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**Unformatted text preview: **Variability Sample Range Variance InterQ Range Probability Model Set Algebra Specify prob Title Page JJ II J I 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 http://legacy.orie.cornell.edu/ ∼ sid sir1@cornell.edu August 30, 2007 Variability Sample Range Variance InterQ Range Probability Model Set Algebra Specify prob Title Page JJ II J I 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 JJ II J I 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. Variability Sample Range Variance InterQ Range Probability Model Set Algebra Specify prob Title Page JJ II J I 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 JJ II J I 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. • W n i =1 | x i- ¯ x | = max 1 ≤ i ≤ n | x i- ¯ x | , ( L ∞ norm) Not bad but less convenient than p = 2. Variability Sample Range Variance InterQ Range Probability Model Set Algebra Specify prob Title Page JJ II J I Page 6 of 24 Go Back Full Screen Close Quit Just right: p = 2....

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