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SEMI E89-1104 E © SEMI 1999, 2004 13 RELATED INFORMATION 2 TESTING MEASUREMENT DISTRI BUTIONS FOR NORMALITY AND EQUAL REPEATABILTY NOTICE : This related information is not an o fficial pa rt of SEMI E89. It was deri ved …

SEMI E89-1104

RELATED INFORMATION 1

MEASUREMENT UNCERTAINTY CALCULATIONS

NOTICE: This related information is not an official part of SEMI E89. It was derived from task force deliberations

during the revision of SEMI E89-0999 in 2001-2003. This related information was approved by full letter ballot

procedures and was approved for publication by the NA RSC on August 16, 2004.

R1-1.1 Measurement uncertainty depends on the

repeatability of the MS, the reproducibility of the result

over time, the number of measurements in the test

result, and all sources of random and systematic error

that could contribute to disagreement between the mea-

surement result and its reference value.

R1-1.2

Determine the measurement uncertainty

according to the following rules:

 Expressing Uncertainty — Each uncertainty com-

ponent is quantified by a standard deviation.

 Bias — All biases are assumed to be corrected and

any uncertainty due to bias is the uncertainty of the

correction.

 Standard Uncertainty — All uncertainty compo-

nents (standard deviations), whether Type A or

Type B (see Section 5.3.41), are combined as an

RSS sum to arrive at a “standard uncertainty,” u.

This standard uncertainty is the standard deviation

of the reported value, taking into account all

sources of error, both random and systematic, that

affect the measurement result.

 Extended Uncertainty — If the purpose of the

uncertainty statement is to provide coverage with a

high level of confidence, an expanded uncertainty

is computed as U = ku, where k = 2 is typically

chosen for an approximate 95% coverage.

 Uncertainty Interval — If Y is the reported mea-

surement value, then the symmetric interval from



U to Y + U is the uncertainty interval associ-

ated with the measurement value.

R1-1.3 The measurement result for which an

uncertainty is required must be completely specified as

 the number of repetitions that were averaged,

 the test method,

 the environmental conditions,

 the operating conditions over which the repetitions

were made, and

 any calibration uncertainty.

R1-1.4 Obtain Type A uncertainty for a particular

measurement from a determination of reproducibility

(see Section 7) conducted according to the same

specifications as the particular measurement and

including all relevant components of reproducibility.

SEMI E89-1104

RELATED INFORMATION 2

TESTING MEASUREMENT DISTRIBUTIONS FOR NORMALITY AND

EQUAL REPEATABILTY

NOTICE: This related information is not an official part of SEMI E89. It was derived from task force deliberations

during the revision of SEMI E89-0999 in 2001-2003. This related information was approved by full letter ballot

procedures and was approved for publication by the NA RSC on August 16, 2004.

R2-1 Testing for Normality

R2-1.1 Testing for normality can be achieved by a

commonly applied statistical hypothesis test. In this

test, the condition under consideration is that the

sample being examined comes from the normal distri-

bution. While there are several ways of implementing

this procedure, in each case assume that the

measurement results come from a normal distribution

and then calculate the probability of obtaining the

particular observed sample. Before calculating this

probability value (

p-value), an a priori decision is made

as to how small this p-value has to be to warrant the

conclusion that the hypothesis is unreasonable and

hence it is unlikely that the observed distribution is

normal. A level of 0.05 is commonly chosen and the

sample is then tested. If the

p-value is greater than 0.05

(or the chosen test level, if different from 0.05), one

accepts that the underlying distribution is normal.

R2-1.2 Statistical tests for normality can be found in

most commercial statistical software. Four common

tests are Andersen-Darling, Pearson’s Chi Square,

Kolmogorov-Smirnov, and Shapiro-Wilk.

R2-1.3

As an example, the 25 observations shown in

Table R2-1 are a random sample taken from a manu-

facturing process. A commonly used statistical package

was used to generate the statistical tests mentioned

above and the results obtained are shown in Table R2-2.

Table R2-1 Measurement Data

481 476 471 477 479

483 477 480 475 479

480 473 479 484 473

481 475 478 472 480

485 480 478 475 481

R2-1.4 Note that if we require a p-value is less than or

equal to 0.05 as a rejection criterion, none of these

statistical procedures would reject normality for the

manufacturing process measurements although the

calculated

p-values are all different.

R2-1.5 For some of these tests, software packages may

rely on a combination of tabled values and interpolation

to arrive at the answer. For that reason, the numbers

given for the same test may vary depending on the

software used.

Table R2-2 Calculated Normality Statistics

Test Test Statistic

Value

Calculated

p-Value

Anderson-Darling 0.28726858 >0.250

Chi-Square 2.86169631 0.239

Kolmogorov-Smirnov 0.11888191 >0.150

Shapiro-Wilk 0.973356 0.731

R2-2 Testing Two Measurement Systems for

Equal Repeatability

R2-2.1 When comparing two MSs, it is possible to test

if the repeatabilities are equal. To do so, calculate the

ratio of the repeatability of each MS. When forming

this ratio, place the larger value in the numerator. The

hypothesis under consideration is that the repeatabilities

are equal. When this is true, the ratio follows an

distribution with



degrees of freedom for the

numerator and



degrees of freedom for the

denominator.

R2-2.2 As a simple example, compare MS 1, which

demonstrated a repeatability of 9.243 based on 25

observations, and MS 2, which demonstrated a

repeatability of 7.658 based on 23 observations. The

ratio is 1.207 with



= 24 degrees of freedom for the

numerator and



= 22 degrees of freedom for the

denominator.

R2-2.3 The probability of seeing an F value of 1.207 or

larger, based on an F distribution with these degrees of

freedom is 0.330. Because this p-value is larger than

the usual pre-selected value of 0.05, there is no reason

to reject the hypothesis of equal reproducibilities, given

the measurement sample data from the two MSs.

R2-2.4 Extension of these calculations to other types of

conditions is beyond the scope of this guide, but

additional information on this topic can be found in the

literature

7 Miliken, G. A., and Johnson, D. E., Analysis of Messy Data:

Volume I Designed Experiments (Chapman & Hall, New York,

1992).

SEMI E89-1104

RELATED INFORMATION 3

MODELS FOR MEASUREMENT SYSTEM ANALYSIS

NOTICE: This related information is not an official part of SEMI E89. It was derived from task force deliberations

during the revision of SEMI E89-0999 in 2001-2003. This related information was approved by full letter ballot

procedures and was approved for publication by the NA RSC on August 16, 2004.

R3-1 Introduction

R3-1.1 This related information section provides a

more detailed theoretical explanation that extends ideas

described earlier in this guide.

R3-1.2 A measurement model expresses each observed

measurement value as the sum of a “true value” plus

“bias” plus “random error.” The “true value” may be

the actual value for the object being measured or the

average population value if a sample of objects is

measured in the MSA. The random error term is

further broken up into contributions from all the

significant sources of error under investigation. If an

MSA is well designed, standard statistical techniques

(such as ANOVA) can be used to estimate the standard

deviation contribution from each source of error, and

from this the repeatability, reproducibility and precision

of the MS are determined.

R3-1.3

Note that any MS that is composed of several

measurement subsystems (such as spectrometry and

ellipsometry), designed for the measurement of

different characteristics or types of samples, should be

treated as composed of separate MSs that need to be

analyzed independently.

R3-1.4

Writing an appropriate measurement model is

the starting point, after which many software packages

can be used to design the MSA and calculate ANOVA

estimates from the resulting measurement data.

R3-1.4.1

The model must indicate whether each source

of variation, i.e., factor, is fixed or random. Variance

components are only available for random factors.

R3-1.4.2 The model must indicate which factors are

crossed and which are nested.

R3-2 Example of a Measurement Model

R3-2.1 A wafer oxide thickness MS is under

investigation. A wafer is placed in the gauge, which

then automatically positions the wafer and takes a

measurement. While the wafer is positioned, the

measurement can be repeated as many times as desired.

Sources of variability that might affect a measurement

include basic repeatability on a positioned wafer,

variability due to the automatic positioning process,

variability over time and variability from measuring

several different wafers (to cover a range of “true”

measurement values).

R3-2.2

An MSA that separates out all these possible

sources of variability is the following: 10 wafers are

chosen at random from normal production. Each wafer

is put in the MS in a random order and measured 3

times at a single site (3 repeats) on day 1 (cycle 1).

Immediately after the 30 measurements are recorded,

the same measurement process is repeated on the 10

wafers (cycle 2). On day 2, the same wafers are again

measured using exactly the same procedure. This is

repeated for a total of 5 days, yielding 3 × 10 × 2 × 5 =

300 measurements.

R3-2.3

Assuming the MS has been calibrated (or bias

is not a concern at the moment), a model for variability

is constructed as follows: let h = 1, 2, …, 5, index the

days; let k = 1, 2, …, 10, index the wafers; let j = 1, 2,

index the repositions (cycles) each day and let i = 1, 2,

3, index the repeat measurements (taken under

repeatability conditions) on a positioned wafer.

R3-2.4 The model for the i

measurement at the j

repositioning on the k

wafer on the h

day is:

hkji



+ d

+ w

+ p

hkj

+ r

hkji

(R3-1)

where:

hkji

= measurement taken on day h, wafer k, cycle j,

and repeat i,



= true oxide thickness average value for the

population of wafers,

= error term associated with the h

day (due to

day-to-day stability variation),

= offset from the average due to the true

thickness of the film on wafer k,

hkj

= (short term) error due to the j

positioning of

the k

wafer on the h

day, and

hkji

= repeatability error term.

R3-2.5 In this model, the quantities d

, p

hkj

, and r

hkji

are

random error terms. They are usually assumed to have

a normal distribution with zero mean and standard

deviations



, and



, respectively. The wafer term

may be either a random term or a “fixed effect,”

depending on whether we consider the wafers measured

to be a random sample from the entire population of

possible wafers or a fixed population used for

experimental purposes. In either case, the estimates of

repeatability and reproducibility will be the same.