Lec3VariabilityProbModels

# Lec3VariabilityProbModels - ORIE 270 Engineering...

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Variability Sample Range Variance InterQ Range Probability Model Set Algebra Specify prob Title Page Page 1 of 24 Go Back Full Screen Close Quit ORIE 270 Engineering Probability and Statistics Lecture 3: Variability, Probability Models Sidney Resnick School of Operations Research and Information Engineering Rhodes Hall, Cornell University Ithaca NY 14853 USA sid [email protected] August 30, 2007

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Variability Sample Range Variance InterQ Range Probability Model Set Algebra Specify prob Title Page Page 2 of 24 Go Back Full Screen Close Quit 1. Measures of Variability: Overview How tightly packed is the data about a central value? Measures of variability: sample range, sample variance, sample standard deviation, interquartile range.
Variability Sample Range Variance InterQ Range Probability Model Set Algebra Specify prob Title Page Page 3 of 24 Go Back Full Screen Close Quit 2. Sample Range Features: Crude measure of spread; only gives extremes. Simple to compute: range=biggest observation –smallest observa- tion. Note R lists the smallest and biggest observation by typing range(dataset). This information also available by typing sum- mary(dataset). Examples: data = {- 1 , 1 } ; range=2. data = {- 2 , 2 } ; range=4.

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Variability Sample Range Variance InterQ Range Probability Model Set Algebra Specify prob Title Page Page 4 of 24 Go Back Full Screen Close Quit Danish fire insurance data: > range(danishall) [1] 0.3134041 263.2503660 Miles per gallon of 19 cars: > mpg [1] 30.0 32.9 33.2 33.6 36.3 36.5 36.6 36.7 36.8 36.9 37.1 37.2 37.3 37.4 37.5 [16] 41.0 41.2 42.1 44.9 > range(mpg) [1] 30.0 44.9 Mile run data (ending in the 1980’s): > range(mile) [1] 47.33 106.00
Variability Sample Range Variance InterQ Range Probability Model Set Algebra Specify prob Title Page Page 5 of 24 Go Back Full Screen Close Quit 3. Variance and Standard Deviation. Measure variability by quantifying spread about central point. If data is { x 1 , . . . , x n } deviation from the mean of x i is x i - ¯ x. How to quantify for whole data set? Possibilities: n i =1 ( x i - ¯ x ) . BUT this sum is 0. (oops) n i =1 | x i - ¯ x | . ( L 1 norm) BUT not that easy to deal with. n i =1 | x i - ¯ x | p , for some p > 1 ( L p norm) BUT not that easy to deal with unless p = 2. n i =1 | x i - ¯ x | = max 1 i n | x i - ¯ x | , ( L norm) Not bad but less convenient than p = 2.

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Variability Sample Range Variance InterQ Range Probability Model Set Algebra Specify prob Title Page Page 6 of 24 Go Back Full Screen Close Quit Just right: p = 2. Definition: Sample variance : s 2 := 1 n - 1 n i =1 ( x i - ¯ x ) 2 . Sample standard deviation : s = s 2 . Rules for calculating; important in the old days when calculations were done by hand:: 1. Enough to compute ¯ x and sums of squares: s 2 = 1 n - 1 n i =1 x 2 i - n ¯ x 2 . .
Variability Sample Range Variance InterQ Range Probability Model Set Algebra Specify prob Title Page Page 7 of 24 Go Back Full Screen Close Quit 2. If y i = c + x i then ¯ y = c + ¯ x, s 2 y = s 2 x .

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• Standard Deviation, Probability theory, Interquartile range, Algebraic structure, Variability Sample Range

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