Basic Statistics

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Basic Statistics
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Cornerstones of a successful use of 6?
Results
World Class Business Performance
Change Management
Methodology
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Different types of data

• Continuous Data
• Often obtained by use of a measuring system e.g.
  dB, Watts, volts etc.
• The usefulness of the data depends on the quality
  of the measurement system

• Discrete Data
• Includes percentages, attribute counts
• Percentages The proportion of items with a
  given characteristic need to be able to
  count both occurrences and and
  non-occurrences, e.g. Yield
• Attribute data Gives only conforming or non
  conforming information, such as Pass/Fail,
  Red/Green, 1 / 0, etc.
• Counts Number of events per hour, per shift or
  other delimitations
• Occurrences must be independent

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Yield
Lower Tolerance Limit
Upper Tolerance Limit
Yield
Defects
Yield Pass / Trials p(d) (1- Yield)/100
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Discrete data - First Time Right (First Time
Yield) Measures the units that avoid the hidden
costs.
Yes
Yes
Step A
Step B
Good?
Good?
Ship It!
No
No
No
No
SCRAP
Fix It?
Fix It?
Yes
Yes
Rework
Rework
COPQ
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Discrete data - Rolled Thru Yield

• Most processes are complex interrelationships of
  many sub-processes. The overall performance is
  usually of interest to us.

Rolled yield is a realistic assessment of the
cumulative effect of sub-processes
FTY First Process 99
First Process
FTY Second Process 89
Second Process
FTY Third Process 95
Third Process
First pass yield or rolled through yield for
these three processes is 0.99 x 0.89 x 0.95
.837, almost 84
Terminator
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YIELD
(process yield)no of operations
Yield \No of op. 3 10 100 1000 10000 0,8 0,
512000 0,107374 0,000000 0,000000 0,000000 0,95 0
,857375 0,214639 0,005921 0,000000 0,000000 0,9999
0,999700 0,999000 0,990049 0,904833 0,367861 0,9
99997 0,999991 0,999970 0,999700 0,997004 0,970445

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Is it fair to compare processes and products that
have different levels of complexity?

• DPO - Defects Per Opportunity

DPO

• DPMO - Defects Per Million Opportunities

DPMO

• Measurable
• The number of opportunities for a defect to
  occur, is related to the complexity involved.

Opportunity
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Yeild to DPMO?

• Ye(-dpu)
• dpu-lnY

dpu defects per unit DPMO(opportunities/unit)
/1 000 000
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Product yield vs dpmo
100 opp.
1000 opp.
10000 opp.
100000 opp.
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The Normal Curve

• Context
• The normal distribution provides the basis for
  many statistical tools and techniques.
• Definition
• A probability distribution where the most
  frequently occurring value is in the middle and
  other probabilities tail off symmetrically in
  both directions. This shape is sometimes called a
  bell-shaped curve.
• Characteristics
• Curve theoretically does not reach zero thus the
  sum of all finite areas total less than 100
• Curve is symmetric on either side of the most
  frequently occurring value
• The peak of the curve represents the center, or
  average, of the process
• For all practical purposes, the area under the
  curve represents virtually 100 of the product
  the process is capable of producing

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Variation
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Calculations
Arithmetic Mean Average
For a Sample
For the whole Population
Median
Middle value, so that half of the data are
above and half of thedata are below the median.
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• Every Normal Curve can be defined by two numbers
• Mean a measure of the center
• Standard deviation a measure of spread

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Observation value X1 0,4 X2 0,3 X3 0,4
X4 0,6 X5 0,5 X6 0,4 X7 0,2 X8 0,3
X9 0,5 X10 0,4
6
?
4
?
2

0
0,1 0,2 0,3 0,4 0,5 0,6 0,7
0,8
???????x-m)2 ?????? ??????n-1
The range method Nlt10 Range/3 Ngt10 Range/4
sample n-1 population n
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Parameters to describe the spread (variability)
Range
Difference between highest and lowest value of
the distribution Influenced by Outliers
Variance (s2)
Average squared difference of data point from
the average
StandardDeviation
Square root of the variance Commonly used
parameter for variability
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How to calculate
Range
Sample
Population
Variance (s2)

StandardDeviation
s2 Variance of entire population, or true
variance S2 Variance of sample, or best
estimate for s2 s Standard deviation of
entire population, or true standard
deviation. S Standard deviation of sample, or
best estimate for s
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ExerciseCalculate Range, Variance and Standard
deviation. Draw a normal probability plot of the
result.
Formulas
Data
R
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Average, Range Spread
Each diagram has an average of 10, range of 18
and a variation of approx. 5,8. Imagine only
looking at the result and not on the graphs.
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The normal distribution
m
s
-6s -5s -4s -3s -2s -1s 0 1s
2s 3s 4s 5s 6s
68.27
95.45
99.73
99.9937
99.999943
99.9999998
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The Z-table
Area under the normal curve is equal to the
probability (p, also named dpo) of getting an
observation beyond Z (see the Z-table)
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Normalizing standard deviations
The expected probability of having a specific
value Observed value - Mean Value
Z-value Standard deviation
( the Z-table gives the
probability occurrence)
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Z-VALUES AND PROBABILITIES
-1???1??
68,3
95,4
-2???2??
99,7
-3???3??
99,999997
-6???6??
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Capability
CP Tolerance width divided by 6 times the
standard deviation. A CP value greater than 2 is
good (thumb rule)
Tolerance width
?
TÖ - TU CP ----------- 6 ?
6
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Capability
Cpk Difference between nearest tolerance limit
and average, divided by 3 times the standard
deviation. A Cpk value greater than 1,5 is good
(thumb rule)
TU
TÖ
Min(TÖ?? alt. ? ?TU) CPK --------------------
-- 3 ?
?
3
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• Continuous data and possible Pitfalls
• Can be divided in to two types of variation
• Common cause (e.g. within batch variation)
• Special cause -The shift between and
  (e.g. batch variation)
• -Outliers or non-rare occasions will appear
  and may ruin the analyze

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Short-Term Capabilities (within group variation)
Shift Happens
Time 1
(between group variation)
Time 2
Time 3
Time 4
Long-Term Capability
(all variation)
Target
USL
LSL
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Z long term and Z short term
The sample and the population sigma are often
almost the same, but the average will probably
differ. Therefore is zST (zB ) and shift drift
preferably used to estimate the true fault
rate.
Shift Drift Zshort term - Zlong term
What will the long term fault rate be in
exercise 5 with a SD of 1.5??
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ZB
Lower Tolerance Limit
Upper Tolerance Limit
PtotPupperPlower
ZB From table with Ptot
Rev C Peter Häyhänen 9805
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Is Six Sigma corresponding to a defect level of
3,4ppm?
USL
LSL
1.5s
Short-term
Short-term
-6s -5s -4s -3s -2s -1s
0 1s 2s 3s 4s 5s
6s
99.9999998 or 0.002 ppm
99.99966 or 3.4 ppm
Yes, with a SD of 1,5!!
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Shift Drift
Z short term in a typical process 4,02 (based on
approx. 30 values).
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Shift Drift
Z long term in a typical process 3,03
(measurments from one and a half year of
production, all values)
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Shift Drift
Poverall 1200ppm Þ Z 3,03s
Psample 29ppm Þ Z 4,02s
Shift Drift Zshort term - Zlong term
Shift Drift 4,02s - 3,03s
Shift Drift 0,99s
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Minitab Capability Output
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Nomenclature
dpmo - defects per million opportunities Yield -
of the number of approved units divided by the
total number of units p(d) - probability for
defects (1-Yield) Fty - First time yield, the
yield when the units are tested for the first
time TpY - Throughput yield, the yield in every
unique process step Yrt - Yield rolled through,
multiplied throughput yield DPU - Defects per
units DPO - Defects per opportunity Opp -
Opportunity, measurable opportunity for defect
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Nomenclature
Zst - Single side short term capability,
calculated with the help of the target Zb - An
estimate of the overall short term capability,
used to calculate Zlt Zlt - A rating of the
long term capability, normally based on SD Zb
pl - Probability for defect beneath lower
specification limit pu - Probability for defect
above upper specification limit p - Summarized
probability for defect, pl pu SD - An
approximation of the drift in average,
fundamentally 1,5? LSL - Lower specification
limit USL - Upper specification limit