Transcript Tutorial for Basics Statistics in Scilab FISL 13 2012 Professora Dra

Tutorial for Basics Statistics
in Scilab
FISL 13 2012
Professora Dra Ariane Ferreira
IPRJ/UERJ
LabMacambira.sf.net
Promovendo a Programação de Software Livre
Basics Concepts of Statistics
Normal Probability Distribution
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A normal distribution has a bell-shaped
density curve described by its mean  and
standard deviation  .
The density curve is symmetrical, centered
about its mean, with its spread determined by
its standard deviation.
Sample - Population
Population
- The population is the entire group of individuals
that we want information about.
Sample : Empirical data
- The sample is the data set that are showed from
a population of units of interest.
- A set of elements drawn from and analyzed to
estimate the characteristics of a population;
Random Sample
- A sample in which every element in the
population has an equal chance of being
selected;
Sample – Population Example
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Process : Oxidation furnace
–
–
–
Population : all wafers coming out of the furnace
after the operation
Sample : 5 measurements taken on 4 selected
wafers = 20 measurements
Individual : Every measurement that could be
taken on each wafer of the batch
Sampling x Empirical Data
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Sampling
–
–
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The act of choosing or selecting;
Random sampling : the selection of a random
sample; each element of the population has an
equal chance of been selected.
Empirical data
–
–
Sample data collected from process.
Empirical data normally follow a distribution of
probability with known density.
Sampling concepts
Random Sample :
- The random variables X1, X2, …, Xn are a
random sample of size n if:
1. The Xi’s are independent random variables;
2. Every Xi has the same probability distribution.
Statistic:
- A statistic is any function of the observations in
a random sample (sample mean, sample
variance).
Sampling Distribution:
- That are the probability distribution of a statistic.
What are probability distributions?
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Concept: a mathematical model that relates
the value of a random variable with its
probability of occurrence.
Two types:
–
Discrete Probability Distributions
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To describe random variables that can only take on
certain specific values: number of defects on a
semiconductor wafer, rotations to failure, number of
landings to failure, etc.
Continuous Probability Distributions
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To describe random variables that can have any value
on a continuous scale: linewidth in a sample population
of interconnect, calendar time.
Main Discrete Probability Distributions
Choice of important distributions, only 2 :
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Binomial Distribution
Poisson Distribution
Distributions in details... to see :
Freund, J.E. & Perles B. M. Modern Elementary Statistics, Pearson International
Edition, 12th edition , 2007.
Montgomery, D.C. & Runger, G.C. Applied Statistics and Probability for Engineers,
John Wiley & Sons, Inc., 4th edition, 2006.
Most common continuous distributions
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Exponential
Weibull
Normal Distributions in details... to follow
Gamma
Lognormal
FDistributions in details... to see :
Freund, J.E. & Perles B. M. Modern Elementary Statistics, Pearson International
Edition, 12th edition , 2007.
Montgomery, D.C. & Runger, G.C. Applied Statistics and Probability for Engineers,
John Wiley & Sons, Inc., 4th edition, 2006.
Normal Distribution
Probability density function a normally distributed variable x:
 1  x   2 
1
f X ( x) 
exp  
 
 2
 2    
Notation x ~N(,): x is normally distributed with mean 
and standard-deviation .
The normal distribution has a symmetric bell-shape.
●
The density function of normal distribution with the mean
value 0 with respect to different standard-deviations  .
Cumulative normal distribution
a
P ( x  a )  FX (a ) 
f
X
( x).dx
This integral cannot be
evaluated in closed form.

Standard normal distribution
a

a 
P( x  a)  P  z 

  
 

  
x
z

 = Standard distribution =
normal distribution with   = 0
and  = 1.
Data from any normal distribution may be
transformed into data following the standard normal
distribution by subtracting the mean  and dividing
by the standard deviation  .
Standard Normal curve
●
If a dataset follows a normal distribution, then
about:
–
68% of the observations will fall within  of the
mean , which in this case is with the interval (1,1).
–
95% of the observations will fall within 2 standard
deviations of the mean, which is the interval (2,2) for the standard normal, and
–
99.7% of the observations will fall within 3
standard deviations of the mean, which
corresponds to the interval (-3,3) in this case.
Standard Normal curve
Empirical Data
Table of data : m rows and n columns.
X j ,k
is the measurement
corresponding to the kth
items of the jth sample.
Normality of data must be checked : Anderson-Darling test, Shapiro-Wilk test
or Q-Plot test.
Statistics from empirical data
From the probability distribution of the empirical data,
the information can be derived :
Mean : X
–
Position Characteristics :
–
Dispersion Characteristics : Standard  deviation : s
~
Median : X
Range : R
Position Characteristics
•Mean :
•Median :
1 n
X1  X 2    X n
X   Xk 
n k 1
n
 X  ( n 1) 
  2 
~ 
X  X n X n
 




1



  2   2 

2
If n is odd number
If n is even number
Mean x Median
The median is more robust than the mean.
If X1, X2, …, Xn
–
–
–
are n independents normal (,) random variables,
then
Mean : X
~
Median : X
are two unbiased estimators of .
Dispersion Characteristics
•Standard-deviation s>0
n

1 n
1
2
2
2
  X k  nX 
s
(Xk  X ) 

n  1 k 1
n  1  k 1

•Range R>0
R  X  n   X  1
•Variance s2
Remark: s and R are not the
same scale
Unbiased estimators of 
For samples with size n<30
If X1, X2, …, Xn
–
–
are n independents normal (,) random
variables,
then
s
R
Correction terms for
dispersion
characteristics
–
K s  n
are two unbiased estimators of .
K R  n
Correction terms for dispersion
●
Where:
Don't worry!!!
Be
Ha
pp
y
Table for Ks(n) and KR(n)
Machine (M) / Process (P)
Variability
Machine variability “1M”
Process variability “5M”
How to Compute position/ Dispersion
Characteristics ?
For the jth sample, one can compute :
The mean: X j
~
The median: X j
The standard-deviation: s j
The range:
Rj
How to Compute position/ Dispersion
Characteristics ?
To compute the process characteristics:
The process mean:
1 m
X  Xj
m j 1
The process median:
~ 1 m ~
X  Xj
m j 1
The process standard-deviation:
The process range:
m n
1
2
 X j ,k  X 
sp 

mn  1 j 1 k 1
R p  max( X j ,k )  min( X j ,k )
How to Compute position/ Dispersion
Characteristics ?
To compute the machine characteristics:
The machine standard-deviation:
The machine range:
1 m
sM   s j
m j 1
1 m
RM   R j
m j 1
Example: Table of data
Normality test for Data
In order to test the normality of the process,
then consider the m samples of size n as one
sample of size mn .
In order to test the normality of the machine, we
have to remove the machine effect by making the
following transformation:
X j ,k  X j ,k  X j
Normality tests
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Anderson-Darling test;
Shapiro-Wilk test;
Skewness and Kurtosis tests;
Normal Q-plot test;
in details... to see :
Freund, J.E. & Perles B. M. Modern Elementary Statistics, Pearson
International Edition, 12th edition , 2007.
Montgomery, D.C. & Runger, G.C. Applied Statistics and Probability for
Engineers, John Wiley & Sons, Inc., 4th edition, 2006.
Skewness and Kurtosis tests
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Skewness
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A term for asymmetry usually employed with
respect to a histogram of data or a probability
distribution.
Kurtosis
–
A measure of the degree to with a unimodal
distribution is peaked.
Example Normal Q-Plot Test
Estimation of the Population
Parameters
We assume:
X j ,k ~ N   ,  
Estimation of :
~
ˆ  X
Estimation of M:
sM
RM

or ˆ M 
K s ( n)
K R ( n)
Estimation of p:
ˆ M
ˆ p  s p