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National Guard
Black Belt Training
Module 22
Process Measurement
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CPI Roadmap – Measure
8-STEP PROCESS
6. See
1.Validate 2. Identify 3. Set 4. Determine 5. Develop 7. Confirm 8. Standardize
Counter-
the Performance Improvement Root Counter- Results Successful
Measures
Problem Gaps Targets Cause Measures & Process Processes
Through
Define Measure Analyze Improve Control
TOOLS
•Process Mapping
ACTIVITIES
• Map Current Process / Go & See •Process Cycle Efficiency/TOC
• Identify Key Input, Process, Output Metrics •Little’s Law
• Develop Operational Definitions •Operational Definitions
• Develop Data Collection Plan •Data Collection Plan
• Validate Measurement System •Statistical Sampling
• Collect Baseline Data •Measurement System Analysis
• Identify Performance Gaps •TPM
• Estimate Financial/Operational Benefits •Generic Pull
• Determine Process Stability/Capability •Setup Reduction
• Complete Measure Tollgate •Control Charts
•Histograms
•Constraint Identification
•Process Capability
Note: Activities and tools vary by project. Lists provided here are not necessarily all-inclusive. UNCLASSIFIED / FOUO
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Learning Objectives
Understand the importance of measurement to
process improvement
Apply measures of central tendency and variation to
process data
Apply the concepts of common and special cause
variation
Apply Sigma Quality Level to processes
Know how to measure the Voice of the Customer and
Voice of the Business
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Measurement Fundamentals
Definition: The assignment of numbers to
observations according to certain decision rules
Measurement is the beginning of any science or
discipline
Without measurements, we do not know where we
are going or if we ever got there – we do not even
know where we are now!
If it is important to the customer, we should measure
it
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Measurement Example
The following data is the number of minutes it took Soldiers to
resolve their AKO issues when calling the AKO Helpdesk. Take a
few minutes to examine the data:
Time of Day Minutes To Resolve Issue
0730 00
0731 11
0800 06
0845 14
0903 11
0925 58
0940 47
1006 16
1120 09
1145 48
1158 43
1205 53
1214 49
1310 09
1400 10
How should we summarize and present this data to understand
the AKO Helpdesk’s overall performance?
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Calculating the Mean
An easy way of summarizing data is to
calculate the arithmetic average (or “mean”) of
the column of numbers
Mathematically, we can express this as
follows:
n
X
i 1 X i
n
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Mean Example
Lets go back to our AKO Helpdesk data:
0, 11, 6, 14, 11, 58, 47, 16, 9, 48, 43, 53,
49, 9, 10
What is the mean value?
X-bar = (0 + 11 + 6 + 14 + 11 + 58 + 47
+ 16 + 9 + 48 + 43 + 53 + 49 + 9 + 10)
/ 15 = 25.6 minutes
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Measures of Central Tendency
The Mean is a measurement of central tendency, that is, where
the “center” of most of the data is. Another measure of central
tendency is the Median.
The Median is calculated by listing the data in ascending order
and then finding the value that is in the middle of the list
If we re-order our AKO Helpdesk data in ascending order, we get
the following list:
0, 6, 9, 9, 10, 11, 11, 14, 16, 43, 47, 48, 49, 53, 58
The value which occurs in the middle of the list is 14 minutes –
this is the Median
The Median can be a fraction or decimal even if the data is all
integers. If we had fourteen instead of fifteen data points (no
58) the median would have been (11 + 14) / 2 = 12.5 minutes
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Central Tendency – The Whole Story?
While it is important to know where the “center” of
our data is, does it tell the whole story?
What does this tell us about the AKO Helpdesk’s
performance? What does is not tell us?
Why is there a difference between the mean and
median in our example?
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Measures of Central Tendency
Mean, Median and
Summary for Minutes
Mode A nderson-Darling N ormality Test
A -S quared 1.29
Mode - most
P -V alue < 0.005
M ean 25.600
frequently
S tDev 20.870
V ariance 435.543
S kew ness 0.43325
occurring data Kurtosis
N
-1.78559
15
point M inimum
1st Q uartile
M edian
0.000
9.000
14.000
3rd Q uartile 48.000
A histogram
0 10 20 30 40 50 60 M aximum 58.000
95% C onfidence Interv al for M ean
shows data by
14.043 37.157
95% C onfidence Interv al for M edian
frequency of
9.374 47.626
95% C onfidence Interv al for S tDev
9 5 % C onfidence Inter vals
occurrence. It
15.279 32.914
Mean
also shows the Median
“distribution” and
10 20 30 40 50
“spread” and of
the data
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Measuring Variation
Another important way of summarizing our data is by
measuring the average “spread” or variation between
each data point and the mean
While the center of our process is important, knowing
the spread is particularly important in service because
each user is an individual and deserves to be provided
with acceptable service
Do you care that the average wait is 26 minutes if you
are the one who had to wait 58 minutes?
A commonly used term in statistics for measuring this
variation is the standard deviation
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Calculating the Standard Deviation
The standard deviation gives us a feel for the
overall consistency of our data set
Mathematically, it is calculated as follows:
n
( X i X )2
s i 1
n 1
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Standard Deviation Example
From the previous example we know that the sample
mean, x-bar, is 25.6 minutes
Find the sample standard deviation:
( 0 - 25.6)2 = 655.36
(11 - 25.6)2 = 213.16 Subtotal (Sum of Squares) = 6097.60
( 6 – 25.6)2 = 384.16 Divided by (n-1) = 14
(14 – 25.6)2 = 134.56
(11 – 25.6)2 = 213.16 Variance = 435.54 min2
(58 – 25.6)2 =1049.76
(47 – 25.6)2 = 457.96 Standard Deviation = 20.86 min
(16 – 25.6)2 = 92.16 (Square Root of Variance)
( 9 – 25.6)2 = 275.56
(48 – 25.6)2 = 501.76
(43 – 25.6)2 = 302.76
(53 – 25.6)2 = 750.76
(49 – 25.6)2 = 547.56
( 9 – 25.6)2 = 275.56
(10 – 25.6)2 =243.36
Subtotal = 6097.60
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Variance
If we square the standard deviation, we get the Variance
The Variance of a Data Sample is defined as follows:
Sample = 2
Variance
s
The Variance of the Population from which the sample is
drawn is defined as:
Population
Variance
2
The Variance is useful since we cannot add Standard Deviations
together, but we can add Variances (more on this in future
modules)
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Min, Max, and Range
A simple way of measuring the amount of consistency
in a data set is by calculating Min, Max, and Range
The Min is the smallest value in our data set; the Max
is the largest value
The Range is the difference between the Max and Min
and gives us a feel for the “spread” in our data
Using our AKO Helpdesk data, the Min = 0 minutes,
the Max = 58 minutes, the Range is 58 - 0 = 58
minutes
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Central Tendency and Variation
A key concept in Lean Six Sigma is understanding how
central tendency and variation work together to describe a
process by summarizing its data:
Central Tendency is where the “middle” of the
process is – this is where we would expect most of the
data points to be
Variation tells us how much “spread” there is in the
data – the smaller the variation, the more consistent
the process
Both a measure of central tendency and variation are
necessary to describe a data set
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Understanding Variation
There will always be some variation present in all processes:
Nature – Shape/size of leaves, snowflakes, etc.
Human – Handwriting, tone of voice, speed of walk, etc.
Mechanical – Weight/size/shape of product, etc.
We can tolerate this variation if:
The process is on target
The variation is small compared to the process
specifications
The process is stable over time
We need to recognize that sources of variation (especially
Special Cause variation) should be minimized or, if possible,
eliminated
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Variation – Its Impact on Business
Variation is the Enemy of Improvement Efforts
In the 1998 GE annual report, Chief Executive Jack
Welch clearly articulated a concern that had been
troubling other CEOs:
“We have tended to use all our energy and Six Sigma
science to “move the mean”… The problem is, as has
been said, “the mean never happens,” and the
customer is still seeing variances in when the deliveries
actually occur – a heroic 4-day delivery time on one
order, with an awful 20-day delay on another, and no
real consistency… Variation is Evil.”
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Types of Variation
Common Cause Variation
This is the consistent, stable, random variability within the process
We will have to make a fundamental improvement to reduce common
cause variation
Is usually hard to reduce
Special Cause Variation
This is due to a specific cause that we can isolate
Special cause variation can be detected by spotting outliers or
patterns in the data
Usually easy to eliminate
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Exercise: Your Signature
First, write your name 5 times
Next, write your name 5 times with the other hand
Is the variability common or special cause?
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Special Cause Variation
Examples of Special Cause variation are:
Uncommon occurrence or circumstance
Soldiers out for training holiday or flu
epidemic Xbar Chart of Supp2
Convoy vehicle flat tire 603 1
1
UCL=602.474
Procedure change 602
Base-wide electrical
Sample Mean
601
power outage 600
_
_
X=600.23
599
Control Chart showing
Special Cause variation 598 LCL=597.986
2 4 6 8 10 12 14 16 18 20
Sample
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Common Cause Variation
Some examples of Common Cause variation are:
Experience of individual Soldiers
Internet server speed fluctuations
Soldier out on sick-call Xbar Chart of Supp1
600.5
Day to day unit issues UCL=600.321
600.0
Sample Mean
Control Chart showing 599.5
_
_
X=599.548
variation due only to
Common Cause
599.0
LCL=598.775
2 4 6 8 10 12 14 16 18 20
Sample
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Understanding Accuracy and Precision
If the pictures to the right
represent weapons training by
two recruits, which one is
better?
Green?
Blue? (Green)
Which one is more accurate
(better average)?
Which one is more precise
(more consistency)?
(Blue)
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Weapons Training Example
On average, the green target is
centered on the bulls-eye,
therefore more accurate
Accuracy is a measure of
“average distance from the
target”
(Green)
However, the blue target is
more consistent, therefore
more precise
Precision is a measure of
“average distance from each
other”
(Blue)
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Weapons Training Example
How could the recruit using the
green target improve
performance?
How could the recruit using the
blue target improve
performance? (Green)
Which recruit do you think has a
better chance of becoming an
expert shooter?
Typically, it is easier to shift the mean than to
reduce variation (Blue)
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Goal: Shift the Mean / Reduce Variation
Too Much Spread Off Center
Centered (Blue)
(Green)
Reduce On-Target Center
Spread Process
Result: Improved Customer Satisfaction and Reduced Costs
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Introducing The Distribution
Upto now, we have been using the mean
and standard deviation to summarize the
data generated from a process
Another way we can summarize the data
is by showing its distribution
Thedistribution shows us the number of
times (“frequency count”) a particular
data value appears in our data set
The “peak” of the distribution shows its
central tendency; the “spread” of the
distribution tells us about the degree of
variation present in the data
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The Distribution
By examining the distribution, we can
see patterns that are difficult to see in
a simple table of numbers
Different processes and phenomena
will generate different distribution
patterns
Both common and special cause
variation will be present in the
distribution
The examples shown are different
types of distributions
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Histogram
A common graphical tool used to portray
the distribution is the histogram
The histogram is constructed by taking the
# difference between the min and max
observation and dividing it up into evenly
spaced intervals
The number of observations in each
Histogram
interval are then counted and their
frequency plotted as the height of each bar
The histogram is, in essence, a simplified
view of the distribution that generated the
plotted data
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Exercise: Build a Histogram
Note the height in inches of the tallest
Frequency
and shortest students in the class
12
Divide this range into 5 equally sized
10
intervals
8
Make a bar to show the number of
6
students in class who’s height falls
within each interval
4
The resulting chart is a histogram
2
How would you describe the shape of
Height our histogram?
56 60 64 68 72 76 (Inches)
How much variation is present in our
data? Common or Special Cause?
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Interpreting Histogram Data
Ifthe variation is common cause, it reflects the
natural variation inherent in the process and
will show higher frequencies around the central
tendency and taper off toward the edges of the
distribution. The underlying process generating
Common Cause Variation the data is stable, and the value of each data
point is random and consistent with the rest
of the distribution.
Outlier
Ifthe variation is special cause, an observation
will not “fit” the rest of the distribution (i.e, it is
Special Cause - Outlier an outlier), or there will be a “pattern” in our
data. In other words, there is an identifiable
reason for why this variation exists.
Special Cause - Bimodal
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Exercise: Minitab
Let’s use Minitab to help us analyze some data
Open the Minitab data set called Red Beads Data.mtw
Four teams of four people each sampled 50 beads from
the same bead box
Each team member drew 10 samples of 50
The samples were randomly drawn and the beads
randomly replaced after drawn
The data collected was the number of “red” beads
counted out of the fifty beads sampled
What do you think the histogram of this data will look like?
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Exercise: Minitab
1. Let’s make a histogram of the data
Select: Stat>
Basic Statistics>
Display Descriptive Statistics
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Exercise: Minitab
2. Double click (select)
C3 Red Beads to place
it in the Variables box
3. Click on Graphs
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Exercise: Minitab
4. Check the box for
Histogram of
Data
5. Click OK
6. Click OK
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Exercise: Minitab
This is a frequency
histogram that shows us,
for the entire 160 samples
run, how many red beads 22
remained in the paddle
each time
For example, 22 times out
of 160, there were 11 red
beads in the paddle
What type of variation is
present? Common or
Special Cause? 11
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Exercise: Minitab
Notice that the Data in Session Window gives us information
on both Central Tendency and Variation
Descriptive Statistics: Red Beads
Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3
Red Beads 160 0 10.684 0.239 3.029 4.000 8.026 11.000 13.000
Variable Maximum
Red Beads 19.000
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Other Information from Minitab
Standard Error of the Mean (SEMEAN) s
Gives the standard error of the mean. It is calculated as n .
Quartiles
Every group of data has four quartiles. If you sort the data from
smallest to largest, the first 25% of the data is less than or equal to the
first quartile. The second quartile takes all the data up to the median.
The first 75% of the data is less than or equal to the third quartile and
25% of the data is greater than or equal to the third quartile – the
fourth quartile.
The Inter Quartile Range equals Q3 - Q1, spanning 50% of the data
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Exercise: Minitab
1. Let’s make a Box Plot of
the data
Select:
Stat>
Basic Statistics>
Display Descriptive Statistics
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Exercise: Minitab
2. When this dialog box comes up,
double click on C3-Red Beads
to place it in the Variables box.
Then double click on C1-Teams to
place it in the By Variables box.
Finally, click on Graphs.
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Exercise: Minitab
3. Select Boxplot of Data,
click on OK and then
click on OK again
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Exercise: Minitab
Displayed are 4
boxplots, one for each
team
One way of
interpreting a box plot
is “looking down at the
top of a histogram”
This is a good way
to see how spread and
centering differ from
one team to another
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Exercise: Minitab
Notice that Team 1
has 2 Outliers
Notice also that team 4
has a slightly wider
spread (i.e., larger
standard deviation) than
team 1 with a narrower
spread (i.e., smaller std.
deviation)
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Exercise: Minitab
As before, the session Descriptive Statistics: Red Beads
window gives us all the Variable Team N N* Mean SE Mean StDev Minimum Q1 Median
numbers Red Beads 1 40 0 11.325 0.426 2.693 6.000 10.000 11.000
2 40 0 10.200 0.482 3.048 4.000 8.000 10.000
As we would have
3 40 0 10.700 0.495 3.131 5.000 8.000 11.000
4 40 0 10.511 0.509 3.220 5.185 7.307 10.802
figured from the box Variable Team Q3 Maximum
plot, team 4 has a Red Beads 1 13.000 19.000
slightly larger standard
2 12.750 17.000
3 13.000 16.000
deviation than team 1 4 12.987 16.573
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Exercise: Minitab
1. Now let’s make a Dotplot
of the data
Select Graph> Dotplot
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Exercise: Minitab
2. Next select One Y and
With Groups, since we have
only one Y variable but four
teams.
Then click on OK.
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Exercise: Minitab
3. Double click on
C3-Red Beads to
place it in the
Graph Variables box
4. Double click on
C1-Team to place it in
the Categorical
Variables box.
Then click on OK.
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Exercise: Minitab
Displayed are the Four Dotplots, one for each team
This is a good way to see how spread and centering differ from
one team to another. Also, the scale remains the same.
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Introducing the Normal Distribution
In our Red Bead example, you may have noticed that the data in our
histogram took on the shape of a bell shaped curve
If we measure process performance over time, many processes tend to
follow a Normal Distribution or bell shaped curve:
Average
ƒ(x) = Y
Frequency
Variation
x
The Normal distribution is important in statistics because of the relationship
between the shape of the curve and the standard deviation ()
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Properties of the Normal Distribution
One way of demonstrating the relationship between the standard deviation
sigma () and the shape of the curve is to use sigma as a “measuring rod” to
describe how far we are away from the mean
The special properties of the normal distribution allow us to calculate the area
underneath the curve based upon how many sigmas (or standard deviations)
we are away from the mean:
-3 -2 -1 +1 +2 +3
+/-1 =68.27%
+/-2 =95.45%
+/-3 =99.73%
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Properties of the Normal Distribution
Another property of the normal distribution is the area under the curve gives
us the probability of a data point being drawn from this portion of the
distribution
This special property enables us to predict process performance over time
Essentially all of the area (99.73%) of the normal distribution is contained
between -3 sigma and +3 sigma from the mean. Only 0.27% of the data falls
outside 3 standard deviations from the mean:
-3 -2 -1 +1 +2 +3
+/-3 = 99.7%
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Process Performance
There are two aspects to process performance:
Efficiency – Time and cost associated with executing the
process
Cycle time (processing time, on-time delivery, responsiveness, etc.)
Cost (number of resources required, capital equipment, etc.)
Effectiveness – Quality of the output of the process
Level of output (calls answered, orders processed, etc.)
Defects (accuracy, mistakes, errors, etc.)
Customer Satisfaction
Improving both the efficiency and effectiveness of process
performance will enable us to reduce costs and better
satisfy customers
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Process Capability
Process capability measures whether or not a process is
capable of meeting customer requirements
It is a quantifiable comparison of a process’
performance (Voice of the Process) vs. the customer
requirements or “specifications” (Voice of the
Customer)
Most measures have some desired value (“target”) and
some acceptable limit of variation around the desired
value
The extent to which the “expected” values fall within these
limits determines how capable the process is of meeting
its requirements
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Understanding Acceptable Performance
“Acceptable Performance” by definition is that which is
acceptable to the customer:
Target – The desired or nominal value of a
characteristic
Tolerance – An allowable deviation from the target
value where performance is still acceptable to the
customer
Specifications – Boundaries where performance
outside of these limits is not acceptable to the customer
LSL = Lower Spec Limit
USL = Upper Spec Limit
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A Graphical View of Process Performance
LSL Target USL
USL
Traditional View
Tolerance • Target
• Pass/Fail
Cost
LSL Target USL Correct View
• Target
• Service Break Points
– Less than 1: Delighted
Cost
– 1 to 2: Very Satisfied
– 2 to 3: Satisfied
3 2 1 1 2 3
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Satisfying Customer Requirements
Specification Limits establish the boundaries for acceptable
process performance. Performance outside these boundaries is
“unacceptable.” They are “defects.”
They are typically described by an Upper Specification Limit
(USL) and Lower Specification Limit (LSL)
For example, what are the spec limits for the temperature of this
room?
How LOW can the temperature get before
you become uncomfortable or dissatisfied?
This is the LSL.
How HIGH can the temperature get before
you become uncomfortable or dissatisfied?
This is the USL.
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What Reduced Variation Looks Like
SQL = 3.0 SQL = 6.0
Target
Target 1 Standard Deviation
1 Standard Deviation
LSL USL
LSL USL
Process Process
Center Center
3 3
3 6
Current process has 3 standard Improved process (reduced
deviations between target and USL variation) has 6 standard deviations
between target and USL
NOTE: Illustrations do not include the 1.5 Sigma Shift, the discussion of which is beyond the scope of this lesson
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Process Capability Is Sigma Quality Level
Sigma is a Greek letter and a statistical unit of
measurement that describes the variability or
spread of data (the standard deviation of a
population)
Six Sigma refers to a methodology of continuous
improvement where the goal is to improve process
performance to meet customers’ requirements
Sigma Quality Level is a measure of process
performance with respect to customer
requirements
Note: Another approach to measuring process capability,
Cp and Cpk, is shown in the Appendix and will be discussed
in a future module.
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Sigma Quality Level and Process Capability
If we measure the performance of a process…
Mean
Standard Deviation
…and know the customer’s Specification Limits, then:
We can calculate Sigma Quality Level…which tells us how
many “defects” we can expect over time (process
capability)
Understanding process capability will help us:
Establish a baseline of current performance
Measure on-going performance to determine level of
improvement and then monitor and control performance to
maintain the gain
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Sigma Quality Level (SQL) and
Defects per Million Opportunities (DPMO)
SQL Yield DPMO
2 69.2% 308,537
3 93.32% 66,807
4 99.379% 6,210
5 99.977% 233
6 99.9997% 3.4
A 3 SQL process will fail to meet customer requirements 7% of the time
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Process Capability: Invoice Example
Errors made in preparing customer invoices has led to unacceptable delays
in receiving customer payments
A review of the last 100 prepared invoices revealed
that 15 of them required corrections
before they could be sent to customers
However, there were three types of errors associated
with the invoices and several invoices had more than
one error:
Incorrect address
Wrong amount
Mismatch of account number
In total, there were 19 different defects on the 15 faulty invoices
What is the Sigma Quality Level?
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Process Capability: Invoice Example
There are two ways to determine Sigma Quality Level
(SQL), depending on the type of data measured:
Continuous or Variable Data – Data that can take
on any value (e.g., average cycle time of a process or
room temperature)
We calculate SQL using mean, standard deviation, and
specification limits
Discrete or Attribute Data – Data that typically can
result in one of two possible outcomes (e.g., pass/fail,
defective/acceptable)
For this Invoice Example case, we calculate Defects Per Million
Opportunities
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Calculating Sigma Quality Level Based on
Defects Per Million Opportunities (DPMO)
1. Determine number of defect or error
opportunities per unit O= 3
2. Determine number of units
N= 100
processed
3. Determine total number D= 19
of defects made
4. Calculate Defects D
DPO = = 0.063
per Opportunity NxO
5. Calculate DPMO DPMO = DPO x 1,000,000 = 63,333
6. Look up the S.Q.L. in the Sigma Quality Level = ~3
Table (see next slide)
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How Do We Improve a Process?
Desired
Current
LSL USL
Let's say that you have this situation
How do you go about improving it?
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Shift Mean and Reduce Variability
CPI Improvements Reduce Process Cycle Time and Improve
Lean Six Sigma Reduces Process Cycle Time, Improving
Consistency
On- Time Delivery Performance for Tier One Auto Supplier
(Average Cycle Time Reduced from 14 days to 2 days,
(Average Reduced from 14 Days to 2 Days, Variance from 2 Days to 4 Hours)
Variation Reduced from 2 days to 4 hours)
90%
80%
70%
60%
% Distribution
50%
40%
Distribution
30%
Mean Delivery Time Reduced
20%
Time Variation Reduced
10%
0%
0 2 4 6 8 10 12 14 16 18 20
Lead-Time (days)
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“As Is” Baseline Statistics Template
Summary for Workdays
A nderson-Darling N ormality Test
A -S quared 12.65
The current process
P -V alue < 0.005
M ean 44.814
- Example -
S tDev
V ariance
61.251
3751.674 has a non-normal
distribution with the
S kew ness 2.87329
Kurtosis 9.54577
N 118
M inimum 1.000
P-Value < 0.05
1st Q uartile 12.000
Mean = 44 days
M edian 22.000
0 60 120 180 240 300 360
3rd Q uartile 52.000
M aximum 365.000
Median = 22 days
95% C onfidence Interv al for M ean
33.647 55.981
95% C onfidence Interv al for M edian
17.000 29.123
95% C onfidence Interv al for S tDev
Std Dev = 61 days
9 5 % C onfidence Inter vals
54.308 70.246
Mean Range = 365 days
Median
20 30 40 50 60
Required Deliverable
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Takeaways
Now you should be able to:
Explain the importance of measurement to process
improvement
Given process data, calculate a measure of central
tendency and variation and describe what they tell us
Identify and contrast special cause and common
cause variation
Given process data, calculate a Sigma Quality Level
and describe what it tells us
Explain what is meant by VOC and VOP
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What other comments or questions
do you have?
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National Guard
Black Belt Training
APPENDIX
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Process Capability Ratio – Cp
Process Capability Ratio (Cp) is the ratio of total variation
allowed by the specification to the total variation actually
measured from the process
Use Cp when:
The mean can easily be adjusted (i.e., in many transactional processes the
resource level(s) can easily be adjusted with no/minor impact on quality),
AND
The mean is monitored (so operators will know when adjustment is
necessary – doing control charting is one way of monitoring)
Typical goals for Cp are greater than 1.33 (or 1.67 for high risk
or high liability items)
If Cp < 1 then the variability of the process
is greater than the specification limits.
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Process Capability Ratio – Cp (Cont.)
Cp = Allowed variation (Specification) or Cp = USL – LSL
Normal variation of the Process 6
Where is “within”
rather than pooled
99.7% of values
-3 +3
Process Width
LSL T USL
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Process Capability Ratio – Cpk
This index accounts for the dynamic mean shift in the
process – the amount that the process is off target
USL x x LSL Where is “within”
C pk Min or rather than pooled
3σ 3σ
Calculate both values and report the smaller number
Notice how this equation is similar to the Z-statistic
Z xx
Cpk Z
3 s
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Process Capability Ratio – Cpk (Cont.)
Ratio of 1/2 total variation allowed by spec. to ½ the actual variation,
with only the portion closest to a spec. limit being counted
Use when the mean cannot be easily adjusted (i.e., in transactional
processes where there is little flexibility, that is, where certain
skill/expertise is not readily adjusted)
Typical goals for Cpk are greater than 1.33 (or 1.67 if safety related)
For sigma estimates use:
R/d2 [short term] (calculated from X-bar and R chart)
s= S (xi – x)2 [long term] (calculated from all data points)
n-1
Long term: When the data has been collected over a time period and
over enough different sources of variation that over 80% of the
variation is likely to be included
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