ENGR 610
Applied Statistics
Fall 2007 - Week 2
Marshall University
CITE
Jack Smith
Overview for Today
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Homework problems 1.25, 2.54, 2.55
Review of Ch 3
Homework problems 3.27, 3.31
Probability and Discrete Probability
Distributions (Ch 4)
Homework assignment
Homework problems
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1.25
2.54
2.55
Chapter 3 Review
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Measures of…
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Central Tendency
Variation
Shape
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Skewness
Kurtosis
Box-and-Whisker Plots
Measures of
Central Tendency
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Mean (arithmetic)
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Median
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Most popular (peak) value(s) - can be multi-modal
Midrange
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Middle value - 50th percentile (2nd quartile)
Mode
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Average value:
1 N
Xi

N i
(Max+Min)/2
Midhinge
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(Q3+Q1)/2 - average of 1st and 3rd quartiles
Measures of Variation
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Range (max-min)
Inter-Quartile Range (Q3-Q1)
Variance
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Sum of squares (SS) of the deviation from mean divided by the
degrees of freedom (df) - see pp 113-5
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df = N, for the whole population
df = n-1, for a sample
2nd moment about the mean (dispersion)
(1st moment about the mean is zero!)
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Standard Deviation
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Square root of variance (same units as variable)
Sample (s2, s, n) vs Population (2, , N)
Quantiles
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Equipartitions of ranked array of observations
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Percentiles - 100
Deciles - 10
Quartiles - 4 (25%, 50%, 75%)
Median - 2
Pn = n(N+1)/100 -th ordered observation
Dn = n(N+1)/10
Qn = n(N+1)/4
Median = (N+1)/2 = Q2 = D5 = P50
Measures of Shape
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Symmetry
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Skewness - extended tail in one direction
3rd moment about the mean
Kurtosis
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Flatness, peakedness
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Leptokurtic - highly peaked, long tails
Mesokurtic - “normal”, triangular, short tails
Platykurtic - broad, even
4th moment about the mean
Box-and-Whisker Plots
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Graphical representation of five-number summary
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Min, Max (full range)
Q1, Q3 (middle 50%)
Median (50th %-ile)
Shows symmetry (skewness) of distribution
Other Resources
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SPSS Tutorial at Statistical Consulting
Services
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MathWorld
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http://www.stats-consult.com/tutorials.html
http://mathworld.wolfram.com
See Probability and Statistics
Wikipedia
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http://en.wikipedia.org/wiki/Category:Probability_a
nd_statistics
Homework Problems
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3.27
3.31
Chapter 4
Probability
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Introduction to Probability
Rules of Probability
Discrete Probability Distributions
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Probability Distributions
Binomial Distribution
Poisson Distribution
Hypergeometric, Negative Binomial, Geometric
Distributions
Introduction to Probability
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Probability - numeric value representing the chance,
likelihood, or possibility that an event will occur
 Classical, theoretical
 Empirical
 Subjective
Elementary event - a distinct individual outcome
Event - a set of elementary events
Joint event - defined by two or more characteristics
Rules of Probability
1.
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5.
A probability P(A) for event A is between 0 (null
event) and 1 (certain event)
The complement of P(A) is the probability that A will
not occur, and P(not-A) = 1- P(A)
Two events are mutually exclusive if
P(A and B) = 0
If two events are mutually exclusive, then
P(A or B) = P(A) + P(B)
If set of events are mutually exclusive and
collectively exhaustive, then
 P(A )  1
i
i
Rules of Probability
6.
7.
8.
9.
If two events are not mutually exclusive, then
P(A or B) = P(A) + P(B) - P(A and B), where
P(A and B) is the joint probability of A and B.
The conditional probability of B occurring, given
that A has occurred, is given by
P(B|A) = P(A and B)/P(A)
If two events are independent, then
P(A and B) = P(A) x P(B) and
P(A) = P(A|B) and P(B) = P(B|A)
If two events are not independent, then
P(A and B) = P(A) x P(B|A)
Probability Distributions
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A probability distribution for a discrete random
variable is complete set of all possible distinct
outcomes and their probabilities of occurring, whose
sum is 1.
The expected value of a discrete random variable is
its weighted average over all possible values where
the weights are given by the probability distribution.
E(X)   X i P(X i )
i
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Probability Distributions
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The variance of a discrete random variable is the
weighted average of the squared difference between
each possible outcome and the mean over all
possible values where the weights (frequencies) are
given by the probability distribution.
 X2  (X i  X ) 2 P(X i )
i
The standard deviation (X) is then the square root
of the variance.
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Binomial Distribution
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Each elementary event is a Bernoulli event, with one
of two mutually exclusive and collectively exhaustive
possible outcomes.
The probability of “success” (p) is constant from trial
to trial, and the probability of “failure” is 1-p.
The outcome for each trial is independent of any
other trial
The proportion of trials resulting in x successes, out
of n trials, with a constant probability of p, is given by:
n!
P(X  x | n, p) 
p x (1 p) nx
x!(n  x)!
Binomial Distribution, cont’d
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Binomial coefficients follow Pascal’s Triangle
1
11
121
1331
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Distribution nearly bell-shaped for large n and p=1/2.
Skewed right (positive) for p<1/2, and
left (negative) for p>1/2
Mean () = np
Variance (2) = np(1-p)
Poisson Distribution
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Probability for a particular number of discrete events
over a continuous interval (area of opportunity)
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Assumes a Poisson process (“isolable” event)
Limit case of Binomial distribution for large n
Based only on expectation value ()
e  x
P(X  x | ) 
x!
Poisson Distribution, cont’d
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Mean () = variance (2) = 
Right-skewed, but approaches symmetric bell-shape
as  gets large
Other Discrete Probability Distributions
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Hypergeometric (pp 159-160)
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Negative Binomial (pp 162-163)
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Bernoulli events, but selected from finite population
without replacement
p  A/N, where A number of successes in population N
Approaches binomial for n < 5% of N
Number of trials (n) until xth success
Binomial with last trial constrained to be a success
Geometric (pp 164-165)
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Special case of negative binomial for x = 1 (1st success)
Cumulative Probabilities
P(X<x) = P(X=1) + P(X=2) +…+ P(X=x-1)
P(X>x) = P(X=x+1) + P(X=x+2) +…+ P(X=n)
Homework
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Ch 4
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Appendix 4.1
Problems: 4.57,60,61,64
Read Ch 5
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Continuous Probability Distributions and
Sampling Distributions