FH MAINZ – MSC. INTERNATIONAL BUSINESS

Statistical Process Control Application of Classical Shewhart Control Charts

February Amelia Curry

Matrikel-Nr.: 903738

Prepared for: Prof. Daniel Porath

Due Date: January 6, 2010

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Table of Contents

I. Introduction ..................................................................................................................... 3

II. Statistical Process Control................................................................................................. 3

III. Control Charts .................................................................................................................. 4

IV. Application of Control Charts ............................................................................................ 6

IV.1. Background .............................................................................................................. 6

IV.2. Assumptions ............................................................................................................ 6

IV.3. Subgroup Analysis .................................................................................................... 9

IV.3.1. Shewhart s and Control Charts ..................................................................................... 9

IV.3.2. R and Control Charts ..................................................................................................13

V. Conclusion ..................................................................................................................... 16

Bibliography ..................................................................................................................................... 17

Appendix 1. Overall Measurement Data ............................................................................................ 18

Appendix 2. Subgroup Data and Calculation ...................................................................................... 23

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I. Introduction

Recent economic recession has necessitated hard and soft savings for many organizations. Yet,

some see this as an opportunity for quality engineers to establish the effects of good quality

management (Nichols & Houry, 2009). Strategically, it can assist organizations in orientating itself to the

changing external environment; specifically by focusing on customer needs through continuous process

improvement.

Continuous process improvement in manufacturing involves defect reduction (Arbogast, 1997)

which can be achieved by employing scientific method in quality and process control. Quality is defined

as “characteristics that a product or service must have” (Anderson & et.al, 2007). Quality control is a

series of inspections and measurements to determine whether quality standards are being met

(Anderson & et.al, 2007). It has had a long history; however the effective application of statistic to

quality control just began in the 1920s as a consequence of the development of sampling theory

(NIST/SEMATECH). The general consensus is that quality in the manufacturing industry is not limited to

the products; even more important is the quality of the processes involved in producing the goods.

In order to control the process, several tools of Statistical Process Control (SPC), namely

histograms, check sheets, Pareto charts, cause and effect diagrams, scatter diagrams and control charts

(NIST/SEMATECH) are employed to determine whether the process is in control or out of control. This

paper attempts to illustrate the application of SPC method, specifically of different control charts of

measurement variables in manufacturing process.

II. Statistical Process Control

Process control is the continuous adjustment of the process based on the information supplied

by the monitoring tools such as the SPC (NIST/SEMATECH). These tools are applied to examine the

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variations in the output quality. By assuming that the production process is a continuous one (Anderson

& et.al, 2007), the variation can be categorized as (Lind, Marchal, & Wathen, 2008):

(1) Assignable variation which is not a random one and may be the result of worn out tools,

improper setting of machinery, poor quality raw materials or human error. As the consequence,

the process is labeled as out of control process.

(2) Chance variation which is random and cannot be completely eliminated since it is caused by

random variations in manufacturing processes such as temperature, pressure, humidity, etc.

Such processes are considered to be in statistical control.

The testing methodology in statistical process control is summarized in the following table:

The outcomes of statistical process control

State of production process

Decision H0 True - Process in Control H0 False – Process out of Control

Continue Process Correct decision Type II error (allowing an out-of-

control process to continue)

Adjust Process Type I error (adjusting an in-

control process)

Correct decision

Table 1. The Outcomes of Statistical Process Control (Anderson & et.al, 2007)

III. Control Charts

Fundamentally, control chart is a plot of the selected sample with its associated measurement to

determine whether the process involved in producing the said sample is within the specification limits of

an in control process. In other words, every time a point is plotted on the control chart, we are carrying

out a hypothesis test to determine whether the process is in control (Lind, Marchal, & Wathen, 2008). A

population model and the associated Upper Control Limit (UCL) and Lower Control Limit (LCL) are

determined from the historical data of how the process typically performed (NIST/SEMATECH).

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Measurements that fall outside the control limits are examined to see if they belong to the same

population as the specified model (NIST/SEMATECH); in other words, we attempt to separate the

assignable variation from the chance variation.

There are two basic types of control charts (NIST/SEMATECH): univariate control chart which is

based on one quality characteristic and multivariate control chart which represents more than one

quality characteristics. Univariate control charts include variable control chart which depicts

measurements and requires the interval or the ratio scale of measurements and attribute control chart

which classifies a product or service as either acceptable or unacceptable (Lind, Marchal, & Wathen,

2008). Variable control charts can be subdivided into (NIST/SEMATECH): (1) Shewhart s and R charts for

subgroup measurement, (2) Moving range for individual measurement, (3) Cumulative Sum (CUSUM)

control chart, and (4) Exponentially Weighted Moving Average (EWMA) control chart.

For each of the control chart, there are two main features: (1) the center line which corresponds

to the target value when the process is in control (2) the Upper Control Limit (UCL) and Lower Control

Limit (LCL) which determine whether the process is in control or out of control. The basic configuration of

control chart can be illustrated as follows:

Figure 1. The Construction of a Simple Control Chart (Moameni & Zinck, 1997)

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It can be seen, that UCL and LCL are basically the critical values which represent the probability

of a data point falling beyond the limits by assuming that only chance variation is present

(NIST/SEMATECH). ±3σ limits indicate that for in control process, the probability of data falling within the

limits is 99.73% (Levinson, 1999) or a data point is expected to fall beyond the limits every 370 times.

The general rule of thumb for control charts is that an in control process is indicated by data points fall

within the control limits and exhibits random pattern (NIST/SEMATECH).

IV. Application of Control Charts

IV.1. Background

The data selected for the application of control charts in this paper is 450 continuous random

variables from lithography process in semiconductor industry (Source: (NIST/SEMATECH))1. The quality

characteristics applied in control charts are the width of lines measured from five different sites of a

single wafer. There are 30 lots, each with three wafers. The charts designed in the analysis are Shewhart

variable control charts.

IV.2. Assumptions

One of the assumptions in applying control charts is rational sub grouping of the samples, which

means that the sampling is performed consecutively from the process output (Frank, 2003). It is

therefore presumed that the line width measurements were taken from the lithography process in

sequence. Another assumption that has to be accepted due to the limitation of the paper is the non-

correlativity of the measurements.

To apply 3σ limit, the underlying assumption is that the chance variations are normally

distributed (NIST/SEMATECH). To test whether this assumption holds, normal probability plot is

generated using SPSS. The result can be seen below:

1 See Appendix for Data

7

Figure 2. Normal Probability Plot

It appears that the overall line width is normally distributed and as a consequence, the ±3σ

control limits can be applied. The normality of the overall distribution is also important to establish the

applicability of central limit theory in the subsequent subgroup analysis. It has been suggested that

effective application of normal probability distribution in estimating the population parameters for

subgroups with small size applies for primary distribution that does not differ significantly from

normality (Levinson, 1999).

The next step is to investigate the shape of the distribution. Skewness increases the risk of

finding a chance variation above the UCL and below the LCL (NIST/SEMATECH). The histogram of the

data is illustrated below:

Observed Cum Prob

1,00,80,60,40,20,0

Exp

ecte

d C

um

Pro

b

1,0

0,8

0,6

0,4

0,2

0,0

Normal P-P Plot of Line Width

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Figure 3. Histogram Plot

The primary distribution is somewhat positively skewed with coefficient 0.461. To separate

chance variations from assignable variations, some studies have attempted to determine the level of

significance for skewness coefficient. Levinson (1999) has listed the values of coefficients which are

considered to be within the random statistical variations with 95% and 99% confidence intervals for

sample size of 25 to 100. However, the size of the overall measurement in this analysis is 450 data

points; therefore it cannot be concluded whether the skewness of distribution results from significant

assignable variations.

Line Width

6,0000005,0000004,0000003,0000002,0000001,0000000,000000

Fre

qu

en

cy

60

50

40

30

20

10

0

Mean =2,530062Std. Dev. =0,692111

N =450

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IV.3. Subgroup Analysis

Lithographic processes consist of inherent variations resulted from variations in materials,

environmental parameters which may affect equipments, and human error (Levinson, 1999). From the

histogram in the previous section, it is not conclusive whether the variations in line width measurements

are due to these chance variations or that the process is out of control and therefore further

investigation and adjustments are warranted.

Shewhart control charts are designed to answer this question. The sub-grouping of the 450 data

points can be carried out in three different ways: (1) single measurements of line width i.e. subgroups

with sample size (n) = 1, (2) subgroup of 90 wafers with n = 5 i.e. measurements from different sites in a

single wafer, and (3) subgroup of 30 lots with n = 15. Due to the limitation of the paper, the subsequent

analysis is based on sub-grouping of 90 wafers.

IV.3.1. Shewhart s and Control Charts

Since the population σ is unknown, an unbiased estimator of standard deviation is calculated

using:

Equation 1

The average of the m subgroups standard deviations:

Equation 2

With si represents the standard deviation of ith subgroup and the constant:

Equation 3

The control limits and the center line of the s chart are:

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Equation 4

Equation 5

Equation 6

The population mean μ also has to be estimated by a target or the average of subgroup means,

i.e. grand mean:

Equation 7

is the mean of the ith subgroup. The control limits and center line of the chart are:

Equation 8

Equation 9

Equation 10

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Based on the equations above, the results of the calculation are given in the table below:

Parameters Results

227.705591

m 90

2.530062122

C4 0.9399

n 5

0.408420821

UCLx 3.113053964

LCLx 1.94707028

UCLs 0.853538032

LCLs -0.03669639

Table 2. The Parameters for Shewhart s and Control Charts

The subgroup estimation of standard deviation (

= 0.434536463) is smaller than the

overall standard deviation (0.69211) shown in histogram because the overall standard deviation also

contains the between-wafer variations (Levinson, 1999). The control charts generated by SPSS are shown

below:

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Figure 4. Control Chart Based on Standard Deviation

Figure 5. Shewhart s Control Chart

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IV.3.2. R and Control Charts

The R and control charts can assist in determining whether the variability in a process is in

control or whether shifts are occurring over time (Berenson & Levine, 1999), i.e. the shift of current

parameters from the initial values. The control limits and central line for R chart are given as follows:

Equation 11

Equation 12

Equation 13

The average range is:

Equation 14

The range of ith subgroup is and the control limits and center line of the chart are:

Equation 15

Equation 16

Equation 17

A2, d2, and d3 are constants which values depend on the size of the subgroup and can be found in

statistic table (See (NIST/SEMATECH)).

The results of the calculation for R and control charts are:

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Parameters Results

1.051011633

2.530062122

n 5

A2 0.577

D3 0

D4 2.115

UCLx 3.136495835

LCLx 1.92362841

UCLR 2.222889605

LCLR 0

Table 3. The Parameters for and R Control Charts

Using SPSS, the control charts calculated using average range can be seen below:

Figure 6. Control Chart Based on Range

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Figure 7. R Control Chart

R and s control charts measure the within- subgroup variation (May & Spanos, 2006), i.e. the

variability of line width from different sites in a single wafer. From the charts above, all of data points are

within the standard deviation and range control limits. This can be interpreted as the variation within a

single wafer is in statistical control, which means the variations present are of the chance or process-

inherent nature. R control chart is regarded to be effective for small sample size (n ≤ 10). For n = 5 as in

the case with the line width measurements in this paper, the relative efficiency of range approach to

standard approach is 0.955 (NIST/SEMATECH).

On the other hand, both control charts examine the between-subgroup variability (May &

Spanos, 2006) i.e. the variability of line width from 90 sequences of wafer. It can be observed that

several data points fall beyond the control limits of both charts. Based on Western Electric Rules

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(WECO), this can be interpreted as the lithography process to be out of control and shifts of the

population parameters have occurred over time (NIST/SEMATECH). It is also detected that there are

more than 8 consecutive points (subgroup 20 – 30) that fall below the target value. The probability for

this pattern to happen is 0.39% and therefore signals mean shifts (Levinson, 1999).

From the results, it appears that the initial assumption of rational sub grouping holds for these

control charts because the indication of rational sampling is the minimizing of within-subgroup variation

and maximizing of between-subgroup variation in the presence of assignable causes (May & Spanos,

2006).

V. Conclusion

The analysis of wafer subgroups indicates that the lithography process is out of control and

investigation of materials, equipment and operators is necessary to find the assignable causes.

Nevertheless, the analysis is limited to wafer sub-grouping which represents the within-lot variation. It is

recommended to design control charts based on between-lot variation (NIST/SEMATECH) with larger

standard deviation and wider distance between control limits and the target value.

Additionally, the application of Shewhart control charts is assigned to non-correlated variables.

For further study, it is crucial to perform autocorrelation test on the data set. Since the charts employ

±3σ limits, the sensitivity of the charts in detecting the shifts of the parameters also needs to be

determined by generating OC Curve.

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Bibliography

Anderson, D. R., & et.al. (2007). Statistics for Business and Economics. London: Thomson Learning.

Arbogast, G. W. (1997). A Case Study: Statistical Analysis in a Production Quality Improvement Project.

Journal of Quality Management: 2(2) , 267-277.

Berenson, M. L., & Levine, D. M. (1999). Basic Business Statistics: Concepts and Application. Prentice Hall.

Frank, P. (2003). Control Charts in Quality Control - Shewhart Charts Application. Retrieved January 02,

2010, from The Faculty of Electrical Engineering and Communication - Brno University of Technology:

www.feec.vutbr.cz/EEICT/2003/fsbornik/03.../09-frank_petr.pdf

Kurekova, E. (2001). Measurement Process Capability: Trends and Approaches. Measurement Science

Review: 1(1) , 43-46.

Levinson, H. J. (1999). Lithography Process Control. SPIE Press Book.

Lind, D. A., Marchal, W. G., & Wathen, S. A. (2008). Statistical Techniques in Business & Economics with

Global Data Sets. McGraw-Hill Irwin.

May, G. S., & Spanos, C. J. (2006). Fundamentals of Semiconductor Manufacturing and Process Control.

John Wiley & Sons Inc.

Moameni, A., & Zinck, J. A. (1997). Application of SQC Charts and Geostatistics to Soil Quality Assessment

in a Semi-Arid Environment of South-Central Iran. ITC Journal: 3(4) , 28p.

Nam, K. H., Kim, D. K., & Park, D. H. (2001). Large-Sample Interval Estimators for Process Capability

Indices. Quality Engineering: 14(2) , 213-221.

Nichols, M. D., & Houry, K. (2009). Adapting to Troubled Times: Versatility is Key if Quality is to Come to

the Forefront. Quality Progress: January 2009 , 8-9.

NIST/SEMATECH. (n.d.). 6. Process or Product Monitoring and Control. Retrieved November 21, 2009,

from e-Handbook of Statistical Methods: http://www.itl.nist.gov/div898/handbook/

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Appendix. 1 Overall Measurement Data

Lot Wafer Site Line Width Lot Wafer Site Line Width 1 1 Top 3.199275 16 1 Top 3.131536

1 1 Left 2.253081 16 1 Left 2.405975

1 1 Center 2.074308 16 1 Center 2.20632

1 1 Right 2.418206 16 1 Right 3.012211

1 1 Bottom 2.393732 16 1 Bottom 2.628723

1 2 Top 2.654947 16 2 Top 2.802486

1 2 Left 2.003234 16 2 Left 2.18501

1 2 Center 1.861268 16 2 Center 2.161802

1 2 Right 2.136102 16 2 Right 2.10256

1 2 Bottom 1.976495 16 2 Bottom 1.961968

1 3 Top 2.887053 16 3 Top 3.330183

1 3 Left 2.061239 16 3 Left 2.464046

1 3 Center 1.625191 16 3 Center 1.687408

1 3 Right 2.304313 16 3 Right 2.043322

1 3 Bottom 2.233187 16 3 Bottom 2.570657

2 1 Top 3.160233 17 1 Top 3.352633

2 1 Left 2.518913 17 1 Left 2.691645

2 1 Center 2.072211 17 1 Center 1.94241

2 1 Right 2.28721 17 1 Right 2.366055

2 1 Bottom 2.120452 17 1 Bottom 2.500987

2 2 Top 2.063058 17 2 Top 2.886284

2 2 Left 2.21722 17 2 Left 2.292503

2 2 Center 1.472945 17 2 Center 1.627562

2 2 Right 1.684581 17 2 Right 2.415076

2 2 Bottom 1.900688 17 2 Bottom 2.086134

2 3 Top 2.346254 17 3 Top 2.554848

2 3 Left 2.172825 17 3 Left 1.755843

2 3 Center 1.536538 17 3 Center 1.510124

2 3 Right 1.96663 17 3 Right 2.257347

2 3 Bottom 2.251576 17 3 Bottom 1.958592

3 1 Top 2.198141 18 1 Top 2.622733

3 1 Left 1.728784 18 1 Left 2.321079

3 1 Center 1.357348 18 1 Center 1.169269

3 1 Right 1.673159 18 1 Right 1.921457

3 1 Bottom 1.429586 18 1 Bottom 2.176377

3 2 Top 2.231291 18 2 Top 3.313367

3 2 Left 1.561993 18 2 Left 2.559725

3 2 Center 1.520104 18 2 Center 2.404662

3 2 Right 2.066068 18 2 Right 2.405249

3 2 Bottom 1.777603 18 2 Bottom 2.535618

3 3 Top 2.244736 18 3 Top 3.067851

3 3 Left 1.745877 18 3 Left 2.490359

3 3 Center 1.366895 18 3 Center 2.079477

3 3 Right 1.615229 18 3 Right 2.669512

3 3 Bottom 1.540863 18 3 Bottom 2.105103

4 1 Top 2.929037 19 1 Top 4.293889

4 1 Left 2.0359 19 1 Left 3.888826

4 1 Center 1.786147 19 1 Center 2.960655

4 1 Right 1.980323 19 1 Right 3.618864

4 1 Bottom 2.162919 19 1 Bottom 3.56248

4 2 Top 2.855798 19 2 Top 3.451872

4 2 Left 2.104193 19 2 Left 3.285934

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4 2 Center 1.919507 19 2 Center 2.638294

4 2 Right 2.019415 19 2 Right 2.91881

4 2 Bottom 2.228705 19 2 Bottom 3.076231

4 3 Top 3.219292 19 3 Top 3.879683

4 3 Left 2.90043 19 3 Left 3.342026

4 3 Center 2.171262 19 3 Center 3.382833

4 3 Right 3.04125 19 3 Right 3.491666

4 3 Bottom 3.188804 19 3 Bottom 3.617621

5 1 Top 3.051234 20 1 Top 2.329987

5 1 Left 2.50623 20 1 Left 2.400277

5 1 Center 1.950486 20 1 Center 2.033941

5 1 Right 2.467719 20 1 Right 2.544367

5 1 Bottom 2.581881 20 1 Bottom 2.493079

5 2 Top 3.857221 20 2 Top 2.862084

5 2 Left 3.347343 20 2 Left 2.404703

5 2 Center 2.53387 20 2 Center 1.648662

5 2 Right 3.190375 20 2 Right 2.115465

5 2 Bottom 3.362746 20 2 Bottom 2.63393

5 3 Top 3.690306 20 3 Top 3.305211

5 3 Left 3.401584 20 3 Left 2.194991

5 3 Center 2.963117 20 3 Center 1.620963

5 3 Right 2.945828 20 3 Right 2.322678

5 3 Bottom 3.466115 20 3 Bottom 2.818449

6 1 Top 2.938241 21 1 Top 2.712915

6 1 Left 2.526568 21 1 Left 2.389121

6 1 Center 1.94137 21 1 Center 1.575833

6 1 Right 2.765849 21 1 Right 1.870484

6 1 Bottom 2.382781 21 1 Bottom 2.203262

6 2 Top 3.219665 21 2 Top 2.607972

6 2 Left 2.296011 21 2 Left 2.177747

6 2 Center 2.256196 21 2 Center 1.246016

6 2 Right 2.645933 21 2 Right 1.663096

6 2 Bottom 2.422187 21 2 Bottom 1.843187

6 3 Top 3.180348 21 3 Top 2.277813

6 3 Left 2.849264 21 3 Left 1.76494

6 3 Center 1.601288 21 3 Center 1.358137

6 3 Right 2.810051 21 3 Right 2.065713

6 3 Bottom 2.90298 21 3 Bottom 1.885897

7 1 Top 2.169679 22 1 Top 3.126184

7 1 Left 2.026506 22 1 Left 2.843505

7 1 Center 1.671804 22 1 Center 2.041466

7 1 Right 1.66076 22 1 Right 2.816967

7 1 Bottom 2.314734 22 1 Bottom 2.635127

7 2 Top 2.912838 22 2 Top 3.049442

7 2 Left 2.323665 22 2 Left 2.446904

7 2 Center 1.854223 22 2 Center 1.793442

7 2 Right 2.39124 22 2 Right 2.676519

7 2 Bottom 2.196071 22 2 Bottom 2.187865

7 3 Top 3.318517 22 3 Top 2.758416

7 3 Left 2.702735 22 3 Left 2.405744

7 3 Center 1.959008 22 3 Center 1.580387

7 3 Right 2.512517 22 3 Right 2.508542

7 3 Bottom 2.827469 22 3 Bottom 2.574564

8 1 Top 1.958022 23 1 Top 3.294288

8 1 Left 1.360106 23 1 Left 2.641762

8 1 Center 0.971193 23 1 Center 2.105774

8 1 Right 1.947857 23 1 Right 2.655097

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8 1 Bottom 1.64358 23 1 Bottom 2.622482

8 2 Top 2.357633 23 2 Top 4.066631

8 2 Left 1.757725 23 2 Left 3.389733

8 2 Center 1.165886 23 2 Center 2.993666

8 2 Right 2.231143 23 2 Right 3.613128

8 2 Bottom 1.311626 23 2 Bottom 3.213809

8 3 Top 2.421686 23 3 Top 3.369665

8 3 Left 1.993855 23 3 Left 2.566891

8 3 Center 1.402543 23 3 Center 2.289899

8 3 Right 2.008543 23 3 Right 2.517418

8 3 Bottom 2.13937 23 3 Bottom 2.862723

9 1 Top 2.190676 24 1 Top 4.212664

9 1 Left 2.287483 24 1 Left 3.068342

9 1 Center 1.698943 24 1 Center 2.872188

9 1 Right 1.925731 24 1 Right 3.04089

9 1 Bottom 2.05744 24 1 Bottom 3.376318

9 2 Top 2.353597 24 2 Top 3.223384

9 2 Left 1.796236 24 2 Left 2.552726

9 2 Center 1.24104 24 2 Center 2.447344

9 2 Right 1.677429 24 2 Right 3.011574

9 2 Bottom 1.845041 24 2 Bottom 2.711774

9 3 Top 2.012669 24 3 Top 3.359505

9 3 Left 1.523769 24 3 Left 2.800742

9 3 Center 0.790789 24 3 Center 2.043396

9 3 Right 2.001942 24 3 Right 2.929792

9 3 Bottom 1.350051 24 3 Bottom 2.935356

10 1 Top 2.825749 25 1 Top 2.724871

10 1 Left 2.502445 25 1 Left 2.239013

10 1 Center 1.938239 25 1 Center 2.341512

10 1 Right 2.349497 25 1 Right 2.263617

10 1 Bottom 2.310817 25 1 Bottom 2.062748

10 2 Top 3.074576 25 2 Top 3.658082

10 2 Left 2.057821 25 2 Left 3.093268

10 2 Center 1.793617 25 2 Center 2.429341

10 2 Right 1.862251 25 2 Right 2.538365

10 2 Bottom 1.956753 25 2 Bottom 3.161795

10 3 Top 3.07284 25 3 Top 3.178246

10 3 Left 2.291035 25 3 Left 2.498102

10 3 Center 1.873878 25 3 Center 2.44581

10 3 Right 2.47564 25 3 Right 2.231248

10 3 Bottom 2.021472 25 3 Bottom 2.302298

11 1 Top 3.228835 26 1 Top 3.320688

11 1 Left 2.719495 26 1 Left 2.8618

11 1 Center 2.207198 26 1 Center 2.238258

11 1 Right 2.391608 26 1 Right 3.12205

11 1 Bottom 2.525587 26 1 Bottom 3.160876

11 2 Top 2.891103 26 2 Top 3.873888

11 2 Left 2.738007 26 2 Left 3.166345

11 2 Center 1.668337 26 2 Center 2.645267

11 2 Right 2.496426 26 2 Right 3.309867

11 2 Bottom 2.417926 26 2 Bottom 2.542882

11 3 Top 3.541799 26 3 Top 2.586453

11 3 Left 3.058768 26 3 Left 2.120604

11 3 Center 2.187061 26 3 Center 2.180847

11 3 Right 2.790261 26 3 Right 2.480888

11 3 Bottom 3.279238 26 3 Bottom 1.938037

12 1 Top 2.347662 27 1 Top 4.710718

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12 1 Left 1.383336 27 1 Left 4.082083

12 1 Center 1.187168 27 1 Center 3.533026

12 1 Right 1.693292 27 1 Right 4.269929

12 1 Bottom 1.664072 27 1 Bottom 4.038166

12 2 Top 2.38532 27 2 Top 4.237233

12 2 Left 1.607784 27 2 Left 4.171702

12 2 Center 1.230307 27 2 Center 3.04394

12 2 Right 1.945423 27 2 Right 3.91296

12 2 Bottom 1.90758 27 2 Bottom 3.714229

12 3 Top 2.691576 27 3 Top 5.168668

12 3 Left 1.938755 27 3 Left 4.823275

12 3 Center 1.275409 27 3 Center 3.764272

12 3 Right 1.777315 27 3 Right 4.396897

12 3 Bottom 2.146161 27 3 Bottom 4.442094

13 1 Top 3.218655 28 1 Top 3.972279

13 1 Left 2.91218 28 1 Left 3.883295

13 1 Center 2.336436 28 1 Center 3.045145

13 1 Right 2.956036 28 1 Right 3.51459

13 1 Bottom 2.423235 28 1 Bottom 3.575446

13 2 Top 3.302224 28 2 Top 3.024903

13 2 Left 2.808816 28 2 Left 3.099192

13 2 Center 2.340386 28 2 Center 2.048139

13 2 Right 2.79512 28 2 Right 2.927978

13 2 Bottom 2.8658 28 2 Bottom 3.15257

13 3 Top 2.992217 28 3 Top 3.55806

13 3 Left 2.952106 28 3 Left 3.176292

13 3 Center 2.149299 28 3 Center 2.852873

13 3 Right 2.448046 28 3 Right 3.026064

13 3 Bottom 2.507733 28 3 Bottom 3.071975

14 1 Top 3.530112 29 1 Top 3.496634

14 1 Left 2.940489 29 1 Left 3.087091

14 1 Center 2.598357 29 1 Center 2.517673

14 1 Right 2.905165 29 1 Right 2.547344

14 1 Bottom 2.692078 29 1 Bottom 2.971948

14 2 Top 3.76427 29 2 Top 3.371306

14 2 Left 3.46596 29 2 Left 2.175046

14 2 Center 2.458628 29 2 Center 1.940111

14 2 Right 3.141132 29 2 Right 2.932408

14 2 Bottom 2.816526 29 2 Bottom 2.428069

14 3 Top 3.217614 29 3 Top 2.941041

14 3 Left 2.758171 29 3 Left 2.294009

14 3 Center 2.345921 29 3 Center 2.025674

14 3 Right 2.773653 29 3 Right 2.21154

14 3 Bottom 3.109704 29 3 Bottom 2.459684

15 1 Top 2.177593 30 1 Top 2.86467

15 1 Left 1.511781 30 1 Left 2.695163

15 1 Center 0.746546 30 1 Center 2.229518

15 1 Right 1.49173 30 1 Right 1.940917

15 1 Bottom 1.26858 30 1 Bottom 2.547318

15 2 Top 2.433994 30 2 Top 3.537562

15 2 Left 2.045667 30 2 Left 3.311361

15 2 Center 1.612699 30 2 Center 2.767771

15 2 Right 2.08286 30 2 Right 3.388622

15 2 Bottom 1.887341 30 2 Bottom 3.542701

15 3 Top 1.923003 30 3 Top 3.184652

15 3 Left 2.124461 30 3 Left 2.620947

15 3 Center 1.945048 30 3 Center 2.697619

22

15 3 Right 2.210698 30 3 Right 2.860684

15 3 Bottom 1.985225 30 3 Bottom 2.758571

23

Appendix. 2 Subgroup Data and Calculation

Wafer 1 2.4677204 1.124967 0.431260212 0.00388649

2 2.1264092 0.793679 0.311204278 0.162935681

3 2.2221966 1.261862 0.4558566 0.09478118

4 2.4318038 1.088022 0.443100272 0.009654698

5 1.8676984 0.744275 0.29613395 0.4387257

6 2.0547646 0.809716 0.321700782 0.225907734

7 1.6774036 0.840793 0.330784882 0.727026555

8 1.8314118 0.711187 0.311191794 0.488112272

9 1.70272 0.877841 0.332518671 0.684494987

10 2.1788652 1.14289 0.440765711 0.123339278

11 2.2255236 0.936291 0.370170221 0.092743711

12 2.9042076 1.04803 0.42903305 0.139984839

13 2.51151 1.100748 0.391454829 0.000344181

14 3.258311 1.323351 0.47617135 0.530346428

15 3.29339 0.744478 0.327475868 0.582669449

16 2.5109618 0.996871 0.383615592 0.000364822

17 2.5679984 0.963469 0.394715722 0.001439161

18 2.6687862 1.57906 0.614129994 0.01924437

19 1.9686966 0.653974 0.294299125 0.315131249

20 2.3356074 1.058615 0.383286486 0.037812639

21 2.6640492 1.359509 0.494109255 0.017952537

22 1.5761516 0.986829 0.418684578 0.909945284

23 1.7648026 1.191747 0.532222103 0.585622136

24 1.9931994 1.019143 0.372181894 0.288221582

25 2.0320546 0.58854 0.230931202 0.248011492

26 1.7826686 1.112557 0.398268296 0.558597077

27 1.535844 1.22188 0.50852267 0.988469674

28 2.3853494 0.88751 0.321873954 0.020941772

29 2.1490036 1.280959 0.526897883 0.145205597

30 2.346973 1.198962 0.468064349 0.033521627

31 2.6145446 1.021637 0.39112425 0.007137289

32 2.4423598 1.222766 0.472072342 0.007691697

33 2.9714254 1.354738 0.518592186 0.194801543

34 1.655106 1.160494 0.439885575 0.765548215

35 1.8152828 1.155013 0.428850234 0.510909479

36 1.9658432 1.416167 0.517730297 0.318342992

37 2.7693084 0.882219 0.375611891 0.057238782

38 2.8224692 0.961838 0.341094132 0.085501899

39 2.6098802 0.842918 0.357791604 0.006370926

24

40 2.9332402 0.931755 0.363112473 0.162552563

41 3.1293032 1.305642 0.515823458 0.35908987

42 2.8410126 0.871693 0.34297938 0.0966902

43 1.439246 1.431047 0.51529903 1.189879812

44 2.0125122 0.821295 0.299672065 0.267857922

45 2.037687 0.287695 0.12444798 0.242433261

46 2.676953 0.925216 0.392522057 0.02157693

47 2.2427652 0.840518 0.32468548 0.082539521

48 2.4191232 1.642775 0.618798066 0.012307444

49 2.570746 1.410223 0.516607855 0.001655178

50 2.2615118 1.258722 0.460307626 0.072119275

51 2.0073508 1.044724 0.410828239 0.273227126

52 2.042183 1.453464 0.549859739 0.238026038

53 2.6437242 0.908705 0.3811776 0.012919068

54 2.4824604 0.988374 0.413075983 0.002265924

55 3.6649428 1.333234 0.488451685 1.287954153

56 3.0742282 0.813578 0.316872072 0.29611672

57 3.5427658 0.537657 0.216565612 1.025568739

58 2.3603302 0.510426 0.200336064 0.028808925

59 2.3329688 1.213422 0.472035495 0.038845778

60 2.4524584 1.684248 0.639463277 0.006022338

61 2.150323 1.137082 0.442912352 0.144201801

62 1.9076036 1.361956 0.516024459 0.387454612

63 1.8705 0.919676 0.345654387 0.435022193

64 2.6926498 1.084718 0.404200808 0.026434753

65 2.4308344 1.256 0.476601961 0.009846141

66 2.3655306 1.178029 0.457322028 0.027070622

67 2.6638806 1.188514 0.421656464 0.017907385

68 3.4553934 1.072965 0.410596551 0.856237974

69 2.7213192 1.079766 0.41590097 0.03657927

70 3.3140804 1.340476 0.534231985 0.61468466

71 2.7893604 0.77604 0.322779366 0.067235597

72 2.8137582 1.316109 0.479509327 0.080483465

73 2.3263522 0.662123 0.244999015 0.041497732

74 2.9761702 1.228741 0.500966 0.199012417

75 2.5311408 0.946998 0.377300532 1.16355E-06

76 2.9407344 1.08243 0.42585389 0.16865172

77 3.1076498 1.331006 0.539494769 0.333607526

78 2.2613658 0.648416 0.266791285 0.072197713

79 4.1267844 1.177692 0.425402544 2.549522033

80 3.8160128 1.193293 0.479530076 1.653669146

81 4.5190412 1.404396 0.525616259 3.956037773

25

82 3.598151 0.927134 0.365579148 1.140813851

83 2.8505564 1.104431 0.45642535 0.102716582

84 3.1370528 0.705187 0.262725413 0.368437683

85 2.924138 0.978961 0.407363063 0.155295798

86 2.569388 1.431195 0.580155629 0.001546525

87 2.3863896 0.915367 0.347214828 0.020641794

88 2.4555172 0.923753 0.370524127 0.005556945

89 3.3096034 0.77493 0.318641253 0.607684604

90 2.8244946 0.563705 0.219600768 0.086690484

227.705591

94.591047

36.75787393 30.31417377

=0.434536