Basic Statistics

DFSS Basic Staistics EAB/JN Stefan Andresen2004-09-27 1

Basic Statistics


Cornerstones of a successful use of 6

Change Management

Results

Methodology

World Class World Class Business PerformanceBusiness Performance


Yield

Defects

LowerToleranceLimit

UpperTolerance

Limit

Yield

Yield = Pass / Trials

p(d) = (1- Yield)/100


Discrete data - First Time Right Discrete data - First Time Right (First Time Yield)(First Time Yield)

Measures the units that avoid the hidden costs.

Step AStep A Step BStep B

SCRAPSCRAP

ReworkRework

Ship It!Ship It!

ReworkRework

Fix It?

Good?

Fix It?

Good?

COPQ

Yes

Yes

No No

No

Yes

Yes

No


Discrete data - Rolled Thru YieldDiscrete data - Rolled Thru Yield

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

First ProcessFirst Process

SecondProcess

SecondProcess

Third ProcessThird Process

TerminatorTerminator

FTYFirst Process

99%

FTYSecond Process

89%

FTYThird Process

95%First pass yield or rolled throughyield for these threeprocesses is 0.99 x 0.89 x 0.95 = .837,almost 84%

Rolled yield is a realisticassessment of the

cumulativeeffect of sub-processesRework

Rework

Rework


YIELD

Yield \No of op.Yield \No of op. 33 1010 100100 10001000 10000100000,80,8 0,5120000,512000 0,1073740,107374 0,0000000,000000 0,0000000,000000 0,0000000,0000000,950,95 0,8573750,857375 0,2146390,214639 0,0059210,005921 0,0000000,000000 0,0000000,0000000,99990,9999 0,9997000,999700 0,9990000,999000 0,9900490,990049 0,9048330,904833 0,3678610,3678610,9999970,999997 0,9999910,999991 0,9999700,999970 0,9997000,999700 0,9970040,997004 0,9704450,970445

(process yield)(process yield)no of operationsno of operations


• DPMO - Defects Per Million Opportunities

0000001*

iesopportunitdefects

DPMO

DPMODPMO

Measurable The number of opportunities for a defect to occur, is

related to the complexity involved.

OpportunOpportunityity

• DPO - Defects Per Opportunity

iesopportunit

defectsDPO

DPODPO

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


Yeild to DPMO?

Y=e(-dpu)

dpu=-lnY

dpu = defects per unit = DPMO*(opportunities/unit)/1 000 000


0 1 1 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0

0

5 0

1 0 0

dpm o

pro

du

ct

yie

ld

66 55 44 33

100 opp.1000 opp.10000 opp.

Product yield vs dpmo

Th

e au

tom

atio

n w

all

Th

e au

tom

atio

n w

all

Th

e D

esig

n &

su

pp

ly w

all

Th

e D

esig

n &

su

pp

ly w

all

100000 opp.


VariationVariation

Output power

Measurement no

Average, Mean-value (x, m or µ, M)

Standard deviation (std, s, )

Common causevariation

Special causevariation


Every Normal Curve can be defined by two numbers:

•Mean: a measure of the center

•Standard deviation: a measure of spread


0

2

4

6

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

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

x-m)2

n-1

sample = n-1population = n

The range method:N<10: Range/3N>10 Range/4


Value (xi-x) (xi-x)2

5657697868

Sumn-1VarianceStd. dev.

Exercise

Calculate Range, Variance and Standard deviation. Draw a normal probability plot of the result.

minmax XX

ni xixn

s 12

1

12

1

12

n

ni xix

s

R =

Formulas Data


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.

Average, Range & Spread

Diagram 3

02468

101214161820

0 2 4 6 8 10 12 14

Number

Fau

lts

Diagram 2

02468

101214161820

0 2 4 6 8 10 12 14

Number

Fau

lts

Diagram 1

02468

101214161820

0 2 4 6 8 10 12 14

Number

Fau

lts


The normal distribution

99.9999998%

99.999943%99.9937%

99.73%

95.45%

68.27%

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

6


Z

Area under the normal curve is equal to the probability (p,

also named dpo) of getting an observation beyond Z (see

the Z-table)

The Z-table


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 = Z


Z-VALUES AND PROBABILITIES Z-VALUES AND PROBABILITIES

-1+168,3%

-2+295,4%

-3+399,7%

-6+699,999997%


Z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090.0 5.00E-01 4.96E-01 4.92E-01 4.88E-01 4.84E-01 4.80E-01 4.76E-01 4.72E-01 4.68E-01 4.64E-010.1 4.60E-01 4.56E-01 4.52E-01 4.48E-01 4.44E-01 4.40E-01 4.36E-01 4.33E-01 4.29E-01 4.25E-010.2 4.21E-01 4.17E-01 4.13E-01 4.09E-01 4.05E-01 4.01E-01 3.97E-01 3.94E-01 3.90E-01 3.86E-010.3 3.82E-01 3.78E-01 3.75E-01 3.71E-01 3.67E-01 3.63E-01 3.59E-01 3.56E-01 3.52E-01 3.48E-010.4 3.45E-01 3.41E-01 3.37E-01 3.34E-01 3.30E-01 3.26E-01 3.23E-01 3.19E-01 3.16E-01 3.12E-010.5 3.09E-01 3.05E-01 3.02E-01 2.98E-01 2.95E-01 2.91E-01 2.88E-01 2.84E-01 2.81E-01 2.78E-010.6 2.74E-01 2.71E-01 2.68E-01 2.64E-01 2.61E-01 2.58E-01 2.55E-01 2.51E-01 2.48E-01 2.45E-010.7 2.42E-01 2.39E-01 2.36E-01 2.33E-01 2.30E-01 2.27E-01 2.24E-01 2.21E-01 2.18E-01 2.15E-010.8 2.12E-01 2.09E-01 2.06E-01 2.03E-01 2.01E-01 1.98E-01 1.95E-01 1.92E-01 1.89E-01 1.87E-010.9 1.84E-01 1.81E-01 1.79E-01 1.76E-01 1.74E-01 1.71E-01 1.69E-01 1.66E-01 1.64E-01 1.61E-01

1.0 1.59E-01 1.56E-01 1.5 39E01 1.52E-01 1.49E-01 1.47E-01 1.45E-01 1.42E-01 1.40E-01 1.38E-011.1 1.36E-01 1.34E-01 1.31E-01 1.29E-01 1.27E-01 1.25E-01 1.23E-01 1.21E-01 1.19E-01 1.17E-011.2 1.15E-01 1.13E-01 1.11E-01 1.09E-01 1.08E-01 1.06E-01 1.04E-01 1.02E-01 1.00E-01 9.85E-021.3 9.68E-02 9.51E-02 9.34E-02 9.18E-02 9.01E-02 8.85E-02 8.69E-02 8.53E-02 8.38E-02 8.23E-021.4 8.08E-02 7.93E-02 7.78E-02 7.64E-02 7.49E-02 7.35E-02 7.21E-02 7.08E-02 6.94E-02 6.81E-021.5 6.68E-02 6.55E-02 6.43E-02 6.30E-02 6.18E-02 6.06E-02 5.94E-02 5.82E-02 5.71E-02 5.59E-021.6 5.48E-02 5.37E-02 5.26E-02 5.16E-02 5.05E-02 4.95E-02 4.85E-02 4.75E-02 4.65E-02 4.55E-021.7 4.46E-02 4.36E-02 4.27E-02 4.18E-02 4.09E-02 4.01E-02 3.92E-02 3.84E-02 3.75E-02 3.67E-021.8 3.59E-02 3.52E-02 3.44E-02 3.36E-02 3.29E-02 3.22E-02 3.14E-02 3.07E-02 3.01E-02 2.94E-021.9 2.87E-02 2.81E-02 2.74E-02 2.68E-02 2.62E-02 2.56E-02 2.50E-02 2.44E-02 2.39E-02 2.33E-02

2.0 2.28E-02 2.22E-02 2.17E-02 2.12E-02 2.07E-02 2.02E-02 1.97E-02 1.92E-02 1.88E-02 1.83E-022.1 1.79E-02 1.74E-02 1.70E-02 1.66E-02 1.62E-02 1.58E-02 1.54E-02 1.50E-02 1.46E-02 1.43E-022.2 1.39E-02 1.36E-02 1.32E-02 1.29E-02 1.26E-02 1.22E-02 1.19E-02 1.16E-02 1.13E-02 1.10E-022.3 1.07E-02 1.04E-02 1.02E-02 9.90E-03 9.64E-03 9.39E-03 9.14E-03 8.89E-03 8.66E-03 8.42E-032.4 8.20E-03 7.98E-03 7.76E-03 7.55E-03 7.34E-03 7.14E-03 6.95E-03 6.76E-03 6.57E-03 6.39E-032.5 6.21E-03 6.04E-03 5.87E-03 5.70E-03 5.54E-03 5.39E-03 5.23E-03 5.09E-03 4.94E-03 4.80E-032.6 4.66E-03 4.53E-03 4.40E-03 4.27E-03 4.15E-03 4.02E-03 3.91E-03 3.79E-03 3.68E-03 3.57E-032.7 3.47E-03 3.36E-03 3.26E-03 3.17E-03 3.07E-03 2.98E-03 2.89E-03 2.80E-03 2.72E-03 2.64E-032.8 2.56E-03 2.48E-03 2.40E-03 2.33E-03 2.26E-03 2.19E-03 2.12E-03 2.05E-03 1.99E-03 1.93E-032.9 1.87E-03 1.81E-03 1.75E-03 1.70E-03 1.64E-03 1.59E-03 1.54E-03 1.49E-03 1.44E-03 1.40E-03

Z – Table Area


3.0 1.35E-03 1.31E-03 1.26E-03 1.22E-03 1.18E-03 1.14E-03 1.11E-03 1.07E-03 1.04E-03 1.00E-033.1 9.68E-04 9.35E-04 9.04E-04 8.74E-04 8.45E-04 8.16E-04 7.89E-04 7.62E-04 7.36E-04 7.11E-043.2 6.87E-04 6.64E-04 6.41E-04 6.19E-04 5.98E-04 5.77E-04 5.57E-04 5.38E-04 5.19E-04 5.01E-043.3 4.84E-04 4.67E-04 4.50E-04 4.34E-04 4.19E-04 4.04E-04 3.90E-04 3.76E-04 3.63E-04 3.50E-043.4 3.37E-04 3.25E-04 3.13E-04 3.02E-04 2.91E-04 2.80E-04 2.70E-04 2.60E-04 2.51E-04 2.42E-043.5 2.33E-04 2.24E-04 2.16E-04 2.08E-04 2.00E-04 1.93E-04 1.86E-04 1.79E-04 1.72E-04 1.66E-043.6 1.59E-04 1.53E-04 1.47E-04 1.42E-04 1.36E-04 1.31E-04 1.26E-04 1.21E-04 1.17E-04 1.12E-043.7 1.08E-04 1.04E-04 9.97E-05 9.59E-05 9.21E-05 8.86E-05 8.51E-05 8.18E-05 7.85E-05 7.55E-053.8 7.25E-05 6.96E-05 6.69E-05 6.42E-05 6.17E-05 5.92E-05 5.68E-05 5.46E-05 5.24E-05 5.03E-053.9 4.82E-05 4.63E-05 4.44E-05 4.26E-05 4.09E-05 3.92E-05 3.76E-05 3.61E-05 3.46E-05 3.32E-054.0 3.18E-05 3.05E-05 2.92E-05 2.80E-05 2.68E-05 2.57E-05 2.47E-05 2.36E-05 2.26E-05 2.17E-054.1 2.08E-05 1.99E-05 1.91E-05 1.82E-05 1.75E-05 1.67E-05 1.60E-05 1.53E-05 1.47E-05 1.40E-054.2 1.34E-05 1.29E-05 1.23E-05 1.18E-05 1.13E-05 1.08E-05 1.03E-05 9.86E-06 9.43E-06 9.01E-064.3 8.62E-06 8.24E-06 7.88E-06 7.53E-06 7.20E-06 6.88E-06 6.57E-06 6.28E-06 6.00E-06 5.73E-064.4 5.48E-06 5.23E-06 5.00E-06 4.77E-06 4.56E-06 4.35E-06 4.16E-06 3.97E-06 3.79E-06 3.62E-064.5 3.45E-06 3.29E-06 3.14E-06 3.00E-06 2.86E-06 2.73E-06 2.60E-06 2.48E-06 2.37E-06 2.26E-064.6 2.15E-06 2.05E-06 1.96E-06 1.87E-06 1.78E-06 1.70E-06 1.62E-06 1.54E-06 1.47E-06 1.40E-064.7 1.33E-06 1.27E-06 1.21E-06 1.15E-06 1.10E-06 1.05E-06 9.96E-07 9.48E-07 9.03E-07 8.59E-074.8 8.18E-07 7.79E-07 7.41E-07 7.05E-07 6.71E-07 6.39E-07 6.08E-07 5.78E-07 5.50E-07 5.23E-074.9 4.98E-07 4.73E-07 4.50E-07 4.28E-07 4.07E-07 3.87E-07 3.68E-07 3.50E-07 3.32E-07 3.16E-075.0 3.00E-07 2.85E-07 2.71E-07 2.58E-07 2.45E-07 2.32E-07 2.21E-07 2.10E-07 1.99E-07 1.89E-075.1 1.80E-07 1.71E-07 1.62E-07 1.54E-07 1.46E-07 1.39E-07 1.31E-07 1.25E-07 1.18E-07 1.12E-075.2 1.07E-07 1.01E-07 9.59E-08 9.10E-08 8.63E-08 8.18E-08 7.76E-08 7.36E-08 6.98E-08 6.62E-085.3 6.27E-08 5.95E-08 5.64E-08 5.34E-08 5.06E-08 4.80E-08 4.55E-08 4.31E-08 4.08E-08 3.87E-085.4 3.66E-08 3.47E-08 3.29E-08 3.11E-08 2.95E-08 2.79E-08 2.64E-08 2.50E-08 2.37E-08 2.24E-085.5 2.12E-08 2.01E-08 1.90E-08 1.80E-08 1.70E-08 1.61E-08 1.53E-08 1.44E-08 1.37E-08 1.29E-085.6 1.22E-08 1.16E-08 1.09E-08 1.03E-08 9.78E-09 9.24E-09 8.74E-09 8.26E-09 7.81E-09 7.39E-095.7 6.98E-09 6.60E-09 6.24E-09 5.89E-09 5.57E-09 5.26E-09 4.97E-09 4.70E-09 4.44E-09 4.19E-095.8 3.96E-09 3.74E-09 3.53E-09 3.34E-09 3.15E-09 2.97E-09 2.81E-09 2.65E-09 2.50E-09 2.36E-095.9 2.23E-09 2.11E-09 1.99E-09 1.88E-09 1.77E-09 1.67E-09 1.58E-09 1.49E-09 1.40E-09 1.32E-09

6.0 1.25E-09 1.18E-09 1.11E-09 1.05E-09 9.88E-10 9.31E-10 8.78E-10 8.28E-10 7.81E-10 7.36E-10

Z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Z – Table Area


* 6* 6

CP Tolerance width divided by 6 times the standard deviation. A CP value greater than 2 is good (thumb rule)

TÖ - TU

CP = -----------

6

Tolerance width

CapabilityCapability


* 3* 3

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)

Min(TÖ alt. TU)CPK = ----------------------

3

TÖTU

CapabilityCapability


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

Output power

10

12

14

16

18

20

22

0 5 10 15 20 25 30

Number

Eff

ec

t (d

Bm

)


Long-Term Capability

Target USLLSL

Time 1

Time 2

Time 3

Time 4

Short-TermCapabilities

(within group variation)

(between group

variation)

(all variation)

„Shift Happens“


Z long term and Z short term

B

termShort

termlong

Zoverallp

SidedSingles

TTolZ

sidedSingleTol

Z

__

__

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?


ZB LowerToleranceLimit

UpperTolerance

Limit

ZB – From table with Ptot

Rev C Peter Häyhänen 9805

Ptot=Pupper+Plower


Short-term

LSL USL

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

5 6

Short-term

99.9999998% or 0.002 ppm99.9999998% or 0.002 ppm

1.5

99.99966% or 3.4 ppm99.99966% or 3.4 ppm

Is Six Sigma corresponding to a defect Is Six Sigma corresponding to a defect level of 3,4ppm?level of 3,4ppm?

Yes, with a S&D of 1,5!!


151413121110

LSLLSL

Process Capability Analysis for Z-short term

PPM Total

PPM > USL

PPM < LSL

PPM Total

PPM > USL

PPM < LSL

PPM Total

PPM > USL

PPM < LSL

Ppk

PPL

PPU

Pp

Cpm

Cpk

CPL

CPU

Cp

StDev (LT)

StDev (ST)

Sample N

Mean

LSL

Target

USL

29,08

*

29,08

29,08

*

29,08

0,00

*

0,00

1,34

1,34

*

*

*

1,34

1,34

*

*

0,641863

0,641863

25

12,5804

10,0000

*

*

Expected LT PerformanceExpected ST PerformanceObserved PerformanceOverall (LT) Capability

Potential (ST) Capability

Process DataSTLT

Shift & DriftShift & Drift

Z short term in a typical process 4,02 (based on approx. 30 values).


1413121110

LSLLSL

Process Capability Analysis for Z-long term

PPM Total

PPM > USL

PPM < LSL

PPM Total

PPM > USL

PPM < LSL

PPM Total

PPM > USL

PPM < LSL

Ppk

PPL

PPU

Pp

Cpm

Cpk

CPL

CPU

Cp

StDev (LT)

StDev (ST)

Sample N

Mean

LSL

Target

USL

1200,46

*

1200,46

1200,46

*

1200,46

0,00

*

0,00

1,01

1,01

*

*

*

1,01

1,01

*

*

0,732048

0,732048

161

12,2222

10,0000

*

*

Expected LT PerformanceExpected ST PerformanceObserved PerformanceOverall (LT) Capability

Potential (ST) Capability

Process DataSTLT


Z long term in a typical process 3,03 (measurments from one and a half year of production, “all values”)


Poverall = 1200ppm Z = 3,03

Psample = 29ppm Z = 4,02

Shift & Drift = Zshort term - Zlong term

Shift & Drift = 4,02 - 3,03

Shift & Drift = 0,99



Minitab Capability Output


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


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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