INTRODUCCION ESTADISTICO

7/30/2019 INTRODUCCION ESTADISTICO

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Introductory Statistics

By Peter Woolf ([email protected])

University of Michigan

Michigan Chemical Process

Dynamics and Controls

Open Textbook

version 1.0

Creative commons


2/37

A foolish consistency is the hobgoblin of little

minds R. W. Emerson

But is this always true??


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Consistency

Why might we want consistency?

Integration of products within a larger

system

Examples: want parts to fit together, want

consistent chemical feeds, want consistent

material properties, want consistent energy

content, want consistent flavor


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Consistency What can be the downsides of

consistency?

Make something consistently bad, but

consistent.

Sometimes people trade consistency forquality--this is not the goal.

Examples: Fast food vs home made food

(depends the cook)


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Measures of Quality (or lack there of):

Six Sigma: Number of defects per million

opportunities

Genichi Taguchi: Uniformity around a target valueor The loss a product imposes on society after it

is shipped

Process control is a central tool for reducing

variability by adjusting and correcting for

variations.

Key Questions: How can we know if our

control system is working well enough?

How can we measure variability?


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Process Specific Questions

1) Do recent data indicate that the

process is broken or changed?

2) Is the process out of control?

3) What are the odds that two samples

come from the same distribution?

4) What factors influence this outcome?


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Detecting if a process has

changed

Scenario: You are a small Acai juice

vendor trying to expand to a world

market with a consistent product.


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Acai juice production

Acai berries in the market

Berry

crusher

juice


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Acai juice productionjuice

A key selling point of your acai

juice is that it contains a large

concentration of antioxidants.

With your berry crusher you get agood quality product most of the time,

but not always. You dont want to

waste berries if your crusher is

hurting your product, but how can

you know if it is not working right?

How can you test this?


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Acai juice productionjuiceHow can you test this?

1) Gather many samples from your

current process and measure the

antioxidant concentration

N sample values: 40.1, 41.3, 44.3, 39.3,

38.6,..

How do we summarize this?


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juice

N sample values: 40.1, 41.3, 44.3,

39.3, 38.6,..How do we summarize this?

=1

Nx

i

i=1

N

"Average:

Deviation from

the average:

(std deviation)"=

1

N(x

i#)2

i=1

N

$


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Deviation from

the average:

(std deviation)

"=

1

N

(xi#)2

i=1

N

$

Interpretation: The average distance from the mean

OR the width of the dispersion around the mean

Problem: What if I have only one sample (e.g. N=1)?

=0!!

Does this mean that the underlying process has no

variation or that I have not sampled it sufficiently?

Result:When N is small, the standard deviation willunderestimate the true variation

Solution: sample standard deviations =

1

(N"1)(x

i")2

i=1

N

#


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Population standard

deviation

(Real deviation)

"=

1

N(x

i#)2

i=1

N

$

s =1

(N"1)(x

i")2

i=1

N

#

Sample standard

deviation

(Observed deviation)

With a measure of the mean and standard deviation, you

have enough information to define a Gaussian distribution

Bell curve shape

based on a model of alarge number of random,

uncorrelated changes


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Gaussian or Normal Distribution:

From previous lecture on Noise:

Approximate Gaussian distribution in Excel by:=RAND()+RAND()+RAND()-RAND()-RAND()-RAND()

The approximation is better and better for larger numbers

of pairs of add and subtract

Gaussian distribution is the basis of much of statisticalquality control, six sigma, and quality engineering in general.

2

3

-2

-3

6 How do we mathematically

define a normal distribution?


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mean and standard deviation aresufficient statistics, meaning that they are sufficient to

describe a normal distribution

Mathematically, we can describe a normal distribution by the followingprobability

distribution functi on :

PDF(x |,") =1

" 2#

exp $1

2

x $

"

%

&

'(

)

*

2+

,-

.

/0


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If we want to find the density up to some point, say z or less we can just integrate:

PDF(x |,")dx#$

z

% =1

21+ erf

z #

" 2

&

'(

)

*+

,

-.

/

01

(Note: this just makes one hard problem into another, in that now we have to calculate theerror function)

The error function is defined as:

erf(x) 22

3

exp(#t2)dt0

x

%

How can we calculate this?

Excel:

Error function is Erf(), thus the solution above could be

expressed as

=1/2*(1+erf((z-m)/(s*sqrt(2))))

Mathematica:

Nintegrate[ f(x), {x,start, end}]

Or

N[1/2*(1+Erf[(z-m)/(s*Sqrt[2])])]

General numerical integration

Using analytical solution

with error function


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juice

Acai juice problem revisited

From 100 samples of the

current process we calculate

the following:

Mean=40 units

Standard deviation= 2 units

From these data, what are the

odds that the next batch will

have an antioxidant value of

37.5 or less?

1

" 2#exp $

1

2

x $

"

%&'

()*2+

,-./0$1

37.5

2 dx

=

1

21+ erf

37.5"

# 2

$

%&

'

()

*

+,

-

./


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Mean=40 units

Standard deviation= 2 units

From these data, what are the

odds that the next batch willhave an antioxidant value of

37.5 or less?

1

" 2#exp $

1

2

x $

"

%&'

()*2+

,-./0$1

37.5

2 dx

=

1

21+ erf

37.5"

# 2

$

%&

'

()

*

+,

-

./

In Mathematica:

short hand notation

Answer: ~10% of the time we expect this situation


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Example 1:Say that we have a reactor with a temperature mean of 100 and standard deviation of 5

degree. Calculate the probability of measuring a temperature of 92 or less.

PDF(x |100,5)dx"#

92

$ =1

21+ erf

92 "100

5 2

%

&'

(

)*

+

,-

.

/0=

1

21+ erf "1.13( )[ ] = 0.054

What about 100 or less? -> 0.5

Example 2:

Given this same system, what is the probability that the reactor is within 4 sigma of themean? (e.g. +/- 10 degrees)

PDF(x |100,5)dx"#

110

$ " PDF(x |100,5)dx"#

90

$ =

1

21+ erf

110 "100

5 2

%

&'

(

)*

+

,-

.

/0"

1

21+ erf

90"100

5 2

%

&'

(

)*

+

,-

.

/0= 0.9545


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Example 1:Say that we have a reactor with a temperature mean of 100 and standard deviation of 5

degree. Calculate the probability of measuring a temperature of 92 or less.

PDF(x |100,5)dx"#

92

$ =1

21+ erf

92 "100

5 2

%

&'

(

)*

+

,-

.

/0=

1

21+ erf "1.13( )[ ] = 0.054

What about 100 or less? -> 0.5

Example 2:

Given this same system, what is the probability that the reactor is within 4 sigma of themean? (e.g. +/- 10 degrees)

PDF(x |100,5)dx"#

110

$ " PDF(x |100,5)dx"#

90

$ =

1

21+ erf

110 "100

5 2

%

&'

(

)*

+

,-

.

/0"

1

21+ erf

90"100

5 2

%

&'

(

)*

+

,-

.

/0= 0.9545


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Acai juice production as a function of time

time

Antioxidant

value

Is this process out of control?

Yes: It is unusual to see so many

batches with such a high value--

this is strange and suggestssomething has changed.

No: This is just normal variation--

nothing is fundamentally different.

Key question:

How do we define

unusual


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One definition: Variation

outside of the six sigma

window is unusual

ean and standard deviation aresufficient statistics, meaning that they are sufficient to

describe a normal distribution

athematically, we can describe a normal distribution by the following probability

istribution function :

PDF(x |,") =1

" 2#exp $

1

2

x $

"

%

&'

(

)*2+

,-

.

/0

2 3-2-3

6

What are the odds of finding

something that falls out of this

bound by chance?

Find by integration!

For both tails the probability is ~0.0027

or 1 in 370

Common confusion:

The Six Sigma process

defines unusualas 3.4 defects

out of 1 million, not within 6

standard deviations (more like10.2 deviations)


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Acai juice production as a function of time

time

Antioxidant

value

Is this process out of control?

Translation: if we assume outside of 6 sigma variation is

unusual: Is this pattern expected to happen less than 1 in

370 of our samples?Solution: Control charts!


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Image from wikipedia western_electric_rules

Control charts determine if a process is behaving in an unusual

way.


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Image from wikipedia western_electric_rules


way.

What are the odds?

If each dot is a single measurement, and UCL is +3 sigma then

UCL=Uppercontrol limit

X-bar=

average

LCL=Lowercontrol limit

For both tails the probability is ~0.0027

or 1 in 370

Rule 1:


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way.

What are the odds?

UCL=Uppercontrol limit

X-bar=

average

LCL=Lowercontrol limit

Rule 2: Can do using probability theory.

Assuming each sample is independent, then can find the

total probability of:

2*[P1(out+)P2(out+)P3(out+)+P1(out+)P2(out+)P3(in)+P1(out+)P2(in)P3(out+)+P1(in)P2(out+)P3(out+)]

=P(out+) P(in)=1-P(out+)

=0.00305

or 1 in 326

1 in 370 1 in 326


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What are the odds?Alternative solution by sampling

1 in 370 1 in 326

Approach: Generate

thousands of samples

and test to see how

many satisfy the rule

~ similar to 1 in 370


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What are the odds?

1 in 326

Alternative solution by sampling

Rule 2:

~ similar to 1 in 326

Message: Many complex decision

processes can be evaluated

numerically with good accuracy

(see mathematica code on

website under Lecture 21.nb)


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Odds 1 in 370 Odds 1 in 326

Odds 1 in 256Odds 1 in 180

In all cases these

represent somewhat

rare cases in a

statistical sense, butthey are not all

equally rare.

These are not only

constrained on

statistics though..

e.g. What are the odds

of finding 15 consecutive

samples in zone c?

=Odds 1 in 306

Thus is this system out of control?

Yes, but in a good way.


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Acai juice problem revisited

What if you know that each batch of

berries has some variation, but you

are unsure if the machine is

behaving strangely? Can you still

use your control charts?

Solution: Take samples from each

batch, average them and plotthese average values and

statistics on a control chart.

Problem: The process of

averaging out different samples

will change your odds--averaging

reduces out variation.

Day 1: 40.36, 39.36, 38.43, 39.67

Day 2: 39.96, 40.32, 39.88, 39.75


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Acai process

control using

X-bar charts

Raw Data:

Plotting the raw data, it

is hard to say if

anything is going on..


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To get something

like this need UCL

and LCL

Acai process

control using

X-bar chartsRaw Data:

Data in excel example online

Lecture.21.xls


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To get something

like this need UCL

and LCL

UCL= grand avg+

A3*(avg stdev)

= 39.86+ 1.628*0.55

=40.76


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To get something

like this need UCL

and LCL

UCL= grand avg+

A3*(avg stdev)

= 39.86+ 1.628*0.55

=40.76

Note: If you use A2, you

use the average R. The

result is 40.77--nearly the

same.LCL=grand avg-A3*(avg stdev)

=38.96

UCL represents 3 standard

deviations away from the mean, sothe line between zones A/B is 2

standard deviations away:

A/B line=grand avg+

A3*(avg stdev)*(2/3)= 40.46


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X-bar chart

Is it in control?

Rule 2: fail: points 9 and 10 are in zone A

Rule 1: okay, no points outside of zone A

Rules 3 and 4: okay

Conclusion:

Not in statisticalcontrol.


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Take Home Messages Statistical process control is a method

for systematically identifyinginconsistencies.

Probabilities are often based on aGaussian process

Control charts provide a systematic

method for evaluating if a process isunder control.


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A foolish consistency is the hobgoblin of

little minds

--R. W. Emerson

An intelligent consistency is a virtue in

an integrated global economy

--Anonymous

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