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SEMI M56-1103 © SEMI 2003 8 RELATED INFORMATION 3 BACKGROUND OF TH E METHODOLOGY NOTICE: This relat ed information is not an official part o f SEMI M56. It was derived from ta sk force deliberations during the developm e…

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SEMI M56-1103 © SEMI 2003 7
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
(
x
)
]
Measurement
Variability PDF
[=
Φ
(
u
)
]
Observed
Measurement
Bias
[=
δ
]
True Value
[=
x
]
U
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
SEMI M56-1103 © SEMI 2003 8
RELATED INFORMATION 3
BACKGROUND OF THE METHODOLOGY
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.
R3-1 The model in this practice is used to assign
probabilities (risks) to specification-related events. In
general, there are two probabilities that are of interest.
The first is the probability of getting a measurement
outside of the specification range for a conforming
item. This is often called the producer’s risk because
conforming product would be needlessly rejected in this
situation. The second probability is that of getting a
measurement within specifications for a non-
conforming item. Consumer’s risk is the term applied
to this probability, since there is a chance of non-
conforming product being accepted.
R3-1.1 For the purposes of this standard, the event of
rejecting an item whose true value lies inside of the
specification limits is considered a Type I error with
error rate (probability)
α
.
R3-1.2 The event of accepting an item whose true
value lies outside of the specification limits is
considered a Type II error with error rate (probability)
β
.
R3-1.3 One of the difficulties in formulating the
statistical model is that there are two or more
distributions involved, possibly in an interdependent
fashion. Given a true underlying (fixed) value, x, a
single measurement of that value produces the obser-
vation Y = x + U, where U is a random and unobs-
ervable value attributable to measurement variability.
If one takes n measurements of the same value x, i.e.,
Y
i
= x + U
i
, where i = 1,…, n, Y and U are treated as
random variables from the observation and
measurement variability distributions, respectively.
The situation is further complicated if several x values
are measured. In this situation, the x values come from
the process (or characteristic) distribution X such that
Y
ij
= X
j
+ U
ij
, where i = 1,…, n and j = 1,…, m. The
relationship between X and U is known as a
convolution. Additional distributions may be involved
if one assumes that certain characteristics of the
measurement process do not remain constant over the
measurement range. For example, one may assume that
bias is a function of the measured value. To model this
a third distribution would be needed, but the scope of
this practice is limited to the use of only two
distributions (characteristic and measurement
variability).
R3-2 The symbols in the equations derived for
α
and
β
have the following meanings:
X = random value attributable to the process
characteristic,
U = random and unobservable value attributable
to measurement variability,
f (x) = process characteristic PDF,
G(u) = measurement variability CDF,
USL = upper specification limit,
LSL = lower specification limit, and
Φ(u) = Gaussian CDF.
R3-3 The process characteristic PDF and measurement
variability CDF are assumed to be independent of each
other.
R3-4 For a process characteristic with only a USL:
R3-4.1 Define
α
as a joint probability:
[
]
()()
,
1
USL
PX USLX U USL
GUSL x f xdx
α
−∞
=≤ +>
=− −


∫
R3-4.2 Similarly, define
β
as:
[
]
()()
,
USL
PX USLX U USL
GUSL x f xdx
β
∞
=> +≤
=−
∫
R3-5 For a process characteristic with only an LSL, the
formulae for
α
and
β
are:
()()
LSL
GLSL x f xdx
a
•
=-
Ú
()()
1
LSL
GLSL x f xdx
b
-•
È˘
=- -
Î˚
Ú
SEMI M56-1103 © SEMI 2003 9
R3-6 For a process characteristic with both upper and
lower specifications limits, the formulae for
α
and
β
are:
()()()
()()()
LSL
USL
G USL x G LSL x f x dx
G USL x G LSL x f x dx
b
-•
•
È˘
=---+
Î˚
È˘
-- -
Î˚
Ú
Ú
()()
()()
1
USL
LSL
USL
LSL
GUSL x f xdx
GLSL x f xdx
a
È˘
=- - +
Î˚
-
Ú
Ú
R3-7 To use the above functions in practice, additional
assumptions must be made:
R3-7.1 Equipment bias is treated as constant
throughout the measurement range. This is needed to
avoid a third function in the integral.
R3-7.2 In general, the distribution of the measurement
variability is assumed Gaussian (or normal), with mean,
µ
, and variance,
σ
2
. For measurement gauges,
µ
=
δ
,
the bias, and
σ
2
=
σ
M
2
, the measurement system
variance.
R3-7.3 The distribution of the parameter of interest has
been characterized. This may be done by taking
multiple measurements at each point in the
measurement range and averaging, or by deconvolving
F(x) and G(u).
R3-8 Given the assumptions above about the mea-
surement system, the following relationships apply:
R3-8.1 When only a USL is given, the formulae for
α
and
β
are:
∫
∞−
+−
Φ−=
USL
M
xxf
xUSL
d)(1
σ
δ
α
∫
∞
+−
Φ=
USL
M
xxf
xUSL
d)(
σ
δ
β
R3-8.2 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
R3-8.3 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
R3.9 The equations used in Section 7.4 are those in
Sections R3-8.1 through R3-8.3.