
Φ
+
+−
Φ−=
σ
δ
σ
δ
α
xxf
xLSLxUSL
xxf
xLSLxUSL
USL
MM
LSL
MM
d)(
d)(
∫
∫
∞
∞−
+−
Φ−
+−
Φ
+
+−
Φ−
+−
Φ=
σ
δ
σ
δ
σ
δ
σ
δ
β
NOTE 5: Background information related to the calculation
of
α
and
β
is given in Related Information 3.
7.5 Use a binary decision model. If measurement-
based decisions are labeled as pass or fail and items are
inherently conforming or nonconforming, there are only
four outcomes:
• pass a conforming item,
• fail a conforming item (
α
error),
• pass a nonconforming item (
β
error), and
• fail a nonconforming item.
The probabilities associated with these outcomes are
1−
α
,
α
,
β
, and 1−
β
, respectively.
7.6 To define the cost model, assign costs to each of
the four decisions above on the basis of the business
model used for the manufacturing process as follows:
• c
pc
: cost of passing a conforming item,
• c
fc
: cost of failing a conforming item (
α
error),
• c
pn
: cost of passing a non-conforming item (
β
error), and
• c
fn
: cost of failing a non-conforming item.
7.6.1 Assign zero incremental cost to the two correct
outcomes (c
pc
and c
fn
).
7.6.2 Assign the incremental costs for the error out-
comes, c
fc
and c
pn
, on the basis of the business model
used for the manufacturing process.
7.7 Calculate the cost due to misclassification resulting
from measurement variability based on the assigned
incremental costs given in Section 7.6 and the
frequency of occurrence of
α
and
β
errors as follows:
cost = c
fc
α
+ c
pn
β
, (2)
using the appropriate equations for α and β as given in
Section 7.4, depending on the nature of the
specification (LSL only, USL only, or both).
NOTE 6: See Related Information 4 for an example of this
calculation.
7.8 If it is desired to include costs for correct as well as
incorrect classification, calculate the total cost resulting
from measurement variability based on the assigned
incremental costs given in Section 7.6 and the
frequency of occurrence of
α
and
β
errors as follows:
cost = c
pc
(1
−α
) +
c
fc
α
+
c
pn
β
+ c
fn
(1−
β
), (3)
using the appropriate equations for α and β as given in
Section 7.4, depending on the nature of the
specification (LSL only, USL only, or both).
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
α
and
β
.
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
α
and
β
error rates for each gauge. Let
α
1
,
α
2
, …,
α
n
be the
α
values and
β
1
,
β
2
, …,
β
n
be the
β
values associated with
the n different gauges.
R1-2.3 The overall
α
value,
α
T
, is calculated from the
equation:
α
T
=
∏
−
−−
n
i
i
1
)1(
απ
where:
,d)(
∫
∞−
=
USL
xxf
π
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:
α
T
= 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
β
value,
β
T
, is calculated from the
equation:
β
T
=
∏
=
n
i
i
1
.
β
NOTE 2: This equation is based on the probability P that the
item is nonconforming but that all gauges give a conforming
result:
β
T
= 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
α
i
=
α
and
β
i
=
β
for all i, as the
α
and
β
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.