Introduction to Six Sigma Applications

m

u

C

l

100

50

0

70

80

90

100

110

120

130

This means that we can now

predict the change that

certain values can occur

based on these percentages.

Note here that 50% of the

values are less than our

expected value of 100.

This means that in a future

experiment set up the same

way, we would expect 50%

of the values to be less than

100.

COIN TOSS EXAMPLE

Results from 10,000 people doing a coin toss 200 times.

Count Frequency

y

c

n

e

u

q

e

r

F

600

500

400

300

200

100

0

80

70

100

"Head Count"

Results from 10,000 people doing a coin toss 200 times.

Cumulative Percent

110

120

130

90

t

n

e

c

r

e

P

e

v

i

t

a

u

m

u

C

l

100

50

0

70

80

90

100

"Head Count"

110

120

130

We can now equate a probability to the

occurrence of specific values or groups of

values.

For example, we can see that the

occurrence of a “Head count” of less than

74 or greater than 124 out of 200 tosses

is so rare that a single occurrence was

not registered out of 10,000 tries.

On the other hand, we can see that the

chance of getting a count near (or at) 100

is much higher. With the data that we

now have, we can actually predict each of

these values.

COIN TOSS PROBABILITY DISTRIBUTION

% of population = probability of occurrence

600

PROCESS CENTERED

ON EXPECTED VALUE

If we know where

we are in the

population we can

equate that to a

probability value.

This is the purpose

of the sigma value

(normal data).

y

c

n

e

u

q

e

r

F

500

400

300

200

100

0

SIGMA (s ) IS A MEASURE

OF “SCATTER” FROM THE

EXPECTED VALUE THAT

CAN BE USED TO

CALCULATE A PROBABILITY

OF OCCURRENCE

s

NUMBER OF HEADS

58

65

72

70

80

79

90

100

110

120

130

86

93

100

107

114

121

128

135

142

SIGMA VALUE (Z)

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

CUM % OF POPULATION

.003

.135

2.275

15.87

50.0

84.1

97.7

99.86

99.997

WHAT DOES IT MEAN?

 Common Occurrence

 Rare Event

What are the chances that this

“just happened” If they are small,

chances are that an external

influence is at work that can be

used to our benefit….

Probability and Statistics

• “the odds of Colorado University winning the national

title are 3 to 1”

• “Drew Bledsoe’s pass completion percentage for the last

6 games is .58% versus .78% for the first 5 games”

• “The Senator will win the election with 54% of the popular

vote with a margin of +/- 3%”

• Probability and Statistics influence our lives daily

• Statistics is the universal lanuage 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

sample

population

 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

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

WHAT DOES IT MEAN?

WHAT IF WE MADE A CHANGE TO THE PROCESS?

Chances are very

good that the

process distribution

has changed. In

fact, there is a

probability greater

than 99.999% that

it has changed.

y

c

n

e

u

q

e

r

F

600

500

400

300

200

100

0

And the first 50

trials showed

“Head Counts”

greater than 130?

s

NUMBER OF HEADS

58

65

72

70

80

79

90

100

110

120

130

86

93

100

107

114

121

128

135

142

SIGMA VALUE (Z)

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

CUM % OF POPULATION

.003

.135

2.275

15.87

50.0

84.1

97.7

99.86

99.997

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

Rare

Occurrence

Common

Occurrence

Rare

Occurrence

SO WHAT MAKES A DISTRIBUTION UNIQUE?

CENTRAL TENDENCY

Where a population is located.

DISPERSION

How wide a population is spread.

DISTRIBUTION FUNCTION

The mathematical formula that

best describes the data (we will

cover this in detail in the next

module).

COIN TOSS CENTRAL TENDENCY

s

e

c

n

e

r

r

u

c

c

o

f

o

r

e

b

m

u

N

6 0 0

5 0 0

4 0 0

3 0 0

2 0 0

1 0 0

0

8 0

7 0

1 2 0

What are some of the ways that we can easily indicate

the centering characteristic of the population?

1 1 0

1 0 0

9 0

1 3 0

Three measures have historically been used; the

mean, the median and the mode.

WHAT IS THE MEAN?

The mean has already been used in several earlier modules

and is the most common measure of central tendency for a

population. The mean is simply the average value of the

data.

mean

x

= =

x

i



n

=

2

-

12

= -

17.

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

ORDERED DATA SET

n=12

-5

-3

-1

-1

0

0

0

0

0

1

3

4

Mean

xi = -

2

WHAT IS THE MEDIAN?

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

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

Median

Value

ORDERED DATA SET

50% of data

points

-5

-3

-1

-1

0

0

0

0

0

1

3

4

ORDERED DATA SET

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

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

-5

-3

-1

-1

0

0

0

0

0

1

3

4

MEASURES OF CENTRAL TENDENCY, SUMMARY

ORDERED DATA SET

-5

-3

-1

-1

0

0

0

0

0

1

3

4

ORDERED DATA SET

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

n=12

-5

-3

-1

-1

0

0

0

0

0

1

3

4

n/2=6

Median

n/2=6

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

Mode = 0

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

ORDERED DATA SET

-5

-3

-1

-1

0

0

0

0

0

1

3

4

} Mode = 0

MEAN ( )

X

(Otherwise known as the average)

X

=

X

i



n

=

2

-

12

=

17.

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.

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.

COIN TOSS POPULATION DISPERSION

s

e

c

n

e

r

r

u

c

c

o

f

o

r

e

b

m

u

N

6 0 0

5 0 0

4 0 0

3 0 0

2 0 0

1 0 0

0

7 0

8 0

9 0

1 0 0

1 1 0

1 2 0

1 3 0

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?

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

=

x

MAX

-

x

MIN

(

= - - =

)

5

4

9

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

Min

Max

Range

ORDERED DATA SET

-5

-3

-1

-1

0

0

0

0

0

1

3

4

Range

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.

-.17

=

X

=

X

i



n

=

2

s

=



(

X

n

X

-

1

i

-

-

2

12

)

2

=

.

61 67

12 1

-

=

.

5 6

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

DATA SET

-5

-3

-1

-1

0

0

0

0

0

1

3

4

X

Xi -

-5-(-.17)=-4.83

(-4.83)2=23.32

-3-(-.17)=-2.83

(-2.83)2=8.01

-1-(-.17)=-.83

(-.83)2=.69

-1-(-.17)=-.83

(-.83)2=.69

0-(-.17)=.17

(.17)2=.03

0-(-.17)=.17

(.17)2=.03

0-(-.17)=.17

(.17)2=.03

0-(-.17)=.17

(.17)2=.03

0-(-.17)=.17

(.17)2=.03

1-(-.17)=1.17

(1.17)2=1.37

3-(-.17)=3.17

(3.17)2=10.05

4-(-.17)=4.17

(4.17)2=17.39

61.67

()XXi-2MEASURES OF DISPERSION

ORDERED DATA SET

Min=-5

RANGE (R)

(The maximum data value minus the minimum)

10

R

= - -

X

X

-

=

=

6

4

)

(

max

min

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

X

=

X

i



n

=

2

-

12

-.17

=

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

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

Max=4

X

Xi -

-5-(-.17)=-4.83

(-4.83)2=23.32

-3-(-.17)=-2.83

(-2.83)2=8.01

-1-(-.17)=-.83

(-.83)2=.69

-1-(-.17)=-.83

(-.83)2=.69

0-(-.17)=.17

(.17)2=.03

0-(-.17)=.17

(.17)2=.03

0-(-.17)=.17

(.17)2=.03

0-(-.17)=.17

(.17)2=.03

0-(-.17)=.17

(.17)2=.03

1-(-.17)=1.17

(1.17)2=1.37

VARIANCE (s2)

(Squared deviations around the center point)

X

2



(

X

n

-

i

1

-

)

=

.

61 67

12 1

-

=

.

5 6

2

s

=

3-(-.17)=3.17

ORDERED DATA SET

4-(-.17)=4.17

(3.17)2=10.05

(4.17)2=17.39

61.67

STANDARD DEVIATION (s)

(Absolute deviation around the center point)

s

s=

2

=

.

5 6

=

.

2 37

-5

-3

-1

-1

0

0

0

0

0

1

3

4

DATA SET

-5

-3

-1

-1

0

0

0

0

0

1

3

4

-5

-3

-1

-1

0

0

0

0

0

1

3

4

()XXi-2SAMPLE MEAN AND VARIANCE EXAMPLE

-X

i

X

(

-X

Xi

)2

$=

X

=



X i

N

s

$ 2

=

(

Xi

-X

-2s

n

1

=

2

)

Xi

10

15

12

14

10

9

11

12

10

12

1

2

3

4

5

6

7

8

9

10

S

X i

X

2s

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...

LSL

USL

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

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.

SO WHAT HAPPENED?

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.

Original Study

Historically, this shift and drift

primarily impacts the position of

the mean and shifts it 1.5 s from

it’s original position.

VARIATION FAMILIES

Sources of

Variation

Within Individual

Sample

Piece to

Piece

Time to Time

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.

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

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

PPM ST Cpk PPM LT (+1.5 s)

500,000

460,172

420,740

382,089

344,578

308,538

274,253

241,964

211,855

184,060

158,655

135,666

115,070

96,801

80,757

66,807

54,799

44,565

933,193

919,243

903,199

884,930

864,334

841,345

815,940

788,145

758,036

725,747

691,462

655,422

617,911

579,260

539,828

500,000

460,172

420,740

0.0

0.0

0.1

0.1

0.1

0.2

0.2

0.2

0.3

0.3

0.3

0.4

0.4

0.4

0.5

0.5

0.5

0.6

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.

SHIFT AND DRIFT EXERCISE

We have just completed a project and have presented the

following short term metrics:

•Zst=3.5

•PPMst=233

•Cpkst=1.2

Calculate the long

term values for each

of these metrics.

COMMON PROBABILITY

DISTRIBUTIONS AND

THEIR USES

Why do we Care?

Probability distributions are

necessary to:

•determine whether an event is

significant or due to random

chance.

•predict the probability of specific

performance given historical

characteristics.

IMPROVEMENT ROADMAP

Uses of Probability Distributions

Common Uses

Phase 1:

Measurement

•Baselining Processes

Breakthrough

Strategy

Characterization

Optimization

Phase 2:

Analysis

Phase 3:

Improvement

Phase 4:

Control

•Verifying Improvements

KEYS TO SUCCESS

Focus on understanding the use of the distributions

Practice with examples wherever possible

Focus on the use and context of the tool

PROBABILITY DISTRIBUTIONS, WHERE

DO THEY COME FROM?

Data points vary, but as the data accumulates, it forms a distribution which occurs naturally.

Distributions can vary in:

Location

Spread

Shape

XXXXXCOMMON PROBABILITY DISTRIBUTIONS

Original Population

Continuous Distribution

0

1

2

3

4

5

6

7

Subgroup Average

Normal Distribution

0

1

2

3

4

5

6

7

Subgroup Variance (s2)

c2 Distribution

0

1

2

3

4

5

6

7

4

3

2

1

0

-1

-2

-3

-4

4

3

2

1

0

-1

-2

-3

-4

4

3

2

1

0

THE LANGUAGE OF MATH

SymbolNameStatistic MeaningCommon UsesaAlphaSignificance levelHypothesis Testing,DOEc2Chi SquareProbability DistributionConfidence Intervals, ContingencyTables, Hypothesis TestingSSumSum of Individual valuesVariance Calculationstt, Student tProbability DistributionHypothesis Testing, Confidence Intervalof the MeannSampleSizeTotal size of the SampleTakenNearly all FunctionsNuDegree of FreedomProbability Distributions, HypothesisTesting, DOEBetaBeta RiskSample Size DeterminationDeltaDifference betweenpopulation meansSample Size DeterminationSigmaValueNumber of StandardDeviations a value Existsfrom the MeanProbability Distributions, ProcessCapability, Sample Size DeterminationsPopulation and Sample Symbology

Value

Population

Sample

Mean

Variance

Standard Deviation

Process Capability

Binomial Mean



s2

s

Cp

P

x

s2

s

Cp

P

THREE PROBABILITY DISTRIBUTIONS

t

CALC =

X

• 

s

n

Significant

=

t

CALC



t

CRIT

2

s

calc = 1

F

2

s

2

(

f

e

-

f

e

2

ca,df

=

Significant

=

F

CALC



F

CRIT

2

)

f

a

Significant

=

2

c

CALC

2

c

CRIT

Note that in each case, a limit has been established to determine what is

random chance verses significant difference. This point is called the critical

value. If the calculated value exceeds this critical value, there is very low

probability (P<.05) that this is due to random chance.

Z TRANSFORM

-1s

+1s

-2s

+2s

+/- 1s = 68%

+/- 2s = 95%

2 tail = 32%

1 tail = 16%

2 tail = 4.6%

1 tail = 2.3%

68.26%

95.46%

-3s

+3s

Common Test Values

Z(1.6) = 5.5% (1 tail a=.05)

Z(2.0) = 2.5% (2 tail a=.05)

+/- 3s = 99.7%

99.73%

2 tail = 0.3%

1 tail = .15%

The Focus of Six Sigma…..

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...

Y = f(x)

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.

BUT WHAT DISTRIBUTION

SHOULD I USE?

Characterize

Population

Population

Average

Population

Variance

Determine

Confidence

Interval for

Point Values

Compare 2

Population

Averages

Compare a

Population

Average

Against a

Target Value

•Z Stat (n>30)

Compare 2

Population

Variances

•F Stat (n>30)

Compare a

Population

Variance

Against Target

Value(s)

•F Stat (n>30)

•Z Stat (,n>30)

•Z Stat (p)

•Z Stat (n>30)

•t Stat (,n<30)

•Z Stat (p)

•Z Stat (p)

•F’ Stat (n<10)

• c2 Stat (n>5)

• c2 Stat (s,n<10)

•t Stat (n<30)

•t Stat (n<30)

• c2 Stat (n>5)

• c2 Stat (Cp)

• t Stat (n<10)

• t Stat (n<10)

HOW DO POPULATIONS INTERACT?

These interactions form a new

population which can now be

used to predict future

performance.

HOW DO POPULATIONS INTERACT?

ADDING TWO POPULATIONS

1

2

Population means interact in a simple intuitive manner.

s1

s2

Population dispersions interact in an additive manner

Means Add

1 + 2 = new

new

snew

Variations Add

s1

2 + s2

2 = snew

2

HOW DO POPULATIONS INTERACT?

SUBTRACTING TWO POPULATIONS

1

2

Population means interact in a simple intuitive manner.

s1

s2

Population dispersions interact in an additive manner

Means Subtract

1 - 2 = new

new

snew

Variations Add

s1

2 + s2

2 = snew

2

TRANSACTIONAL EXAMPLE

 Orders are coming in with the following

characteristics:

X = $53,000/week

s = $8,000

 Shipments are going out with the following

characteristics:

X = $60,000/week

s = $5,000

 Assuming nothing changes, what percent of the

time will shipments exceed orders?

TRANSACTIONAL EXAMPLE

Shipments

Orders

X = $53,000 in orders/week

s = $8,000

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:

X

X = $60,000 shipped/week

s = $5,000

shipments orders

$60,

$53,

000

000

000

shipments

$7,

orders

X

X

-

=

=

-

-

=

s

shipments orders

-

=

2

s

shipments

+

2

s

orders

=

(

5000

2

)

+

(

8000

2

)

=

$9434

$7000

$0

Shipments > orders

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

shipments orders

-

=

X

shipments

s

shipments orders

-

=

2

s

shipments

+

X

-

2

s

orders

orders

=

$60,

=

(

5000

000

2

)

+

$53,

-

(

8000

000

2

)

=

=

$7,

000

$9434

$7000

$0

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:

...