Basic Statistics
DFSS Basic Staistics
2004-09-27
1
EAB/JN Stefan Andresen
Cornerstones of a successful use of 6
Results
World Class
Business Performance
Methodology
DFSS Basic Staistics
Change Management
2004-09-27
2
EAB/JN Stefan Andresen
Yield
Lower
Tolerance
Limit
Upper
Tolerance
Limit
Yield
Defects
Yield = Pass / Trials
p(d) = (1- Yield)/100
DFSS Basic Staistics
2004-09-27
4
EAB/JN Stefan Andresen
Discrete data - First Time Right (First Time Yield)
Measures the units that avoid the hidden costs.
Step A
Good?
Yes
Step B
Good?
No
Fix It?
Yes
Ship It!
No
No
SCRAP
No
Fix It?
Yes
Yes
Rework
Rework
COPQ
DFSS Basic Staistics
2004-09-27
5
EAB/JN Stefan Andresen
Discrete data - Rolled Thru Yield
Most processes are complex interrelationships of many sub-processes.
The overall performance is usually of interest to us.
First Process
FTY
First Process
99%
Second
Process
First pass yield or rolled through
yield for these three
processes is
0.99 x 0.89 x 0.95 = .837,
almost 84%
DFSS Basic Staistics
Rolled yield is a realistic
assessment of the cumulative
effect of sub-processes
Rework
FTY
Second Process
89%
Third Process
Rework
FTY
Third Process
95%
Rework
Terminator
2004-09-27
6
EAB/JN Stefan Andresen
YIELD
(process yield)no of operations
Yield \No of op.
0,8
0,95
0,9999
0,999997
3
0,512000
0,857375
0,999700
0,999991
DFSS Basic Staistics
10
0,107374
0,214639
0,999000
0,999970
2004-09-27
100
0,000000
0,005921
0,990049
0,999700
7
1000
0,000000
0,000000
0,904833
0,997004
10000
0,000000
0,000000
0,367861
0,970445
EAB/JN Stefan Andresen
Is it fair to compare processes and products that have different levels of complexity?
DPO
• DPO - Defects Per Opportunity
DPO 
DPMO
 defects
 opportunities
• DPMO - Defects Per Million Opportunities
DPMO 
Opportunity
DFSS Basic Staistics
 defects
 opportunities
*1 000 000
 Measurable
 The number of opportunities for a defect to occur, is
related to the complexity involved.
2004-09-27
8
EAB/JN Stefan Andresen
Yeild to DPMO?
(-dpu)
Y=e
dpu=-lnY
dpu = defects per unit = DPMO*(opportunities/unit)/1 000 000
DFSS Basic Staistics
2004-09-27
9
EAB/JN Stefan Andresen
Product yield vs dpmo
100000 opp.
10000 opp.
1000 opp. 100 opp.
50
0
0
1
10
dpmo
6
DFSS Basic Staistics
100
5
2004-09-27
10
The automation wall
The Design & supply wall
100
1000
10000
100000
4 3
EAB/JN Stefan Andresen
Variation
Output power
Standard deviation
(std, s, )
Special cause
variation
Average, Mean-value
(x, m or µ, M)
Common cause
variation
Measurement no
DFSS Basic Staistics
2004-09-27
12
EAB/JN Stefan Andresen
Every Normal Curve can be defined by two numbers:
•Mean: a measure of the center
•Standard deviation: a measure of spread


DFSS Basic Staistics
2004-09-27
15
EAB/JN Stefan Andresen
Observation

6
4
X1
X2
X3
X4
X5
X6
X7
X8
X9
X10

2
0
0,1
0,2
x-m)2

n-1
DFSS Basic Staistics
0,3 0,4
0,5
0,6
0,7
0,4
0,3
0,4
0,6
0,5
0,4
0,2
0,3
0,5
0,4
0,8
sample = n-1
population = n
2004-09-27
value
16
The range method:
N<10: Range/3
N>10 Range/4
EAB/JN Stefan Andresen
Exercise
Calculate Range, Variance and Standard deviation. Draw a normal
probability plot of the result.
Formulas
Data
Value (xi-x)
R = X max  X min
s2 
s



2
n
i 1 xi  x
n 1
DFSS Basic Staistics
2
5
6
5
7
6
9
7
8
6
8
1
2
n
xi  x

i

1
n 1

(xi-x)
Sum
n-1
Variance
Std. dev.
2004-09-27
19
EAB/JN Stefan Andresen
Average, Range &
Spread
Diagram 1
Faults
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.
20
18
16
14
12
10
8
6
4
2
0
0
2
4
6
8
10
12
14
8
10
12
14
Number
Diagram 3
20
18
16
14
12
10
8
6
4
2
0
Faults
Faults
Diagram 2
0
2
4
6
8
10
12
14
0
Number
DFSS Basic Staistics
20
18
16
14
12
10
8
6
4
2
0
2
4
6
Number
2004-09-27
20
EAB/JN Stefan Andresen
The normal distribution


-6

-5

-4

-3

-2

-1

0
1

2

3

4

5

6

68.27%
95.45%
99.73%
99.9937%
99.999943%
99.9999998%
DFSS Basic Staistics
2004-09-27
21
EAB/JN Stefan Andresen
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)
Z
DFSS Basic Staistics
2004-09-27
22
EAB/JN Stefan Andresen
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)
|x-M|
std
DFSS Basic Staistics
=Z
2004-09-27
23
EAB/JN Stefan Andresen
Z-VALUES AND PROBABILITIES
68,3%
-1+1
95,4%
-2+2
99,7%
-3+3
99,999997%
-6+6
DFSS Basic Staistics
2004-09-27
24
EAB/JN Stefan Andresen
Z – Table
Area
Z
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
5.00E-01
4.60E-01
4.21E-01
3.82E-01
3.45E-01
3.09E-01
2.74E-01
2.42E-01
2.12E-01
1.84E-01
0.01
4.96E-01
4.56E-01
4.17E-01
3.78E-01
3.41E-01
3.05E-01
2.71E-01
2.39E-01
2.09E-01
1.81E-01
0.02
4.92E-01
4.52E-01
4.13E-01
3.75E-01
3.37E-01
3.02E-01
2.68E-01
2.36E-01
2.06E-01
1.79E-01
0.03
4.88E-01
4.48E-01
4.09E-01
3.71E-01
3.34E-01
2.98E-01
2.64E-01
2.33E-01
2.03E-01
1.76E-01
0.04
4.84E-01
4.44E-01
4.05E-01
3.67E-01
3.30E-01
2.95E-01
2.61E-01
2.30E-01
2.01E-01
1.74E-01
0.05
4.80E-01
4.40E-01
4.01E-01
3.63E-01
3.26E-01
2.91E-01
2.58E-01
2.27E-01
1.98E-01
1.71E-01
0.06
4.76E-01
4.36E-01
3.97E-01
3.59E-01
3.23E-01
2.88E-01
2.55E-01
2.24E-01
1.95E-01
1.69E-01
0.07
4.72E-01
4.33E-01
3.94E-01
3.56E-01
3.19E-01
2.84E-01
2.51E-01
2.21E-01
1.92E-01
1.66E-01
0.08
4.68E-01
4.29E-01
3.90E-01
3.52E-01
3.16E-01
2.81E-01
2.48E-01
2.18E-01
1.89E-01
1.64E-01
0.09
4.64E-01
4.25E-01
3.86E-01
3.48E-01
3.12E-01
2.78E-01
2.45E-01
2.15E-01
1.87E-01
1.61E-01
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.59E-01
1.36E-01
1.15E-01
9.68E-02
8.08E-02
6.68E-02
5.48E-02
4.46E-02
3.59E-02
2.87E-02
1.56E-01
1.34E-01
1.13E-01
9.51E-02
7.93E-02
6.55E-02
5.37E-02
4.36E-02
3.52E-02
2.81E-02
1.5 39E01
1.31E-01
1.11E-01
9.34E-02
7.78E-02
6.43E-02
5.26E-02
4.27E-02
3.44E-02
2.74E-02
1.52E-01
1.29E-01
1.09E-01
9.18E-02
7.64E-02
6.30E-02
5.16E-02
4.18E-02
3.36E-02
2.68E-02
1.49E-01
1.27E-01
1.08E-01
9.01E-02
7.49E-02
6.18E-02
5.05E-02
4.09E-02
3.29E-02
2.62E-02
1.47E-01
1.25E-01
1.06E-01
8.85E-02
7.35E-02
6.06E-02
4.95E-02
4.01E-02
3.22E-02
2.56E-02
1.45E-01
1.23E-01
1.04E-01
8.69E-02
7.21E-02
5.94E-02
4.85E-02
3.92E-02
3.14E-02
2.50E-02
1.42E-01
1.21E-01
1.02E-01
8.53E-02
7.08E-02
5.82E-02
4.75E-02
3.84E-02
3.07E-02
2.44E-02
1.40E-01
1.19E-01
1.00E-01
8.38E-02
6.94E-02
5.71E-02
4.65E-02
3.75E-02
3.01E-02
2.39E-02
1.38E-01
1.17E-01
9.85E-02
8.23E-02
6.81E-02
5.59E-02
4.55E-02
3.67E-02
2.94E-02
2.33E-02
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.28E-02
1.79E-02
1.39E-02
1.07E-02
8.20E-03
6.21E-03
4.66E-03
3.47E-03
2.56E-03
1.87E-03
2.22E-02
1.74E-02
1.36E-02
1.04E-02
7.98E-03
6.04E-03
4.53E-03
3.36E-03
2.48E-03
1.81E-03
2.17E-02
1.70E-02
1.32E-02
1.02E-02
7.76E-03
5.87E-03
4.40E-03
3.26E-03
2.40E-03
1.75E-03
2.12E-02
1.66E-02
1.29E-02
9.90E-03
7.55E-03
5.70E-03
4.27E-03
3.17E-03
2.33E-03
1.70E-03
2.07E-02
1.62E-02
1.26E-02
9.64E-03
7.34E-03
5.54E-03
4.15E-03
3.07E-03
2.26E-03
1.64E-03
2.02E-02
1.58E-02
1.22E-02
9.39E-03
7.14E-03
5.39E-03
4.02E-03
2.98E-03
2.19E-03
1.59E-03
1.97E-02
1.54E-02
1.19E-02
9.14E-03
6.95E-03
5.23E-03
3.91E-03
2.89E-03
2.12E-03
1.54E-03
1.92E-02
1.50E-02
1.16E-02
8.89E-03
6.76E-03
5.09E-03
3.79E-03
2.80E-03
2.05E-03
1.49E-03
1.88E-02
1.46E-02
1.13E-02
8.66E-03
6.57E-03
4.94E-03
3.68E-03
2.72E-03
1.99E-03
1.44E-03
1.83E-02
1.43E-02
1.10E-02
8.42E-03
6.39E-03
4.80E-03
3.57E-03
2.64E-03
1.93E-03
1.40E-03
DFSS Basic Staistics
2004-09-27
25
EAB/JN Stefan Andresen
Z – Table
Z
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
0
1.35E-03
9.68E-04
6.87E-04
4.84E-04
3.37E-04
2.33E-04
1.59E-04
1.08E-04
7.25E-05
4.82E-05
3.18E-05
2.08E-05
1.34E-05
8.62E-06
5.48E-06
3.45E-06
2.15E-06
1.33E-06
8.18E-07
4.98E-07
3.00E-07
1.80E-07
1.07E-07
6.27E-08
3.66E-08
2.12E-08
1.22E-08
6.98E-09
3.96E-09
2.23E-09
1.25E-09
0.01
1.31E-03
9.35E-04
6.64E-04
4.67E-04
3.25E-04
2.24E-04
1.53E-04
1.04E-04
6.96E-05
4.63E-05
3.05E-05
1.99E-05
1.29E-05
8.24E-06
5.23E-06
3.29E-06
2.05E-06
1.27E-06
7.79E-07
4.73E-07
2.85E-07
1.71E-07
1.01E-07
5.95E-08
3.47E-08
2.01E-08
1.16E-08
6.60E-09
3.74E-09
2.11E-09
1.18E-09
DFSS Basic Staistics
0.02
1.26E-03
9.04E-04
6.41E-04
4.50E-04
3.13E-04
2.16E-04
1.47E-04
9.97E-05
6.69E-05
4.44E-05
2.92E-05
1.91E-05
1.23E-05
7.88E-06
5.00E-06
3.14E-06
1.96E-06
1.21E-06
7.41E-07
4.50E-07
2.71E-07
1.62E-07
9.59E-08
5.64E-08
3.29E-08
1.90E-08
1.09E-08
6.24E-09
3.53E-09
1.99E-09
1.11E-09
Area
0.03
1.22E-03
8.74E-04
6.19E-04
4.34E-04
3.02E-04
2.08E-04
1.42E-04
9.59E-05
6.42E-05
4.26E-05
2.80E-05
1.82E-05
1.18E-05
7.53E-06
4.77E-06
3.00E-06
1.87E-06
1.15E-06
7.05E-07
4.28E-07
2.58E-07
1.54E-07
9.10E-08
5.34E-08
3.11E-08
1.80E-08
1.03E-08
5.89E-09
3.34E-09
1.88E-09
1.05E-09
0.04
1.18E-03
8.45E-04
5.98E-04
4.19E-04
2.91E-04
2.00E-04
1.36E-04
9.21E-05
6.17E-05
4.09E-05
2.68E-05
1.75E-05
1.13E-05
7.20E-06
4.56E-06
2.86E-06
1.78E-06
1.10E-06
6.71E-07
4.07E-07
2.45E-07
1.46E-07
8.63E-08
5.06E-08
2.95E-08
1.70E-08
9.78E-09
5.57E-09
3.15E-09
1.77E-09
9.88E-10
2004-09-27
0.05
1.14E-03
8.16E-04
5.77E-04
4.04E-04
2.80E-04
1.93E-04
1.31E-04
8.86E-05
5.92E-05
3.92E-05
2.57E-05
1.67E-05
1.08E-05
6.88E-06
4.35E-06
2.73E-06
1.70E-06
1.05E-06
6.39E-07
3.87E-07
2.32E-07
1.39E-07
8.18E-08
4.80E-08
2.79E-08
1.61E-08
9.24E-09
5.26E-09
2.97E-09
1.67E-09
9.31E-10
26
0.06
1.11E-03
7.89E-04
5.57E-04
3.90E-04
2.70E-04
1.86E-04
1.26E-04
8.51E-05
5.68E-05
3.76E-05
2.47E-05
1.60E-05
1.03E-05
6.57E-06
4.16E-06
2.60E-06
1.62E-06
9.96E-07
6.08E-07
3.68E-07
2.21E-07
1.31E-07
7.76E-08
4.55E-08
2.64E-08
1.53E-08
8.74E-09
4.97E-09
2.81E-09
1.58E-09
8.78E-10
0.07
1.07E-03
7.62E-04
5.38E-04
3.76E-04
2.60E-04
1.79E-04
1.21E-04
8.18E-05
5.46E-05
3.61E-05
2.36E-05
1.53E-05
9.86E-06
6.28E-06
3.97E-06
2.48E-06
1.54E-06
9.48E-07
5.78E-07
3.50E-07
2.10E-07
1.25E-07
7.36E-08
4.31E-08
2.50E-08
1.44E-08
8.26E-09
4.70E-09
2.65E-09
1.49E-09
8.28E-10
0.08
1.04E-03
7.36E-04
5.19E-04
3.63E-04
2.51E-04
1.72E-04
1.17E-04
7.85E-05
5.24E-05
3.46E-05
2.26E-05
1.47E-05
9.43E-06
6.00E-06
3.79E-06
2.37E-06
1.47E-06
9.03E-07
5.50E-07
3.32E-07
1.99E-07
1.18E-07
6.98E-08
4.08E-08
2.37E-08
1.37E-08
7.81E-09
4.44E-09
2.50E-09
1.40E-09
7.81E-10
0.09
1.00E-03
7.11E-04
5.01E-04
3.50E-04
2.42E-04
1.66E-04
1.12E-04
7.55E-05
5.03E-05
3.32E-05
2.17E-05
1.40E-05
9.01E-06
5.73E-06
3.62E-06
2.26E-06
1.40E-06
8.59E-07
5.23E-07
3.16E-07
1.89E-07
1.12E-07
6.62E-08
3.87E-08
2.24E-08
1.29E-08
7.39E-09
4.19E-09
2.36E-09
1.32E-09
7.36E-10
EAB/JN Stefan Andresen
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
DFSS Basic Staistics
2004-09-27
27
EAB/JN Stefan Andresen
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
CPK
TÖ
Min(TÖ alt.  TU)
= ----------------------

3
*3
DFSS Basic Staistics
2004-09-27
28
EAB/JN Stefan Andresen
Continuous data and possible Pitfalls
Can be divided in to two types of variation

Common cause

Special cause
(e.g. within batch variation)
-The shift between
and
(e.g. batch variation)
-Outliers or non-rare occasions will appear and may ruin the analyze
Output power
22
20
Effect (dBm)

18
16
14
12
10
0
5
10
15
20
25
30
Number
DFSS Basic Staistics
2004-09-27
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EAB/JN Stefan Andresen
Short-Term
Capabilities
(within group
variation)
„Shift Happens“
Time 1
(between
group
variation)
Time 2
Time 3
Time 4
Long-Term
Capability
(all variation)
LSL
DFSS Basic Staistics
2004-09-27
Target
30
USL
EAB/JN Stefan Andresen
Z long term and Z short term
Z _ long _ term 
Tol  

Tol  T
Z _ Short _ term 
s
p  overall  Z B
Single  sided 
Single  Sided 
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 S&D of 1.5?
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EAB/JN Stefan Andresen
ZB
Lower
Tolerance
Limit
Upper
Tolerance
Limit
Ptot=Pupper+Plower
ZB – From table with Ptot
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Rev C Peter Häyhänen 9805
EAB/JN Stefan Andresen
Is Six Sigma corresponding to a defect level of 3,4ppm?
LSL
USL
 1.5
Short-term
Short-term
-6

-5

-4

-3

-2

-1

0
1

2

3

4

5

6

99.9999998% or 0.002 ppm
99.99966% or 3.4 ppm
Yes, with a S&D of 1,5!!
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EAB/JN Stefan Andresen
Shift & Drift
Z short term inProcess
a typical
process
4,02
on approx. 30 values).
Capability
Analysis
for (based
Z-short term
LSL
Process Data
USL
Target
LSL
Mean
Sample N
StDev (ST)
StDev (LT)
ST
LT
*
*
10,0000
12,5804
25
0,641863
0,641863
Potential (ST) Capability
Cp
CPU
CPL
Cpk
Cpm
*
*
1,34
1,34
*
Overall (LT) Capability
Pp
*
PPU
*
PPL
1,34
Ppk
10
11
Observed Performance
PPM < LSL
0,00
PPM > USL
*
PPM Total
0,00
12
13
Expected ST Performance
PPM < LSL
29,08
PPM > USL
*
PPM Total
29,08
14
15
Expected LT Performance
PPM < LSL
29,08
PPM > USL
*
PPM Total
29,08
1,34
DFSS Basic Staistics
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34
EAB/JN Stefan Andresen
Shift & Drift
Z long term in a Process
typicalCapability
processAnalysis
3,03 for
(measurments
from one and a
Z-long term
half year of production, “all values”)
LSL
Process Data
USL
Target
LSL
Mean
Sample N
StDev (ST)
StDev (LT)
ST
LT
*
*
10,0000
12,2222
161
0,732048
0,732048
Potential (ST) Capability
Cp
CPU
CPL
Cpk
Cpm
*
*
1,01
1,01
*
Overall (LT) Capability
Pp
*
PPU
*
PPL
1,01
Ppk
10
11
Observed Performance
PPM < LSL
0,00
PPM > USL
*
PPM Total
0,00
12
13
Expected ST Performance
PPM < LSL
1200,46
PPM > USL
*
PPM Total
1200,46
14
Expected LT Performance
PPM < LSL
1200,46
PPM > USL
*
PPM Total
1200,46
1,01
DFSS Basic Staistics
2004-09-27
35
EAB/JN Stefan Andresen
Shift & Drift
Poverall = 1200ppm  Z = 3,03
 Z = 4,02
Psample = 29ppm
Shift & Drift = Zshort term - Zlong term
Shift & Drift = 4,02 - 3,03
Shift & Drift = 0,99
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EAB/JN Stefan Andresen
Minitab Capability Output
DFSS Basic Staistics
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EAB/JN Stefan Andresen
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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EAB/JN Stefan Andresen
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 S&D & Zb
pl
- Probability for defect beneath lower specification limit
pu
- Probability for defect above upper specification limit
p
- Summarized probability for defect, pl + pu
S&D
- An approximation of the drift in average, fundamentally 1,5
LSL
- Lower specification limit
USL
- Upper specification limit
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39
EAB/JN Stefan Andresen