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SEMI E89-1104 E © SEMI 1999, 2004 4 measurement process is in a state of statistical control, the precision of the process has no meaning. Since the precision is poor er for greater disper sion of the test results, speci…
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independent of the value of the measurand, it is more
appropriate to use it directly rather than CV.
5.3.6 effect change in the expected value of a given
response due to the change of a given factor from one
level to another. It is a measure of influence that a
particular variable level has on the output variable.
5.3.6.1 fixed effect — variable for which estimates of
the mean are obtained for each level.
5.3.6.2 random effect — variable for which estimates of
the mean are not obtained for each level; rather the
variable is treated as a variance component.
5.3.7 factor — predictor variable whose level is
changed with the intent of assessing its effect on the
response variable (in a designed experiment) [adapted
from ISO 3534-3].
5.3.7.1 crossed factor(s) — two factors are crossed
when every level of one factor appears with every level
of the second factor.
5.3.7.2 fixed factor — factor that has either all of its
levels represented in an experiment or levels selected
by a nonrandom process.
5.3.7.3 nested factor(s) — factor that has a different set
of levels appearing within each level of a second factor.
Factor B is nested in factor A when randomization of
the levels of factor B is restricted to specific levels of
factor A.
5.3.7.4 random factor — factor that has randomly
sampled levels from a population of levels.
5.3.8 gage — alternate spelling of gauge.
5.3.9 gauge — instrument used to assign a value to a
quantitative or qualitative characteristic of a physical
entity or phenomenon.
5.3.10 interaction — effect for which the apparent
influence of one factor on the response variable
depends upon one or more other factors [ISO 3534-3].
5.3.11 level value of a factor (in a designed
experiment) [adapted from ISO 3534-3]. Also called
“setting of a variable.”
5.3.12 linearity — absence of changes in variability or
bias as measurements are made at different points
within the measurement range.
5.3.12.1 Discussion — Traditional definitions of
linearity ignore the fact that variability can change over
the measurement range, as well as bias. The
assumption of constant variability over the
measurement range should be verified during the MS
analysis.
5.3.13 lower specification limit (LSL) — value of an
attribute below which a product is said to be
nonconforming.
5.3.14
matching tolerance (
m
) — difference in bias for
any two measurement systems (MSs) of the same kind
made under the conditions of reproducibility.
5.3.15 measurand — particular attribute of a phenome-
non, body or substance subject to measurement. [VIM]
5.3.16 measurement resolution, of a gauge — smallest
difference in measurand that can be meaningfully
distinguished by the gauge.
5.3.17 measurement subsystem any set of entities,
processes, or conditions that share a common purpose
in the measurement.
5.3.17.1 Discussion — A measurement subsystem may
contain one or more of its own subsystems. For
example, a wafer handling mechanism may be further
composed of wafer loading and wafer positioning
subsystems.
5.3.18 measurement system (MS) — all entities,
procedures, and conditions that can influence the test
result obtained with a given measurement process.
5.3.18.1 Discussion — The MS may include, but is not
limited to, the gauge, operators, setup mechanics,
wafers, locations on a wafer, environmental conditions,
software used by the gauge, measurement method, etc.
The MS may be comprised of measurement
subsystems.
5.3.19 measurement system analysis (MSA) —
procedure in which relevant sources of bias and
variability associated with a measurement system (MS)
are estimated.
NOTE 2: MSA is also sometimes called gauge (or gage)
repeatability and reproducibility (GRR or GR&R).
5.3.20 nested design — experimental design in which
different levels of one factor appear in each level of a
second factor.
5.3.21 population standard deviation (
) — square root
of the population variance.
5.3.22 population variance (
2
) — measure of
dispersion associated with a population distribution.
5.3.22.1 Discussion — For continuous distributions, the
population variance is the second central moment.
5.3.23 precision — general estimator of the variability
of a measurement process about the mean value of the
test results obtained.
5.3.23.1 Discussion — Precision is a random
component of measurement uncertainty. Unless the
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measurement process is in a state of statistical control,
the precision of the process has no meaning. Since the
precision is poorer for greater dispersion of the test
results, specific measures of variability (such as
repeatability and reproducibility) are actually direct
measures of the imprecision of the measurement
process.
5.3.24 precision-to-tolerance (P/T) ratio — ratio of the
precision of a measurement system (MS) to the
tolerance (i.e., absolute magnitude of the full range of
the product specification).
5.3.24.1 Discussion — If the variability associated with
the measurement of a parameter by an MS is very small
compared with the width of the specification range, the
probability of obtaining a test result outside the
specification limits when the value of the parameter
actually lies within the specification limits (or
conversely) is quite small. On the other hand, if the
ratio is too large, the probability of obtaining a false test
result is much greater.
5.3.25 predictor variable — variable that can contribute
to the explanation of the outcome of a designed
experiment. Also called “input variable,” “descriptor
variable,” and “explanatory variable.”
5.3.25.1 Discussion — The term “independent variable”
is not recommended as a synonym due to potential
confusion with independence.
5.3.26 product standard deviation (
Product
)
population standard deviation associated with the
distribution of values of all possible realizations of a
property of an entity manufactured under specified
conditions.
5.3.26.1 Discussion — The product variability may be
estimated by taking a representative sample from the
population and calculating its sample standard deviation
(s
Product
) taking suitable account of MS variation (see
Section 10.2).
5.3.27 reference material — material or substance, one
or more of whose property values are sufficiently
homogeneous and well established to be used for the
calibration of a MS, for the assessment of a measure-
ment method or for assigning values to materials.
5.3.28 repeatability (
r
) — variability associated with
repeated measurements taken under repeatability
conditions.
5.3.29 repeatability conditions — test conditions
involving acquisition of a series of test results with the
same test protocol and MS setup in the same laboratory
by the same operator on the same equipment in the
shortest practical period of time on the same test wafer
without explicit recalibration.
5.3.29.1 Discussion — The acquisition of test data
under repeatability conditions is intended to avoid
influences of long-term drift, operator or MS
differences, material variability, and the like.
Recalibration of the MS is expected to cause
discontinuous differences in test results. However, if
recalibration is required by the test protocol or is
internal to the MS, it is considered to be an allowable
variation in determination of repeatability.
5.3.30 reproducibility (
R
) — variability associated
with the measurement system (MS) when
measurements are made under different (but typical)
conditions.
5.3.30.1 Discussion — Changes associated with
subsystems or test conditions are potential sources of
variation to be estimated. Repeatability is one source of
variation. Other relevant sources of variability may
include time, operator, setup procedure, wafer (of like
variety), measurement location, test instrumentation,
environmental conditions, etc. Although the total
number of contributors to the variance can be
exceedingly large, one typically focuses on a subset that
accounts for a significant portion of the expected MS
variability. For clarity, the selected subset should be
reported together with the reproducibility. If q different
conditions introduce variability into the measurement
independently from one another, the variances add
directly
22
3
2
2
2
1 qR
K
(3)
and they may be separated by the use of judiciously
designed experiments.
5.3.31 response variable — variable representing the
outcome of a designed experiment. Also called “output
variable.”
5.3.31.1 Discussion — The term “dependent variable”
is not recommended as a synonym due to potential
confusion with independence.
5.3.32 root sum of squares (RSS) difference — square
root of the difference of the squares of two numbers.
5.3.33 root sum of squares (RSS) sum — square root of
the sums of the squares of two or more numbers.
5.3.34 sample standard deviation (s) square root of
the sample variance.
5.3.35 sample variance (s
2
) — measure of dispersion
given by the average squared deviation from the mean
for a set of numbers.
5.3.35.1 Discussion — If x
i
is an individual
measurement,
x is the average across all
measurements, and n is the number of measurements,
then
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n
i
i
xx
n
s
1
22
)(
1
1
(4)
The denominator value of n – 1 is used instead of n to
make the sample variance an unbiased estimator of the
population variance.
5.3.36 signal–to-noise ratio (SNR) — ratio of the
variation in the manufactured product to the precision
of the measurement system (MS).
5.3.36.1 Discussion — Because it is difficult to directly
measure the standard deviation of the product without
including variation due to the measurement instrument,
SNR is generally defined as:
2
22
R
RTotal
SNR
(5)
where
Total
2
is an estimate of the total population
variance obtained from appropriate measurements of a
large, representative sample of the product.
5.3.37 stability — absence of additional variability due
to taking measurements over time (typically several
days or longer).
5.3.38 statistical model mathematical function
relating one or more variables to known and
measurable inputs plus one or more unknown stochastic
(error) terms.
5.3.38.1 Discussion — A statistical model consists of
three parts. The first part is the response variable that is
being modeled. The second part is the deterministic or
the systematic part of the model that includes predictor
variables. Finally, the third part is the random error or
stochastic part of the model, which can be quite
elaborate. An example of a statistical model is:
p
i
jiij
exy
1
(6)
where:
y
j
= j
th
measurement,
p = number of input variables,
x
i
= i
th
input variable,
i
= its corresponding coefficient, and
e
j
= error associated with the j
th
measurement.
In many cases the error distribution(s) are specified
before the model is fit (e.g., as normal).
5.3.39 tolerance — absolute magnitude of the full
range of the product specification.
5.3.40 total variance (
2
Total
) sum of the product
variance and the square of the reproducibility.
5.3.41 uncertainty — parameter, associated with a
measurement, that characterizes the dispersion of
values that can be reasonably attributed to the object
being measured.
5.3.41.1 Discussion — Two types of measurement
uncertainty are:
Type A: uncertainty components evaluated by
statistical methods and
Type B: uncertainty components evaluated by
other than statistical methods.
5.3.42 upper specification limit (USL) — value of an
attribute above which a product is said to be
nonconforming.
5.3.43 variable quantitative or qualitative
characteristic of an object, processes, or state that may
take on more than one value.
5.3.43.1 Discussion — When the values occur
unpredictably, it is a random variable.
5.3.44 variance — population variance (see Section
5.3.22).
6 Goals of and Preliminary Steps in Planning
an MSA
6.1 One major goal of an MSA is to determine MS
reproducibility under the desired operating conditions.
This may range from a simple repeatability study, to a
complex determination of the relative contributions of
many sources of MS variability using factorial
experiments and analysis of variance (ANOVA). Some
of the typical types of variability-determination goals to
be met by an MSA include the following:
6.1.1 Determination of repeatability.
6.1.2 Determination of the effect of loading and
unloading the wafer samples between measurements.
6.1.3 Determination of stability.
NOTE 3: Stability may also have a bias component that can
be confounded with the variability component of stability (see
Section 6.2.3).
6.1.4 Assessment of the multiple factors that affect the
MS variability, which requires the use of factorial
experiments and ANOVA.
6.1.5 Assessment of the largest sources of variability in
order to focus improvement efforts on the most critical
sources. This identification can be made from the
analysis of a properly designed and executed factorial
experiment.