ISE-180 Engineering Statistics

1
ISE-180Engineering Statistics

• Prof. Dessouky

2
What is statistics

• To guess is cheap. To guess wrongly is expensive
  - Chinese Proverb
• There are three kinds of lies lies, d--n lies,
  and statistics - Benjamin Disraeli, British PM
• First get your facts, then you can distort them
  at your leisure - Mark Twain
• Statistical Thinking will one day be as necessary
  for efficient citizenship as the ability to read
  and write - H. G. Wells

3
What is statistics

• In our world, through conversation with others,
  through books, TV and sports, we are continually
  confronted with collections of facts or data
• STATISTICS is a branch of mathematics with
  the collection, analysis, interpretation, and
  presentation of masses of numerical data
• Statistics is used to understand variability.

4
What is statistics

• Frequently we wish to acquire information or draw
  conclusions about a population (all individuals
  or objects of a particular type)
• Many times, the data at our disposal is a sample
  (a portion or subject of a population)
• The field of statistics can be broken into 2
  branches
• Descriptive statistics concerned with the
  organization, summation, and presentation of data
  
• Inferential statistics - Value to apply to create
  population or when data consist of a sample

5
What is statistics

• Probability a bridge between the 2 fields by
  studying random variation
• Inferential statistics using sample data to
  draw conclusion about the entire population (
  i.e. making an inference about a group based on a
  sample)
• To do so, split each xi into 2 parts, a stem
  consisting of one or more leading digits and a
  leaf which consists of the remaining digits .

6
Descriptive statistics
7
Descriptive statistics
8
Descriptive statistics
9
Descriptive Statistics

• Histograms
• A graphical summary tool that permits sorting of
  data into cells. It is especially useful for
  finding population tendencies (location and
  dispersion). Requires multiple (20-30)
  observations to allow process responses to
  exhibit their tendencies. Also data specifics
  are lost within the cell boundaries.

10
Descriptive Statistics

• An example of this is the representation of 36
  air pollution readings (units .01 ppm of ozone).
  

11
Descriptive Statistics

• As a set of readings, it is not easy to see any
  clear trend to the data. So, we can construct a
  frequency distribution using the classes 2.0 lt
  3.0, 3.0 - lt 4.0, etc. to 8.0 - lt 9.0 in order
  to get a sense of the distribution. The tallies
  are tabulated below

12
Descriptive Statistics

• These lead to the frequency distribution

13
Descriptive Statistics

• And also lead to the histogram

14
Descriptive Statistics

• The drawback is that within each cell, we lose
  the data point values contained by the cell. For
  example, we would not be able to see from the
  graphical representation that the 3s cell data
  all lie within the range of 3.0 3.4.
  Therefore, cell selection can be an art form
  requiring some care.

15
Descriptive statistics
16
Descriptive statistics
17
Descriptive statistics
18
Descriptive statistics
19
Descriptive Statistics

• Continuing with the data set from before,
• Mean the weighted average of all observations
• Sample Mean
• Median the data point at which half of the
  observations are higher and half are lower in
  value
• Mode the most common data point. There may be
  multiple modes in a distribution. The most
  frequently occurring value is 4.7 (5 times).

20
Descriptive statistics
21
Descriptive statistics
22
Descriptive statistics
23
Descriptive statistics
24
Descriptive Statistics

• Continuing with the example data from before

25
Exploratory data analysis
26
Exploratory data analysis
27
Probability
28
Properties of Probability

• An example would be the use of batting averages
  in baseball. If there is a .300 hitter at the
  plate, the chance of a base hit is 0.300 ( P(A))
  for each official at-bat. This is expected over
  many at-bats, but does not guarantee a base hit
  at any given at-bat.

29
Probability Theory
30
Probability Theory
31
Probability Theory
32
Properties of Probability
33
Properties of Probability
34
Properties of Probability
35
Enumeration or Counting Technologies
36
Enumeration or Counting Technologies
37
Enumeration or Counting Technologies
38
Enumeration or Counting Technologies
39
Enumeration or Counting Technologies
40
Conditional Probability
41
Conditional Probability
42
Conditional Probability

• Continuing the baseball analogy, managers
  frequently attempt to play the percentages by
  setting up relatively favorable match-ups. For
  example, if a particular hitter owns the
  pitcher, is really hot at the time, or bats from
  the correct side of the plate for the situation,
  the manager can expect better odds for success
  than if these facts were ignored (again, over the
  long run).

43
Conditional Probability
44
Independence
45
Independence

• The classic example is a coin flip, whereby using
  an honest coin would not tell us any extra
  information about the next flip or series.
• Other examples abound in industrial and other
  settings. We use the assumption independence to
  ease the estimation of processes and expected
  results.

46
Independence
47
Discrete Random Variables
48
Random Variables
49
Random Variables
50
Random Variable Definition
51
Random Variable Definition
52
Discrete Random Variables
53
Probability Distribution for Discrete R.V.
54
Probability Distribution for Discrete R.V.
55
Probability Distribution for Discrete R.V.

• An example a biotech firm specializing in test
  kits tracks sales and has developed the following
  distribution for forecasting

56
Probability Distribution for Discrete R.V.

• The probability mass function can be described
  by

57
Parameter of a probability distribution
58
Parameter of a probability distribution
59
Cumulative Distribution Function
60
Cumulative Distribution Function
61
Cumulative Distribution Function

• Continuing from the data provided, the CDF for
  the forecast distribution would be

62
Expected value of a Discrete R.V.
63
Expected value of a Discrete R.V.
64
Expected Value for a Discrete R.V.

• Here are two examples, noting that E(x) doesnt
  have to be an allowed distribution value.
• First, the expected value of a one die would be
• Continuing the example of the sales forecast

65
Expected Value for a Discrete R.V.

• A chip maker tracks the number of large particles
  on a chip as part of the inspection process. 100
  sets of data were taken to get an estimate of the
  probability of each number of particles and the
  expected long-run number of particles.

66
Expected Value for a Discrete R.V.

• The long-term expected value can then be
  determined from the observed distribution using
  the summation described previously

67
Rules of Expected Value
68
Expected Value of a Function
69
Variance of a Discrete RV
70
Variance of a Discrete RV
71
Variance of a Discrete R.V.

• Continuing with the chip example, the variance is
  calculated using the estimate for the mean and
  the observations

72
Variance of a Discrete R.V.

• Similarly, the variances for the forecast and die
  situations can also be calculated, and are
• Die
• Variance 2.917 s 1.708
• Forecast
• Variance 0.7875 s 0.8874
• This information is handy for determining the
  likelihood of an observation occurring.

73
Rules of Variance
74
Common Families of Discrete Probability
Distribution
75
Bernoulli Distribution
76
Binomial Distribution
77
Binomial Distribution
78
Binomial Distribution
79
Binomial Distribution
80
Binomial Distribution

• A process uses an inspection plan that calls for
  a sample of 5 units to be checked before
  shipment. One failure rejects the lot. If there
  are 10 defective, what is the probability of lot
  rejection?
• In this example, p 0.10, and n 5, giving the
  following pmf

81
Binomial Distribution

• This yields the following distribution table for
  the possible results
• There is a 59 chance that failing lots would be
  sent out.

82
Geometric Probability Distribution
83
Geometric Probability Distribution
84
Geometric Probability Distribution
85
Poisson Probability Distribution
86
Poisson Probability Distribution
87
Poisson Probability Distribution
88
Poisson Probability Distribution

• A manufacturer checks for contamination on their
  storage disks. The mean value is 0.1
  contaminants per square centimeter, with a disk
  surface of 100 square centimeters. What is the
  probability of five or more contaminants on the
  disks?
• The expected value per disk is
• 100 0.10 10 contaminants per disk

89
Poisson Probability Distribution

• The question asked for 5 or more, which can be
  calculated by difference, e.g.
• 1 P(0) P(1) P(2) P(3) P (4)
• lambda 10, so the basic relation is

90
Poisson Probability Distribution

• Tabulating the results and subtracting from 1
  gives
• This totals 0.0292528, leaving the probability as
  0.9707472 that 5 or more contaminants will be
  found

91
Poisson Probability Distribution
92
Tchebysheffs Theorem
93
Continuous Random Variables
94
Continuous Random Variables
95
Continuous Random Variables
96
Continuous Random Variables
97
Continuous Random Variable

• An example would be the following random
  variable, distributed as follows

98
Continuous Random Variable

• Its density function would be

99
Continuous Random Variables
100
Cumulative Distribution Function
101
Cumulative Distribution Function
102
Cumulative Distribution Function

• The c.d.f. for the previous example is shown
  below

103
Expected Value for a Continuous R.V.
104
Expected Value for a Continuous R.V.
105
Expected Value for a Continuous R.V.

• Continuing with the previous example
  distribution, and integrating from the allowed
  values from 0 to 2, the expected value is

106
Variance of a Continuous R.V.
107
Variance of a Continuous R.V.

• Continuing as before, and integrating from 0 to 2
  as before, the variance of the distribution is

108
Variance of a Continuous R.V.
109
Common families of Continuous Distribution
110
Common Families of Continuous Distributions
111
The Uniform Distribution
112
The Uniform Distribution
113
The Uniform Distribution
114
Exponential Distribution
115
Exponential Distribution
116
Exponential Distribution

• An example A production line has the potential
  to break down, with an average time between
  breakdown events of 10 months (e.g. ? 0.10
  /month). What is the probability of the time
  between breakdowns being one year or less?

117
Normal Distribution
118
Normal Distribution
119
Normal Distribution
120
Normal Distribution
121
Normal Distribution
122
Normal Distribution
123
Normal Distribution
124
Exponential Distribution

• An example A production line has the potential
  to break down, with an average time between
  breakdown events of 10 months (e.g. ? 0.10
  /month). What is the probability of the time
  between breakdowns being one year or less?

125
Normal Distribution

• An example of converting to standard normal
  distribution is given by the data from a dry
  plasma etch study (Lynch and Markle, 1997). The
  data are in angstroms, from the before process
  improvement trial. The mean is 564.11, and the
  standard deviation is 10.747 angstroms.

126
Normal Distribution

• This example demonstrated the yielded values from
  the normalization equation for Z. The values
  obtained can then be used to compare the
  likelihood of occurrence when comparing to other
  data. This is done in hypothesis testing and
  related methods (SPC, etc.)

127
Normal approximates to Binomial
128
Multivariate/ Joint Probability Distributions
129
Sampling distribution
130
Sampling distribution
131
Sampling distribution
132
Central limit theorem
133
Central limit theorem
134
Central limit theorem
135
Central Limit Theorem

• Using the exponential distribution and random
  number generator, it is possible to plot the
  resulting frequency distributions of data.
  Notice the trend towards normality.

136
Central Limit Theorem

• Continuing,

137
T-distribution

• Use of the t-distribution is similar to the use
  of the standard normal distribution, except that
  the degrees of freedom must be accounted for.
  The estimation of the true process mean µ by the
  experimental mean creates the loss of one degree
  of freedom in estimating the true process
  standard deviation s by s.

138
T-Distribution
139
T-Distribution
140
Parameter estimation
Statistical inference process by which
information from samples data is used to draw
conclusions about the population from which the
sample was selected.
141
Parameter estimation
142
Parameter estimation
143
Parameter estimation
144
Confidence Interval
145
Confidence Interval

• Interpreting a confidence interval? is covered
  by interval with confidence 100(1- ?).If many
  samples are taken and a 100(1- ?) CI is
  calculated for each, then 100(1-?) of them will
  contain/ cover the true value for ?.
• Note the larger (wider) a CI, the more confident
  we are that the interval contains the true value
  of ?.
• But, the longer it is, the less we know about ?,
  due to variability or uncertainty ? need to
  balance

146
Confidence Interval
147
Confidence Interval
148
Confidence Interval
149
Confidence Interval
150
Confidence Interval
151
Confidence Interval

• Interpreting a confidence interval? is covered
  by interval with confidence 100(1- ?).If many
  samples are taken and a 100(1- ?) CI is
  calculated for each, then 100(1-?) of them will
  contain/ cover the true value for ?.
• Note the larger (wider) a CI, the more confident
  we are that the interval contains the true value
  of ?.
• But, the longer it is, the less we know about ?,
  due to variability or uncertainty ? need to
  balance

152
Normal Distribution

• An example of converting to standard normal
  distribution is given by the data from a dry
  plasma etch study (Lynch and Markle, 1997). The
  data are in angstroms, from the before process
  improvement trial. The mean is 564.11, and the
  standard deviation is 10.747 angstroms.

153
Confidence Interval

• Continuing,

154
Sample Size Needed

• Suppose we desire a confidence interval
• Based on a preliminary sample of n0, we have an
  estimate of S2 and confidence interval

155
Sample Size needed

• Find n such that
• If n is large,

156
Hypothesis Tests - Review

• Hypothesis Tests
• Objective This section devoted to enabling us
  to
• Construct and test a valid statistical hypothesis
• Conduct comparative statistical tests( t-tests)
• Relate alpha and beta risk to sample size
• Conceptually understand analysis of variance
  (ANOVA)
• Interpret the results of various statistical
  tests
• T-tests, f-tests, chi-square tests.
• Understand the foundation for full and fractional
  factorial
• Compute confidence intervals to assess degree of
  improvements

157
Hypothesis Tests

• Hypotheses defined
• Used to infer population characteristics from
  observed data.
• Hypothesis test A series of procedures that
  allows us to make inferences about a population
  by analyzing samples
• Key question was the observed outcomes the
  result of chance variation, or was it an unusual
  event?
• Hint Frequency Area Probability

158
Hypothesis Tests

• Hypothesis Definition of terms
• Null hypothesis (H0) Statement of no change or
  difference. This statement is tested directly,
  and we either reject H0 or we do not reject H0
• Alternative hypothesis (H1) The statement that
  must be true if H0 is rejected.

159
Hypothesis Tests

• Definition of terms
• Type I error The mistake of rejecting H0 when it
  is true.
• Type II error The mistake of failing to reject
  H0 when it is false.
• alpha risk (?)Probability of a type I error
• beta risk (?) Probability of a type II error
• Test statistic sample value used in making
  decision about whether or not to reject H0

160
Hypothesis Tests

• Definition of terms
• Critical region Area under the curve
  corresponding to test statistic that leads to
  rejection of H0
• Critical value The value that separates the
  critical region from those values that do not
  lead to rejection of H0
• Significance level The probability of rejecting
  H0 when it is true
• Degrees of freedom Referred to as d.f. or ?, and
  n - 1

161
Hypothesis Tests

• Definition of terms
• Type I error Producers risk
• Type II error Consumers risk
• Set so type I is the more serious error type
  (taking action when none is required)
• Levels for ? and ? must be established before the
  test is conducted

162
Hypothesis Tests

• Hypothesis Definition of terms
• Degree of freedom
• Degree of freedom are a way of counting the
  information in an experiment. In other words,
  they relate to sample size. More specifically,
  d.f. n 1
• A degree of freedom corresponds to the number of
  values that are free to vary in the sample. If
  you have a sample with 20 data points, each of
  the data points provides a distinct place of
  information. The data set is described completely
  by these 20 values. If you calculate the mean for
  this set of data, no new information is created
  because the mean was implied by all of the
  information in the 20 data points.

163
Hypothesis Tests

• Hypothesis Definition of terms
• Degree of freedom
• Once the mean is known, though, all of the
  information in the data set can be described with
  any 19 data points. The information in a 20th
  data point is now redundant because the 20th data
  points has lost the freedom to have any value
  besides the one imposed on it by the mean
• We have one less than the total in our sample
  because a sample is at least one less than the
  total population.

164
Hypothesis Tests

• If the population variance is unknown, use s of
  the sample to approximate population variance,
  since under central limit theorem, s ? when n gt
  30. Thus solve the problem as before, using s
• With smaller sample sizes, we have a different
  problem. But it is solved in the same manner.
  Instead of using the z distribution, we use the t
  distribution

165
Hypothesis Tests

• Using t distribution when
• Sample is small (lt30)
• Parent population is essentially normal
• Population variance (?) is unknown
• As n decreases, variation within the sample
  increases, so distribution becomes flatter.

166
Methods to Test a Statistical Hypothesis
167
Methods to Test a Statistical Hypothesis
168
Relationship Between Hypothesis Tests and
Confidence Intervals
169
Relationship Between Hypothesis Tests and
Confidence Intervals
170
Relationship between Hypothesis Tests and
Confidence Intervals

• Using the data from the plasma etch study, can a
  true process mean of 530 angstroms be expected at
  a 95 confidence level?
• The 95 confidence interval (developed earlier in
  detail) runs from 555.85 to 572.37. Since 530 is
  not included in this interval, the null
  hypothesis of µ 530 is rejected.

171
Confidence Interval
172
Prediction Interval
173
Prediction Interval
174
Prediction Interval
175
P - Values

• The P value is the smallest level of
  significance that leads to rejection of the null
  hypothesis with the given data. It is the
  probability attached to the value of the Z
  statistic developed in experimental conditions.
  It is dependent upon the type of test (two-sided,
  upper, or lower tail tests) selected to analyze
  data significance.

176
Confidence Interval
177
P-value
178
P-value
179
Hypothesis Tests

• Compare the means of two samples Steps
• Understand word problem by writing out null and
  alternative hypotheses
• Select alpha risk level and find critical value
• Draw graph of the relation
• Insert data into formula
• Interpret and conclude

180
Test for comparing two means
181
Tests for Comparing Two Means

• A company wanted to compare the production from
  two process lines to see if there was a
  statistically significant difference in the
  outputs, which would then require separate
  tracking. The line data is as follows
• A 15 samples, mean of 24.2, and variance of 10
• B 10 samples, mean of 23.9, and variance of 20
• 95 confidence

182
P - Values

• Using the data developed from the process line
  example, but with line A having a mean of 27.2,
  instead of 24.2, the P-value would be

183
Test for comparing two means
184
Test for comparing two means
185
Tests for Comparing Two Means

• A process improvement by exercising equipment was
  attempted for an etch line. Given that the true
  variances are unknown but equal, determine
  whether a statistically significant difference
  exists at the 95 confidence level.
• Before Mean 564.108, standard deviation
  10.7475, number of observations 9.
• Exercise Mean 561.263, standard deviation
  7.6214, number of observations 9.

186
Tests for Comparing Two Means

• Since the variances are equal, the pooled
  variance is used for creation of the confidence
  interval. If zero is included, there is no
  statistically significant difference.
• There are 16 degrees of freedom, and at the a/2
  0.025 level, the critical value for t is 2.120.

187
Test for comparing two means
188
Tests for Comparing Two Means

• An etch process was improved by recalibration of
  equipment. The values for a determination of
  statistically significant improvement at the 95
  confidence level are given as follows
• Before Mean 564.108, standard deviation
  10.7475, number of observations 9.
• Calibrated Mean 552.340, standard
  deviation 2.3280, number of observations 9.
• The null hypothesis is that µb µc 0

189
Tests for Comparing Two Means

• The first task is to determine the number of
  degrees of freedom and the appropriate critical
  value.

190
Tests for Comparing Two Means

• For 9 degrees of freedom and a/2 0.025, the
  critical value for t is 2.262

191
Test for comparing two means
192
Tests for Comparing Two Means
193
Tests for Comparing Two Means

• Two materials were compared in a wear test as a
  paired, randomized experiment. The coded data
  are tabulated below, as well as the differences
  for each pairing.

194
Tests for Comparing Two Means

• The mean, standard deviation, and 95 confidence
  intervals are constructed below, with nine
  degrees of freedom. If the interval does not
  contain zero, the null hypothesis of no
  difference is rejected.

195
Test for Comparing Two Variances

• The variances can also be used to determine the
  likelihood of observations being part of the same
  population. For variances, the test used is the
  F-test since the ratio of variances will follow
  the F-distribution. This test is also the basic
  test used in the ANOVA method covered in the next
  section.

196
F - Test

• The F distribution is developed from three
  parameters a (level of confidence), and the two
  degrees of freedom for the variances under
  comparison. The null hypothesis is typically one
  where the variances are equal, which would yield
  an allowed set of values that F can be and still
  not reject the null hypothesis.

197
F - Test

• If the observed value of F is outside of this
  range, the null hypothesis is rejected and the
  observation is statistically significant.
• Tables for the F distribution are found in texts
  with a statistical interest. Normally, the ratio
  is tabulated with the higher variance in the
  denominator, the lower variance in the numerator.

198
F - Test

• Compare two sample variances
• Compare two newly drawn samples to determine if a
  difference exists between the variance of the
  samples.(Up until now we have compared samples
  to populations, and sample means)
• For two normally distributed populations with
  equal variances. ?12 ?22
• We can compare the two variances such that s12 /
  s22 Fmax where s12 gt s22

199
F - Test

• Compare two sample variances
• F tests for equality of the variances and uses
  the f-distribution.
• This works just like method used with the t
  distribution critical value is compared to test
  statistics.
• If two variances are equal, F s12 / s22 1
  ,thus we compare ratios of variances.

200
F - Test

• Compare two sample variances
• If two variances are equal, F s12 / s22
  1Thus, we compare ratios of variances
• Large F leads to conclusion variances are very
  different.
• Small F (close to 1) leads to conclusion
  variances do not differ significantly. Thus for F
  test
• H0 s12 s22 H1 s12 ? s22

201
F - Test

• Compare two sample variances
• F tables
• Several exist, depending on alpha level.
• Using F tables requires 3 bits of information.
• Chosen alpha risk
• Degree of freedom (n1 1) for numerator term.
• Degree of freedom (n2 1) for denominator term.

202
Test for comparing two variances
203
F - Test

• An etching process of semiconductor wafers is
  used with two separate gas treatment mixtures on
  20 wafers each. The standard deviations of
  treatments 1 and 2 are 2.13 and 1.96 angstroms,
  respectively. Is there a significant difference
  between treatments at the 95 confidence level?

204
F - Test

• 95 confidence level infers a 0.05, and also
  since this is a two-tailed distribution, a/2
  0.025 is used for F. There are 19 degrees of
  freedom for each set of data.
• Therefore the null hypothesis of no difference
  in treatments cannot be rejected.

205
Test for comparing two proportions
206
Test for comparing two means
207
Test for comparing two means
208
Test for comparing two means
209
Test for comparing two means
210
Test for comparing two means - Paired
211
Test for comparing two means - Paired
212
Test for comparing two means - Paired
213
Test for comparing two variances
214
Test for comparing two variances
215
ANOVA

• Analysis of Variance
• Employs F distribution to compare ratio of
  variances of samples when true population
  variance is unknown
• Compares variance between samples to the variance
  within samples (variance of means compared to
  mean of variances). If ratio of variance of means
  gt mean of variances, the effect is significant
• Can be used with more than 2 group at a time
• Requires independent samples and normally
  distributed population.

216
ANOVA

• ANOVA
• Anova concept

217
ANOVA

• ANOVA Concepts
• All we are saying is
• Assumption that the population variances are
  equal or nearly equal allows us to treat the
  samples from many different populations as if
  they in fact belonged to a single large
  population. If that is true, then variance among
  samples should nearly equal variance within the
  samples
• H0 ?1 ?2 ?3 ?4 ?k

218
ANOVA

• ANOVA Steps
• Understand word problem by writing out null and
  alternative hypotheses
• Select alpha risk level and find critical value
• Run the experiment
• Insert data into Anova formula
• Draw graph of relation
• Interpret and conclude

219
ANOVA

• Analysis of Variance (ANOVA) is a powerful tool
  for determining significant contributors towards
  process responses. The process is a vector
  decomposition of a response, and can be modified
  for a wide variety of models.
• Can be used with more than 2 groups at a time
• Requires independent, normally distributed
  samples.

220
ANOVA

• ANOVA decomposition is an orthogonal vector
  breakdown of a response (which is why
  independence is required), so for a process with
  factors A and B, as tabulated below

221
ANOVA

• The ANOVA values are given by

222
ANOVA

• In this case, we use it to demonstrate how the
  deposition of oxide on wafers as described by
  Czitrom and Reese (1997) can be decomposed into
  significant factors, checking the wafer type and
  furnace location. The effects will be removed in
  sequence and verified for significance using the
  F-test. The proposed model is
• YM W F R, where
• M is the grand mean, W is the effect of a given
  wafer type, F is the effect of a particular
  furnace location, and R is the residual. Y
  denotes the observed value.

223
ANOVA

• F-test in ANOVA
• The estimator for a given variance in ANOVA is
  the mean sum of squares (MS). For a given
  factor, its MS can be calculated as noted before,
  and the ratio of the factor MS and residual MS
  compared to the F-distribution for significance.
  To do so, the level of significance ? must be
  defined to establish the Type I error limit.

224
ANOVA

• In this example, the level of significance is
  selected at 0.10, yielding the following table of
  upper bounds for the F-test. In all cases, the
  higher variance (usually the factor) is divided
  by the lower variance (usually the residual).

225
ANOVA

• The set of means observed in the process, broken
  down by wafer type and furnace location are (in
  mils 0.001 inch) tabulated below. The grand
  mean is 92.1485.

226
ANOVA

• The sum of squares about the grand mean is found
  by adding the squares of all of the deviations in
  the 12 inner cells, and totals 6.7819. One
  degree of freedom is expended in fixing the mean,
  and 11 are left.

227
ANOVA

• Determining the sum of squares for the wafer
  types is done by multiplying the squared
  difference between a type and the grand mean by
  the number of times that type appears in the data
  (e.g. 4squared differences). This is done for
  all types, and totals 3.4764, with 2 degrees of
  freedom.

228
ANOVA

• The residual sum of squares totals 3.3145 with
  nine degrees of freedom, and indicates that the
  wafer type may not be the only significant
  factor. The significance of the wafer type is
  verified using the F-test

229
ANOVA

• As noted before, a residual sum of squares of
  about 50 percent of the total sum of squares may
  indicate the presence of a significant factor.
  The effect of furnace location is then removed
  from the data and tested for significance as
  before. The furnace location sum of squares
  totals 2.7863, with three degrees of freedom.

230
ANOVA

• The remaining residual sum of squares totals
  0.5282 with six degrees of freedom. Repeating
  the F-tests as before for both factors yields

231
ANOVA

• These are very significant values to the ? 0.02
  level. The resulting ANOVA table shows all of
  the factors combined

232
ANOVA

• The last task is to verify that the residuals
  follow a normal distribution about the expected
  value from the model. This is done easily using
  a normal plot and checking that the responses are
  approximately linear. Patterns or significant
  deviations could indicate another significant
  factor affecting the response. The plot that
  follows of the residual responses from all values
  shows no significant indication of non-normal
  behavior, and we fail to reject (on this level)
  the null hypothesis of the residuals conforming
  to a normal distribution around the model
  predicted value.

233
ANOVA
234
DOE Overview

• Focus is on univariate statistics (1 dependent
  variable) vs. multivariate statistics (more than
  one dependent variable)
• Focus is on basic designs, and does not include
  all possible types of designs (I.e. Latin
  squares, incomplete blocks, nested, etc.)

235
DOE Overview

• One key item to keep in clear focus while
  performing a designed experiment
• Why are we doing it?
• According to Taguchi (among others) it is to
  refine our process to yield one of the following
  quality outcomes

236
DOE Overview

• Bigger is better (yields, income, some tensile
  parameters, etc.)
• Smaller is better (costs, defects, taxes,
  contaminants, etc.)
• Nominal is best (Most dimensions, and associated
  parameters, etc.)
• Remember also that whatever is selected as the
  standard for comparison, it must be measured!

237
ISE-130

• The End !
• Have a nice vacation!