Lecture 3 Statistical ProcessControl (SPC).docx

Lecture 3
Statistical Process
Control (SPC)
Data collection for Six SigmaData are simply facts and figures
without context or interpretation.Information refers to useful or
meaningful patterns found in the data.Knowledge represents
information of sufficient quality and/or quantity that actions can
be taken based on the information.If data are not collected and
used wisely, their vary existence can lead to activities that are
ineffective and possibly even counterproductive.An
organization collects data & reacts whenever an out-of-
specification condition occurs.
“Common cause” & “ special cause” variation
There are two causes of process variations:
1) Common cause variation: This variation is due to the process
only. It may not tell you whether the process meets the needs of
the customer unless it is compared with the specification. This
can be improved by focusing on the process.

2) Special cause variation: This variation is due the individual
employee, if the point is beyond specification limits. In this
case the focus should be about what happened relative to the
individual employee as though it were a “special” condition.
Attribute versus Variable Data
Attribute data: It is a data with yes or no decision such
as:whether an iten passed or failed a testpass/fail, go/no go
gaging, true/false, accept/reject. There are no quantifiable
values
Variable data: are related to measurements with quantifiable
values such as:Diameter of a part which has been
machinedlength or thickness of the machined part
The success of Six SigmaThe success of Six Sigma depends
upon knowing the difference between special & common cause
variations and how the organization reacts to the data.If the
management focuses on wrong cause of variation, it can lead to
waste of time (firefighting).It can also effect employee
motivation & morale.Reacting to one data point that do not meet
the specification limit can be counterproductive and very
expensive.Do not use “firefighting” actions just because the
data point is out of specification limits. It must first be
determined whether the condition is common or special cause.

Example of variability due to common causeControl limits are
calculated from the sample data.There are no data points outside
the control limits therefore there are no special causes within
the data.The source of variation in this case is “common cause”
due to process.
Type of firefighting done by management before evaluating the
cause of variabilityProduction supervisors might constantly
review production output by employee, machine, product line,
work shift etc.An administrative assistant’s daily output &
memo’s may be monitored.The average time per call may be
monitored in a call center.The efficiency of computer
programmers may be monitored by tracking “lines of code
produced per day”.
All of these actions would be a waste of time if the cause of
variability is “common cause” and due to the process rather
than individual employees who have no control over it.
Goals for Six Sigma projectsThe success of Six Sigma depends
upon how goals are set & any boundaries that are set relative to
meeting those objectives.Arbitrary goals that are set for
employees can be counterproductive and costly, when the
outcome is beyond the employee’s control. If management
extends the scope of what can be done by employees to fix
problems, real productivity improvement can result.

Organization sometimes can be trapped into “declaring victory”
with one metric at the expense of another metric.
Strategy for Six Sigma successIt should deal with numbers,
information & knowledge.It should quantify & track
measurements that are vital to the success of the organization.A
wisely applied Six Sigma strategy can facilitate productive
utilization of information.It can be used in manufacturing,
service, product design, and transactional applications.Visual
observations, pictures, data analysis and statistical assessments
are the key to the success of a Six Sigma projects.
CONTROL CHARTS
What is a Control Chart ?
A control chart is simply a run chart with confidence intervals
calculated and drawn in. These “Statistical control limits” form
the trip wires which enable us to determine when a process
characteristic is operating under the influence of a “Special
cause”.

+/- 3s =
99% Confidence Interval

Control charts are useful to: determine the occurrence of
“special cause” situations.Utilize the opportunities presented by
“special cause” situations” to identify and correct the
occurrence of the “special causes” .
IMPROVEMENT ROADMAP
Uses of Control Charts
Breakthrough
Strategy
Phase 4:
Control
Characterization
Phase 1:
Measurement

Phase 2:
Analysis
Optimization
Phase 3:
Improvement
Control charts can be effectively used to determine “special
cause” situations in the Measurement and Analysis phases
Common Uses
KEYS TO SUCCESS
Use control charts on only a few critical output characteristics
Ensure that you have the means to investigate any “special
cause”
What is a “Special Cause”?
Remember our earlier work with confidence intervals? Any
occurrence which falls outside the confidence interval has a low
probability of occurring by random chance and therefore is
“significantly different”. If we can identify and correct the
cause, we have an opportunity to significantly improve the

stability of the process. Due to the amount of data involved,
control charts have historically used 99% confidence for
determining the occurrence of these “special causes”
Special cause occurrence.
X
Point Value

+/- 3s = 99% Confidence Band
So how do I construct a control chart?
First things first: Select the metric to be evaluatedSelect the
right control chart for the metricGather enough data to calculate
the control limitsPlot the data on the chartDraw the control
limits (UCL & LCL) onto the chart.Continue the run,
investigating and correcting the cause of any “out of control”
occurrence.
How do I select the correct chart ?
Note: A defective unit can have more than one defect.
What type of data do I have?
Variable
Attribute
Counting defects or defectives?
X-s Chart
IMR Chart
X-R Chart

n > 10
1 < n < 10
n = 1
Defectives
Defects
What subgroup size is available?
Constant Sample Size?
Constant Opportunity?
yes
yes
no
no
np Chart
u Chart
p Chart
c Chart
SPC Charts
Steps for SPC implementation

Processes ideally suited for SPCAre repetitive (produce
simmilar items)Have a high inspection contentHave higher than
desired reject rateItems (parts) with dimensions that are easy to
measure
SPC charts
The most frequently SPC charts are:
1) Xbar:r charts
2) p-charts
Xbar:r charts are used for controlling processes which have
“variable” data.
p-charts are used for controlling processes which have
“attribute” data
Example of Xbar:r chart for a machined part
Example of a process which is in control
Example of a process which is heading out of control

Example of out of control process
(If 7 or more sequential data points are above or below Xbar
line)
Steps to develop Xbar chart
Step 1: Collect 100 different data points as follows:
(20 different subgroups with 5 data points in each subgroup)
Step 2: Calculate the value of the grand average as
follows:Calculate the average value of each of the 20
subgroups.Calculate the average of all 20 subgroups called
Xbar.This is the center-line of the Xbar chart.
Step 3: Calculate the standard deviation (s) of the average
values of 20 subgroups
Step 4: Multiply the standard deviation value by 3
Upper control limit = Xbar + 3s
Lower control limit = Xbar - 3s
Steps to develop r-chart
Step 1: Determine the ranges for each of the 20 subgroups
Step 2: Calculate the value of the grand average as
follows:Calculate the grand average by summing all 20 values
and then dividing the sum by 20. This is called rbar This is the
center-line of the rbar chart.
Step 3: Calculate the standard deviation (s) of the average
r-values values of 20 subgroups
Step 4: Multiply the standard deviation value by 3
Upper control limit = rbar + 3s

Lower control limit = rbar - 3s
Example of out of control process
(If a pattern is repeated in the Xbar chart)
Steps to develop p-chart
Step 1: determine the number of samples “n” to develop the
chart.
Step 2: Inspect each sample and identify the number of
defects in each sample.
Step 3 Calculate the percentage defects “p” for each sample.
Step 4: Calculate the paverage as follows:
paverage = (n1p1+n2p2+ ….. nkpk) / (n1+ n2
+…….nk)
Step 5: Calculate the upper and lower control limits as follows:
Components of a control chart
Cases when the process is “out of control”
and should be stopped for investigation

What do I do when the process is “out of control”?
Time to Find and Fix the cause Look for patterns in the
dataAnalyze the “out of control” occurrenceFishbone diagrams
and Hypothesis tests are valuable “discovery” tools.
Stages of continuous improvement
UCL =
Paverage
+ 3
(
Paverage
)
(
1
-
Paverage
)
n
LCL =
Paverage
-
3
(
Paverage
)
(

1
-
Paverage
)
n
Unit II Essay
Over the course of this unit, we have discussed the importance
of mission and vision statements. As a part of that discussion,
we analyzed mission and vision statements for their
effectiveness. For the Unit II Essay, you will expand on this
topic.
Using your favorite search engine, research the mission and
vision statements of different Fortune 500 companies. Then, you
will write an essay in which you compare and contrast the
mission statements of two companies and the vision statements
of two companies. You may use the same companies for both
the mission and vision comparisons or separate companies.
Within your essay, include the following: Explain the principle
value of two vision statements. Explain the principle value of
two mission statements. Compare and contrast vision
statements of each organization in terms of composition and
importance. Compare and contrast mission statements of each
organization in terms of composition and importance. Do you
think organizations that have comprehensive mission statements
tend to be high performers? How do mission and vision
statements assist in selecting an industry-specific strategy?
Explain why a mission statement should not include monetary
amounts, numbers, percentages, ratios, goals, or objectives.
Your essay should be a minimum of three pages in length or
approximately 750 words, not including the title and reference
pages. You must also include an outside source from the
Waldorf Online Library to support your explanations. Follow
APA standards for formatting and referencing.

Access to the Waldorf Online Library:
https://mywaldorf.waldorf.edu/student/resources/library/
Lecture 2
Review of
basic statistics
IMPROVEMENT ROADMAP
Uses of Probability DistributionsEstablish baseline data
characteristics.
Project UsesIdentify and isolate sources of variation.Use the
concept of shift & drift to establish project
expectations.Demonstrate before and after results are not
random chance.
Breakthrough
Strategy
Phase 4:
Control
Characterization

Phase 1:
Measurement
Phase 2:
Analysis
Optimization
Phase 3:
Improvement
Measurements are critical...If we can’t accurately measure
something, we really don’t know much about it.If we don’t
know much about it, we can’t control it.If we can’t control it,
we are at the mercy of chance.
Types of DataData where the metric is composed of a
classification in one of two (or more) categories is called
Attribute data. This data is usually presented as a “count” or
“percent”.Good/BadYes/NoHit/Miss etc.Data where the metric
consists of a number which indicates a precise value is called
Variable data.TimeMiles/Hr
Use Minitab to have students record the results and have the
students display using Graph..HistogramNote how “rough” the

graph looksRedo using Basic Statistics …. Descriptive Statistics
and display using the Graphical Summary. Walk through the
normal curve transform:Mean (Arithmetic Average)Standard
DeviationSkew (How off center the data is skewed -
=left)Kurtosis (How flat or peaked the data is -=flat)Show the
Box Plot:Quartile (25% of the Data Points)Median (50% of the
Data Points on Each Side)Show the 95% Confidence Interval
and Explain how it relates to the data.
Probability and Statistics Probability and Statistics influence
our lives daily Statistics is the universal language for science
Statistics is the art of collecting, classifying,
presenting, interpreting and analyzing numerical
data, as well as making conclusions about the
system from which the data was obtained.
Population Vs. Sample (Certainty Vs. Uncertainty) A sample is
just a subset of all possible values
population
sample
Since the sample does not contain all the possible values, there
is some uncertainty about the population. Hence any statistics,
such as mean and standard deviation, are just estimates of the
true population parameters.

Descriptive Statistics
Descriptive Statistics is the branch of statistics which
most people are familiar. It characterizes and summarizes
the most prominent features of a given set of data (means,
medians, standard deviations, percentiles, graphs, tables
and charts.
Descriptive Statistics describe the elements of
a population as a whole or to describe data that represent
just a sample of elements from the entire population
Inferential Statistics
Inferential Statistics is the branch of statistics that deals with
drawing conclusions about a population based on information
obtained from a sample drawn from that population.
While descriptive statistics has been taught for centuries,
inferential statistics is a relatively new phenomenon having
its roots in the 20th century.
We “infer” something about a population when only information
from a sample is known.
Probability is the link between
Descriptive and Inferential Statistics

USES OF PROBABILITY DISTRIBUTIONS
Primarily these distributions are used to test for significant
differences in data sets.
To be classified as significant, the actual measured value must
exceed a critical value. The critical value is tabular value
determined by the probability distribution and the risk of error.
This risk of error is called a risk and indicates the probability of
this value occurring naturally. So, an a risk of .05 (5%) means
that this critical value will be exceeded by a random occurrence
less than 5% of the time.

Critical Value
Critical Value
Common Occurrence
Rare Occurrence
Rare Occurrence
WHAT IS THE MEAN?
The mean is the most common measure of central tendency for a
population. The mean is simply the average value of the data.
n=12
Mean

ORDERED DATA SET
-5
-3
-1
-1
0
0
0
0
0
1
3
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
4
x
i
=
-
å
2

mean
x
x
n
i
=
=
=
-
=
-
å
2
12
17
.
WHAT IS THE MEDIAN?
ORDERED DATA SET
-5
-3
-1

-1
0
0
0
0
0
1
3
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
4
If we rank order (descending or ascending) the data set for this
distribution we could represent central tendency by the order of
the data points.
If we find the value half way (50%) through the data points, we
have another way of representing central tendency. This is
called the median value.
Median Value
Median
50% of data points

WHAT IS THE MODE?
If we rank order (descending or ascending) the data set for this
distribution we find several ways we can represent central
tendency.
We find that a single value occurs more often than any other.
Since we know that there is a higher chance of this occurrence
in the middle of the distribution, we can use this feature as an
indicator of central tendency. This is called the mode.
Mode
Mode
ORDERED DATA SET
-5
-3
-1
-1
0
0
0
0
0
1
3
-6
-5

-4
-3
-2
-1
0
1
2
3
4
5
6
4
MEASURES OF CENTRAL TENDENCY, SUMMARY
X
X
n
i
=
=
-
=
å
2
12
17
.
X
MEAN ( )
(Otherwise known as the average)

ORDERED DATA SET
-5
-3
-1
-1
0
0
0
0
0
1
3
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6

4
ORDERED DATA SET
-5
-3
-1
-1
0
0
0
0
0
1
3
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5

6
4
ORDERED DATA SET
-5
-3
-1
-1
0
0
0
0
0
1
3
-6
-5
-4
-3
-2
-1
0
1
2
3
4

5
6
4
n/2=6
n/2=6
}
Mode = 0
Mode = 0
MEDIAN
(50 percentile data point)
Here the median value falls between two zero values and
therefore is zero. If the values were say 2 and 3 instead, the
median would be 2.5.
MODE
(Most common value in the data set)
The mode in this case is 0 with 5 occurrences within this data.
Median
n=12
SO WHAT’S THE REAL DIFFERENCE?
MEAN
The mean is the most consistently accurate measure of central
tendency, but is more difficult to calculate than the other
measures.
MEDIAN AND MODE
The median and mode are both very easy to determine. That’s
the good news….The bad news is that both are more susceptible
to bias than the mean.

SO WHAT’S THE BOTTOM LINE?
MEAN
Use on all occasions unless a circumstance prohibits its use.
MEDIAN AND MODE
Only use if you cannot use mean.
Example of tossing a coin 200 time
(probability of getting heads)
1
3
0
1
2
0
1
1
0
1
0
0
9
0
8
0
7
0

6
0
0
5
0
0
4
0
0
3
0
0
2
0
0
1
0
0
0
Number of occurrences

What are some of the ways that we can easily indicate the
dispersion (spread) characteristic of the population?
Three measures have historically been used; the range, the
standard deviation and the variance.
WHAT IS THE RANGE?
ORDERED DATA SET
-5
-3

-1
-1
0
0
0
0
0
1
3
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
4
Range
Range
x
x
MAX
MIN
=
-
=
-
-
=

4
5
9
(
)
The range is a very common metric which is easily determined
from any ordered sample. To calculate the range simply
subtract the minimum value in the sample from the maximum
value.
Range
Max
Min
WHAT IS THE VARIANCE & STANDARD DEVIATION?
The variance (s2) is a very robust metric which requires a fair
amount of work to determine. The standard deviation(s) is the
square root of the variance and is the most commonly used
measure of dispersion for larger sample sizes.
DATA SET
-5
-3
-1
-1
0
0
0
0
0
1
3
-6

-5
-4
-3
-2
-1
0
1
2
3
4
5
6
4
X
X
i
-
-5-(-.17)=-4.83
-3-(-.17)=-2.83
-1-(-.17)=-.83
-1-(-.17)=-.83
0-(-.17)=.17
0-(-.17)=.17
0-(-.17)=.17
0-(-.17)=.17
0-(-.17)=.17
1-(-.17)=1.17
3-(-.17)=3.17
4-(-.17)=4.17
(-4.83)2=23.32
(-2.83)2=8.01
(-.83)2=.69
(-.83)2=.69
(.17)2=.03
(.17)2=.03
(.17)2=.03

(.17)2=.03
(.17)2=.03
(1.17)2=1.37
(3.17)2=10.05
(4.17)2=17.39
61.67
(
)
s
X
X
n
i
2
2
1
61
67
12
1
5
6
=
-
-
=
-
=
å
.
.

X
X
n
i
=
=
-
=
å
2
12
-.17
MEASURES OF DISPERSION
ORDERED DATA SET
-5
-3
-1
-1
0

0
0
0
0
1
3
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
4
ORDERED DATA SET
-5
-3
-1
-1

0
0
0
0
0
1
3
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
4
Min=-5
R
X
X
=
-
=
-
-
=
max
min
(
)

4
6
10
Max=4
DATA SET
-5
-3
-1
-1
0
0
0
0
0
1
3
-6
-5
-4
-3
-2
-1
0
1

2
3
4
5
6
4
X
X
n
i
=
=
-
=
å
2
12
-.17
(
)
s
X
X
n
i
2
2
1

61
67
12
1
5
6
=
-
-
=
-
=
å
.
.
X
X
i
-
-5-(-.17)=-4.83
-3-(-.17)=-2.83
-1-(-.17)=-.83
-1-(-.17)=-.83
0-(-.17)=.17
0-(-.17)=.17
0-(-.17)=.17
0-(-.17)=.17
0-(-.17)=.17
1-(-.17)=1.17
3-(-.17)=3.17
4-(-.17)=4.17
(-4.83)2=23.32
(-2.83)2=8.01
(-.83)2=.69
(-.83)2=.69

(.17)2=.03
(.17)2=.03
(.17)2=.03
(.17)2=.03
(.17)2=.03
(1.17)2=1.37
(3.17)2=10.05
(4.17)2=17.39
61.67
s
s
=
=
=
2
5
6
2
37
.
.
RANGE (R)
(The maximum data value minus the minimum)
VARIANCE (s2)
(Squared deviations around the center point)

STANDARD DEVIATION (s)
(Absolute deviation around the center point)
SO WHAT’S THE REAL DIFFERENCE?
VARIANCE/ STANDARD DEVIATION
The standard deviation is the most consistently accurate
measure of central tendency for a single population. The
variance has the added benefit of being additive over multiple
populations. Both are difficult and time consuming to calculate.
RANGE
The range is very easy to determine. That’s the good
news….The bad news is that it is very susceptible to bias.
SO WHAT’S THE BOTTOM LINE?
VARIANCE/ STANDARD DEVIATION
Best used when you have enough samples (>10).
RANGE
Good for small samples (10 or less).
SO WHAT IS THIS SHIFT & DRIFT STUFF...
The project is progressing well and you wrap it up. 6 months
later you are surprised to find that the population has taken a
shift.
-12
USL
LSL

Time
All of our work was focused in a narrow time frame. Over
time, other long term influences come and go which move the
population and change some of its characteristics. This is
called shift and drift.
Historically, this shift and drift primarily impacts the position
of the mean and shifts it 1.5 s from it’s original position.
Original Study
VARIATION FAMILIES
Variation is present upon repeat measurements within the same
sample.
Variation is present upon measurements of different samples
collected within a short time frame.
Variation is present upon measurements collected with a
significant amount of time between samples.

Sources of Variation
Within Individual Sample
Piece to Piece
Time to Time
SO WHAT DOES IT MEAN?
To compensate for these long term variations, we must consider
two sets of metrics. Short term metrics are those which
typically are associated with our work. Long term metrics take
the short term metric data and degrade it by an average of 1.5s.
IMPACT OF 1.5s SHIFT AND DRIFT
Z
PPM
ST
C
pk
PPM
LT
(+1.5
s
)
0.0
500,000
0.0
933,193
0.1

460,172
0.0
919,243
0.2
420,740
0.1
903,199
0.3
382,089
0.1
884,930
0.4
344,578
0.1
864,334
0.5
308,538
0.2
841,345
0.6
274,253
0.2
815,940
0.7
241,964
0.2
788,145
0.8
211,855
0.3
758,036
0.9
184,060
0.3
725,747
1.0

158,655
0.3
691,462
1.1
135,666
0.4
655,422
1.2
115,070
0.4
617,911
1.3
96,801
0.4
579,260
1.4
80,757
0.5
539,828
1.5
66,807
0.5
500,000
1.6
54,799
0.5
460,172
1.7
44,565
0.6
420,740
Here, you can see that the impact of this concept is potentially
very significant. In the short term, we have driven the defect
rate down to 54,800 ppm and can expect to see occasional long

term ppm to be as bad as 460,000 ppm.
IMPROVEMENT ROADMAP
Uses of Probability Distributions
Breakthrough
Strategy
Phase 4:
Control
Characterization
Phase 1:
Measurement
Phase 2:
Analysis
Optimization
Phase 3:

Improvement
Baselining Processes
Verifying Improvements
Common Uses
Data points vary, but as the data accumulates, it forms a
distribution which occurs naturally.
Location
Spread
Shape
Distributions can vary in:
PROBABILITY DISTRIBUTIONS,
WHERE DO THEY COME FROM?

Sheet: Sheet1
Sheet: Sheet2
Sheet: Sheet3
Sheet: Sheet4
Sheet: Sheet5
Sheet: Sheet6
Sheet: Sheet7

Sheet: Sheet8
Sheet: Sheet9
Sheet: Sheet10
Sheet: Sheet11
Sheet: Sheet12
Sheet: Sheet13
Sheet: Sheet14
Sheet: Sheet15
Sheet: Sheet16
X
COMMON PROBABILITY DISTRIBUTIONS

-4
-3
-2
-1
0
1
2
3
4
0
1
2
3
4
5
6
7

-4
-3
-2
-1
0
1
2
3
4
0
1
2
3
4

Original Population
Subgroup Average
Subgroup Variance (s2)
Continuous Distribution
Normal Distribution
c2 Distribution
Population and Sample Symbology
s
2
s
2
x
P
P
Cp
Value
Population

Standard Deviation
s
s
Process Capability
Cp
Binomial Mean

68.26%
2 tail = 32%
1 tail = 16%
+/- 1s = 68%

2 tail = 4.6%
1 tail = 2.3%
+/- 2s = 95%
-3s
+3s

99.73%
2 tail = 0.3%
1 tail = .15%
+/- 3s = 99.7%
Common Test Values
Z(1.6) = 5.5% (1 tail a=.05)
Z(2.0) = 2.5% (2 tail a=.05)
The Focus of Six Sigma…..
Y = f(x)
All critical characteristics (Y) are driven by factors (x) which
are “downstream” from the results….
Attempting to manage results (Y) only causes increased costs
due to rework, test and inspection…
Understanding and controlling the causative factors (x) is the
real key to high quality at low cost...
Probability distributions identify sources of causative factors
(x). These can be identified and verified by testing which
shows their significant effects against the backdrop of random
noise.
HOW DO POPULATIONS INTERACT?

mnew
snew
Population means interact in a simple intuitive manner.
m1
m2
Means Add
m1 + m2 = mnew
Population dispersions interact in an additive manner
s1
s2
Variations Add
s12 + s22 = snew2
HOW DO POPULATIONS INTERACT?
SUBTRACTING TWO POPULATIONS

mnew
snew
Population means interact in a simple intuitive manner.
m1
m2
Means Subtract
m1 - m2 = mnew
Population dispersions interact in an additive manner
s1
s2
Variations Add
s12 + s22 = snew2

TRANSACTIONAL EXAMPLEOrders are coming in with the
following characteristics:
Shipments are going out with the following characteristics:
Assuming nothing changes, what percent of the time will
shipments exceed orders?
X = $53,000/week
s = $8,000
X = $60,000/week
s = $5,000
TRANSACTIONAL EXAMPLE
X
X
X
shipments
orders
shipments
orders
-
=
-

=
-
=
$60
,
$53
,
$7
,
000
000
000
(
)
(
)
s
s
s
shipments
orders
shipments
orders
-
=
+
=
+

=
2
2
2
2
5000
8000
$9434
$7000
$0
Shipments > orders
X = $53,000 in orders/week
s = $8,000
X = $60,000 shipped/week
s = $5,000
Orders
Shipments
To solve this problem, we must create a new distribution to

model the situation posed in the problem. Since we are looking
for shipments to exceed orders, the resulting distribution is
created as follows:
The new distribution looks like this with a mean of $7000 and a
standard deviation of $9434. This distribution represents the
occurrences of shipments exceeding orders. To answer the
original question (shipments>orders) we look for $0 on this new
distribution. Any occurrence to the right of this point will
represent shipments > orders. So, we need to calculate the
percent of the curve that exists to the right of $0.
TRANSACTIONAL EXAMPLE, CONTINUED
X
X
X
shipments
orders
shipments
orders
-
=
-
=
-
=
$60
,
$53
,
$7
,
000
000

000
(
)
(
)
s
s
s
shipments
orders
shipments
orders
-
=
+
=
+
=
2
2
2
2
5000
8000
$9434
To calculate the percent of the curve to the right of $0 we need
to convert the difference between the $0 point and $7000 into
sigma intervals. Since we know every $9434 interval from the
mean is one sigma, we can calculate this position as follows:
$0
m
0
74
-
=
-

=
X
s
s
$0
$7000
$9434
.
Look up .74s in the normal table and you will find .77.
Therefore, the answer to the original question is that 77% of the
time, shipments will exceed orders.
$7000
Shipments > orders
CORRELATION ANALYSIS

Correlation Analysis is necessary to:
show a relationship between two variables. This also sets the
stage for potential cause and effect.
IMPROVEMENT ROADMAP
Uses of Correlation AnalysisDetermine and quantify the
relationship between factors (x) and output characteristics (Y)..
Common Uses
Breakthrough
Strategy
Phase 4:
Control
Characterization
Phase 1:
Measurement
Phase 2:
Analysis

Optimization
Phase 3:
Improvement
KEYS TO SUCCESS
Always plot the data
Remember: Correlation does not always imply cause & effect
Use correlation as a follow up to the Fishbone Diagram
Keep it simple and do not let the tool take on a life of its own
WHAT IS CORRELATION?

Input or x variable (independent)
Output or y variable (dependent)
Correlation
Y= f(x)
As the input variable changes, there is an influence or bias on
the output variable.
A measurable relationship between two variable data
characteristics.
Not necessarily Cause & Effect (Y=f(x))
Correlation requires paired data sets (ie (Y1,x1), (Y2,x2), etc)
The input variable is called the independent variable (x or
KPIV) since it is independent of any other constraints
The output variable is called the dependent variable (Y or
KPOV) since it is (theoretically) dependent on the value of x.
The coefficient of linear correlation “r” is the measure of the
strength of the relationship.
The square of “r” is the percent of the response (Y) which is
related to the input (x).
WHAT IS CORRELATION?
TYPES OF CORRELATION

Strong
Weak
None
Positive
Negative
Y=f(x)
Y=f(x)
Y=f(x)
x
x
x
CALCULATING “r”
Coefficient of Linear CorrelationCalculate sample covariance
( )Calculate sx and sy for each data setUse the calculated
values to compute rCALC.Add a + for positive correlation and -
for a negative correlation.
(
)
(
)
s
x
x
y
y
n
xy
i
i
=

-
-
-
å
1
r
s
s
CALC
s
xy
x
y
=
s
xy
While this is the most precise method to calculate Pearson’s r,
there is an easier way to come up with a fairly close
approximation...
APPROXIMATING “r”
Coefficient of Linear Correlation
W
L

Y=f(x)
x
r
W
L
»
±
-
æ
è
ç
ö
ø
÷
1
r
»
-
-
æ
è
ç
ö
ø
÷
=
-
1
6
7
12
6
47

.
.
.
+ = positive slope
- = negative slope
W
L
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1
2
3
4
5
6
7
8

9
10
11
12
13
14
15
6.7
12.6Plot the data on orthogonal axisDraw an Oval around the
dataMeasure the length and width of the OvalCalculate the
coefficient of linear correlation (r) based on the formulas below
HOW DO I KNOW WHEN I HAVE CORRELATION? The
answer should strike a familiar cord at this point… We have
confidence (95%) that we have correlation when |rCALC|>
rCRIT.Since sample size is a key determinate of rCRIT we need
to use a table to determine the correct rCRIT given the number
of ordered pairs which comprise the complete data set.So, in
the preceding example we had 60 ordered pairs of data and we
computed a rCALC of -.47. Using the table at the left we
determine that the rCRIT value for 60 is .26. Comparing
|rCALC|> rCRIT we get .47 > .26. Therefore the calculated
value exceeds the minimum critical value required for
significance. Conclusion: We are 95% confident that the
observed correlation is significant.
Ordered
Pairs
r
CRIT

CENTRAL LIMIT THEOREM
For this module you will need 12 dice and flip charts.
The Central Limit Theorem is: the key theoretical link between
the normal distribution and sampling distributions.the means by
which almost any sampling distribution, no matter how
irregular, can be approximated by a normal distribution if the
sample size is large enough.
IMPROVEMENT ROADMAP
Uses of the Central Limit Theorem
Common Uses
Breakthrough
Strategy
Phase 4:
Control

Characterization
Phase 1:
Measurement
Phase 2:
Analysis
Optimization
Phase 3:
Improvement
The Central Limit Theorem underlies all statistic techniques
which rely on normality as a fundamental assumption
WHAT IS THE CENTRAL LIMIT THEOREM?
Central Limit Theorem
For almost all populations, the sampling distribution of the
mean can be approximated closely by a normal distribution,
provided the sample size is sufficiently large.

Normal
What this means is that no matter what kind of distribution we
sample, if the sample size is big enough, the distribution for the
mean is approximately normal.
This is the key link that allows us to use much of the inferential
statistics we have been working with so far.
This is the reason that only a few probability distributions (Z, t
and c2) have such broad application.
If a random event happens a great many times, the average
results are likely to be predictable.
Jacob Bernoulli

HOW DOES THIS WORK?
As you average a larger and larger number of samples, you can
see how the original sampled population is transformed..

ANOTHER PRACTICAL ASPECT
s
n
x
x
=
s
1
n
This formula is for the standard error of the mean.
What that means in layman's terms is that this formula is the
prime driver of the error term of the mean. Reducing this error
term has a direct impact on improving the precision of our
estimate of the mean.
The practical aspect of all this is that if you want to improve the
precision of any test, increase the sample size.
So, if you want to reduce measurement error (for example) to
determine a better estimate of a true value, increase the sample
size. The resulting error will be reduced by a factor of .
The same goes for any significance testing. Increasing the
sample size will reduce the error in a similar manner.
Team experimentBreak into 3 teams

Team one will be using 2 dice
Team two will be using 4 dice
Team three will be using 6 dice
Each team will conduct 100 throws of their dice and record the
average of each throw.
Plot a histogram of the resulting data.
Each team presents the results in a 10 min report out.
(
)
X
X
i
-
2
X
Lecture 1
Introduction to
Six Sigma
History & experiences with Six SigmaMotorola is a birthplace
of Six Sigma and launched the program in January 15, 1987.GE
expanded the application of Six Sigma from manufacturing to
service and increased the operating income to $750 million in
1998.Other successes of Six Sigma in GE:
a) Produced a 10-fold increase in the life of CT scanner.
b) Quadrupled the ROI in Industrial Diamond business.

c) 62% reduction in turnaround time at repair shop in railcar
leasing process.
d) Added 300 million pounds of new capacity in plastic
business.Since their success, many other fortune 500 companies
have implemented Six Sigma with significant financial benefits.
Pioneers of Six Sigma quality revolution
1) Dr. Edward Deming (USA): Introduced basic quality
philosophy, that productivity improves as variability decreases.
2) William Convey (USA): Introduced the tools to eliminating
waste in processes in order to improve quality.
3) Joseph Juran (USA): Introduced the concept of Total Quality
Management (TQM) by adding “human dimension” to quality.
4) Philip Crosby (USA): Introduced the role of management in
quality improvement by using simple tools & statistical quality
control.
5) Dr. Genichi Taguchi (Japan): Introduced the philosophy of
“design of experiments” to improve quality by introducing
variance-reduction strategies and robust design techniques.
6) Shugeo Shingo (Japan): Introduced the idea of improving
process design so that mistakes are impossible to make or at
least easily detected and corrected (Poka-yoke).
Comments by Jack Welch who started Six Sigma at GE
“In just three years, Six Sigma had saved the company more
than $2 billion”.

“At GE, it’s almost an obsession to keep external customer’s
need in plain sight, driving the improvement effort”.
“We did not invent Six Sigma, but we learned it and applied it
effectively to manufacturing and service departments”.
“The cumulative impact on the company’s numbers is not
anecdotal, nor a product of charts. It is the product of all
employees executing & delivering the results of Six Sigma to
our bottom line”.
Comment by Juran (1988) about “Cost of poor quality”
(COPQ)Quality-related cost is about 20% - 40% of sales in an
organization.Quality costs are incurred not only in
manufacturing but also in support areas (service).There is no
person or department in an organization which is directly
responsible for reducing quality cost. Most of the functions of
the quality department are involved with measuring quality but
not reducing it.
The hidden cost of quality

Examples of Quality costs
1) Prevention 3) Internal Failure
Training Scrap & Rework
Capability Studies Design Changes
Vendor Survey Retyping Letters
Quality Design Excess Inventory Cost
2) Appraisal 4) External Failure
Inspection & Test Warranty cost
Test Equipment & Maintenance Customer
Complaints
Inspection & Test Reporting Returns & Recalls
Other Expense Reviews Liability Suits
Two critical components of Six Sigma
Reduces dependency on “Tribal Knowledge”
- Decisions based on facts and data rather than opinion
Attacks the high-hanging fruit (the hard stuff)
- Eliminates chronic problems (common cause variation)
- Improves customer satisfaction
Provides a disciplined approach to problem solving

- Changes the company culture
Creates a competitive advantage (or disadvantage)
Improves profits!
Why Companies Need Six Sigma
Six Sigma is not another quality programSix Sigma is a
technique which fixes identifiable, chronic problems that
directly impacts the bottom line.Six Sigma projects are selected
to reduce or eliminate waste, which improves productivity.Six
Sigma is not a theory but a tool which defines, measures,
analyzes, improves, and controls the processes that matter most,
to tie quality improvement directly to the bottom-line results.
Motorola ROI
1987-1994
• Reduced in-process defect levels by a factor of 200.
• Reduced manufacturing costs by $1.4 billion.
• Increased employee production on a dollar basis by 126%.
• Increased stockholders share value fourfold.
AlliedSignal ROI
1992-1996
• $1.4 Billion cost reduction.
• 14% growth per quarter.
• 520% price/share growth.
• Reduced new product introduction time by 16%.
• 24% bill/cycle reduction.
Six Sigma ROI

Six Sigma as a Philosophy
Internal &
External
Failure
Costs
Prevention &
Appraisal
Costs
Old Belief
4s
Costs
Internal &
External
Failure Costs
Prevention &
Appraisal
Costs
New Belief
Costs
4s
5s
6s
Quality
Quality
Old Belief
High Quality = High Cost
New Belief
High Quality = Low Cost
s is a measure of how much variation exists in a process

Input-output of the process
Key Process Output Variables (Y) Profits Customer satisfaction
Expense Production cycle time Defect rate Critical dimension
on the part
Key Process Input Variables (X) Actions taken Out of stock
items Amount of WIP inventory Amount of internal rework
Inspection procedure Process temperature
Principles of Six Sigma
1) Focus on the customer
2) Collect data and facts about the processes
3) Continue to improve the performance of the process
4) Act in advance of events rather than reacting to them
5) Collaborate across organizational lines
6) Drive for perfection & tolerate failure
POSITIONING SIX SIGMA
THE FRUIT OF SIX SIGMA
Ground Fruit
Logic and Intuition
Low Hanging Fruit
Seven Basic Tools
Bulk of Fruit
Process Characterization and Optimization
Process Entitlement

Sweet Fruit
Design for Manufacturability
Definition of Six Sigma“Six Sigma is a strategic business
improvement approach that seeks to increase both customer
satisfaction and an organization’s financial health” “Six Sigma
puts customer first and uses facts and data to drive better
solutions”.“Six Sigma is about making every area of the
organization better able to meet the changing needs of
customer, markets, and technologies”.Six Sigma efforts target
three main areas:
a) Improving customer satisfaction
b) Reduce cycle time
c) Reducing defects
Key characteristics of Normal DistributionThe normal curve is
totally independent of the LSL and USL.This curve summarize
the empirical quantification for the variability that exists within
the manufacturing process.The normal distribution in science,
engineering & manufacturing is typically used when the
measuring characteristic falls outside the range of + 3 standard
deviation about the target mean.The process outside of the
customer specification limits are defined as defects, failures, or
nonconformities.+ 3s unit of standard deviation represents
99.73% of the total area under the normal distribution curve.
This corresponds to 0.27% outside the curve or 2700 defects per
million. For 6s the defective rate is 0.002 ppm.

What is a defect?Six Sigma is a statistical concept that
measures a process in terms of defects.A defect is a measurable
characteristic of the process or its output that is not within the
acceptable process specifications.Six Sigma is about practices
that help you eliminate defects.Sigma level is calculated in
terms of number of defects in ratio to the number of
opportunities for defects.
The statistical definition of Six SigmaSpecification limits are
the tolerances or performance ranges that customers demand of
the products or processes they are purchasing.It falls within the
range between customer-specified lower specification limit
(LSL) and upper specification limit (USL).
Definition of Sigma Quality level
To address “typical” shifts of a process mean from a
specification centered value, Motorola added a shift value +
1.5s to the mean. This shift of the mean is used when computing
a process “sigma level” or “sigma quality level”. 3.4 ppm rate
corresponds to 6s quality level.

Defective ppm with & without shift
Spec limitwithout shift with shift
(Defective ppm) (Defective ppm)
+ 1 sigma 317,300 697,700
+ 2 sigma 45,500 308,700
+ 3 sigma 2,700 66,800
+ 4 sigma 63 6,210
+ 5 sigma 0.57 233
+ 6 sigma 0.002 3.4
What is the sigma level for on-time delivery 2 sigma = 68% on-
time delivery 3 sigma = 93% on-time delivery 4 sigma = 99.4%
on-time delivery 5 sigma = 99.98% on-time delivery 6 sigma =
99.9997% on-time delivery
Keep in mind that the sigma level measures how well you are
meeting customer requirements.
3 Sigma vs. 6 Sigma
parameter conformance
How good is good enough?
99.9% is already VERY GOOD

But what could happen at a quality level of 99.9% (i.e., 1000
ppm),
in our everyday lives (about
•
4000 wrong medical prescriptions each year

More than 3000 newborns accidentally falling
from the hands of nurses or doctors each year
•

•
400 letters per hour which never arrive at their destination
•

Two long or short landings at American airports each day
Measurements are critical...If we can’t accurately measure
something, we really don’t know much about it.If we don’t
know much about it, we can’t control it.If we can’t control it,
we are at the mercy of chance.
THE ROLE OF STATISTICS IN SIX SIGMA..
WE DON’T KNOW WHAT WE DON’T KNOW
IF WE DON’T HAVE DATA, WE DON’T KNOW
IF WE DON’T KNOW, WE CAN NOT ACT
IF WE CAN NOT ACT, THE RISK IS HIGH
IF WE DO KNOW AND ACT, THE RISK IS MANAGED
IF WE DO KNOW AND DO NOT ACT, WE DESERVE THE
LOSS.

TO GET DATA WE MUST MEASURE
DATA MUST BE CONVERTED TO INFORMATION
INFORMATION IS DERIVED FROM DATA THROUGH
STATISTICS

Learning ObjectivesHave a broad understanding of statistical
concepts and tools.
Understand how statistical concepts can be used to improve
business processes.
Understand the relationship between the course and the four
step six sigma problem solving process (Measure, Analyze,
Improve and Control).
Three key characteristics of
Six Sigma projects at GE.
1) It must be customer focused.
2) It must produce major returns on investment (ROI).
3) It changes how management operates.
How to identify Six Sigma projects?
1) Projects which have potential for significant monetary
savings.
2) Projects which are of strategic importance.
3) Projects which involve key output issues.
4) Projects which are of vital concern to the customer.
5) Projects which can be quantified with poor quality within

strategic area of business.
Key focus areas for Six Sigma program
1) Measurement: It should focus on statistical measure of the
performance of a process or a product.
2) Goal: It should focus on the goal that reaches near perfection
for performance improvement.
3) System of management: It should focus on a system of
management to achieve lasting business leadership and World
class performance.
Six Sigma Tools
Process MappingTolerance AnalysisStructure TreeComponents
SearchPareto AnalysisHypothesis TestingGauge R &
RRegressionRational SubgroupingDOEBaseliningSPC

Problem Solving Methodology
Breakthrough Strategy
Characterization
Phase 2:
Measure
Phase 3:
Analyze
Optimization
Phase 4:
Improve
Phase 5:
Control
Projects are worked through these 5 main
phases of the Six Sigma methodology.
Phase 1:
Define
Unlocking the hidden factory
VALUE STREAM TO THE CUSTOMER
PROCESSES WHICH PROVIDE PRODUCT VALUE IN THE
CUSTOMER’S EYES

FEATURES OR CHARACTERISTICS THE CUSTOMER
WOULD PAY FOR….
WASTE DUE TO INCAPABLE PROCESSES
WASTE SCATTERED THROUGHOUT THE VALUE STREAM
EXCESS INVENTORYREWORKWAIT TIMEEXCESS
HANDLINGEXCESS TRAVEL DISTANCESTEST AND
INSPECTION
Waste is a significant cost driver and has a major impact on the
bottom line...
Common Six Sigma Project AreasManufacturing Defect
ReductionCycle Time ReductionCost ReductionInventory
ReductionProduct Development and IntroductionLabor
ReductionIncreased Utilization of ResourcesProduct Sales
ImprovementCapacity ImprovementsDelivery Improvements
Six Sigma scorecard
FINANCIALInventory levelsCost per unitHidden
factoryActivity-based costingCost of poor qualityCost of doing
nothing differentOverall project savings
INTERNAL PROCESSDefects: inspection data, DPMO, Sigma
quality levelOn-time shippingRolled throughput yieldSupplier
qualityCycle timeVolume hoursBaseline measurementsKPIVs
CUSTOMERCustomer satisfaction (CTW)On-time
deliveryProduct quality (KPOV)SafetyCommunications

LEARNING AND GROWTHSix Sigma tool utilizationQuality
of trainingMeeting effectivenessLessons learnedNumber of
employees trained in Six Sigma datesNumber of project
completedTotal dollars saved on Six Sigma projects to date
Six Sigma applicationManufacturingServicesEngineering and
R&DSales and marketingHealth careGovernmentCorporate
functions
Typical Six Sigma projects in ManufacturingIncreasing
manufacturing yieldReducing assembly cycle-timeMinimizing
changeover timeReducing variations in machining
processEliminating need for regular testing
Typical Six Sigma projects in ServicesReducing time to open a
new accountEliminating statement errorsMinimize wait time for
customerReducing cycle time for check-inReducing invoice
errors
Typical Six Sigma projects in Engineering and R&DReducing
time for engineering drawing changeReducing the time for
approval and changesReduce the time to procure test
materialsIncreasing the utilization of test equipment
Typical Six Sigma projects in Sales and MarketingOptimizing
sales force allocationMinimizing promotion timeImproving

response rate from customerImproving communication rate of
proposalsReducing time to prepare for complex bids
Typical Six Sigma projects in Health careReducing inpatient
length of stayReducing ER wait timeReducing outstanding
accounts receivableReducing fall and injury claimsImprove and
streamline medical procedure
Typical Six Sigma projects in GovernmentReducing the amount
of waste activitiesIncreasing the number of
inspectionsStreamlining the permit processReducing the number
of change orders in contracts
Typical Six Sigma projects in Corporate functionsReducing
time for monthly closingDecreasing patent-filing timeReducing
time for hiring processIncreasing response rate of employee
surveyIncreasing the accuracy of tracking of assets
Characteristics of Six Sigma success
Very successful programCommitted leadershipUse of top
talentFormal project selection and review processDedicated
resourcesFinancial system integration
Mediocre programSupportive leadershipUse of whoever was
availableNo formal project selection and review processPart-
time resourcesNo financial system integration

Following two approaches are used in the market- place to
improve manufacturing operations
1) Lean Manufacturing
2) Six Sigma
They are dependent upon each other for success.They both
battle “variation” from two different point of views.Generally
Lean manufacturing program is implemented before Six Sigma
program.Lean manufacturing strategy builds a strong foundation
for the effective use of Six Sigma tools.Lean manufacturing
focuses on the entire operation of a factory to reduce
“variation” while Six Sigma focuses on a specific part number
or a specific process to reduce “variation“.
Lean Manufacturing Versus Six Sigma
Lean Manufacturing Focuses on the product flow & on the
operator Standardization of operator method Reduce
communication barriers through flow-focused cells Focuses on
process speed & feedback of information
Six Sigma Uses data driven technique for problem solving It is
a method to measure deviation from the standard Focuses on
part variation & overall process variation Addresses process
variations which builds up over time
Lean manufacturing creates the standard and Six Sigma
analyzes variations from the standard

What is DMAIC?Define the projects and the goals of internal
and external customers.Measure the current performance of the
process.Analyze and determine the root cause(s) of the
defects.Improve the process to eliminate defects.Control the
performance of the process.
USL
T
m
LSL
Six Sigma BME/IHE 6850-90 (lectures 1-3)
Distance case study #1
(Due date: Saturday, September 24, 2016)
Use 12 size font, single spaced
Case study 1.1: The P.T. Company located in Florida makes
electronic rocker-recliner chairs. The success of these chairs
over the past three years has been remarkable. However, for the
past couple of months, sales have been in a dramatic and
alarming slide. Distributors have been flooding the home office
with e-mails about returns and customer complaints. As the
biggest contributor to the company’s profits, any problems with
these chairs (called the E-Rock) created big worries for the
management. The top product management group and the senior
executive team met to discuss strategies for dealing with the
declining sales. “This product has outlived its welcome,”
exclaimed the head of marketing. “We need to get going on a
whole new generation of E-chairs!”
“That’s going too far,” countered the field sales director. “I
can’t believe the entire market suddenly decided that the E-
Rock is outmoded in a period of weeks.” The head of seat
engineering weighed in: “We need to give more incentives to
those distributors or threaten to drop them. It’s clear they’re
getting lazy.”

After a few more minutes of discussion (using some better
meeting management tools) the group was able to agree to get a
Six Sigma DMAIC team started on trying to find out the causes
of the sales decline.
Case study objective: You have been appointed the DMAIC
team leader. Using information in lectures 1-3, describe how
you will proceed to solve the problem of declining sales.
(Do not exceed 1 page, 12 font size, single spaced).
Case study 1.2: Refer to the slide “Common Six Sigma Project
Areas”. There are following 10 areas described in the slide:
1. Manufacturing Defect Reduction
2. Cycle Time Reduction
3. Cost Reduction
4. Inventory Reduction
5. Product Development and Introduction
6. Labor Reduction
7. Increased Utilization of Resources
8. Product Sales Improvement
9. Capacity Improvement
10. Delivery Improvement.
For each of the above project areas, write a statement (one
sentence) about strategic project goal

See the examplebelow:
Six Sigma Project Area 10:
Delivery Improvement
Strategic Project Goal:
Ship customer orders within one week of receiving the order.
Case study 1.3: Refer to the slide “Six Sigma application”,
which lists the 7 areas of application:
1. Manufacturing
2. Services
3. Engineering and R&D
4. Sales and Marketing
5. Healthcare
6. Government
7. Corporate function
Refer to the following 7 slides, which lists the typical Six
Sigma projects in each of the above areas. Each slide has 4 or 5
projects.
Case study objective: Select one project from each slide (area)
and list 2 KPIVs for that project.
See the following example: (note KPIV = Key Process Input
Variable). Note that each KPIV starts with “variation ……..” in
the following example.
Area:
Manufacturing
Project:
Reducing variations in machining process

KPIV-1:
Variation in the vendor material reject rate
KPIV-2
Variation in employee skill level (seniority)
Case study 1.4 and 1.5 : Dave Johnson is the manager of an
insurance claims office and serves on a Six Sigma team that
established a goal of making improvements aimed at speeding
up the number of claims processed per day. To get started, the
team gathered data for just over a month (using good sampling
methods) and plotted the data on a run chart (figure 2). Master
Black Belt showed Dave how to determine the expected amount
of variation in the claims process, and to plot the resulting
“control limits” on the chart. When looking at this control chart
(figure 3), Dave saw that many data points were beyond the
control limits.
Figure 2: Run Chart of Insurance Claims
Figure 3: Control Chart of Insurance Claims
Objective 1. 4: Explain why some points are outside (above and
below) the control limits.
Objective 1.5: What actions will you take to bring the process
within control limits.
UCL
LCL

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Lecture 3 Statistical ProcessControl (SPC).docx