Lect3_2700_s09 - ENGRD 2700 Basic Engineering Probability...

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Measures of Variability Probability Model Title Page JJ II J I Page 1 of 21 Go Back Full Screen Close Quit ENGRD 2700 Basic Engineering Probability and Statistics Lecture 3: Data Summarization; Probability Models David S. Matteson School of Operations Research and Information Engineering Rhodes Hall, Cornell University Ithaca NY 14853 USA [email protected] January 26, 2009
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Measures of Variability Probability Model Title Page JJ II J I Page 2 of 21 Go Back Full Screen Close Quit 1. Measures of Variability How tightly packed is the data about a central value? Measures of variability: sample range sample variance, sample standard deviation interquartile range
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Measures of Variability Probability Model Title Page JJ II J I Page 3 of 21 Go Back Full Screen Close Quit Sample Range Features: Crude measure of spread; only gives extremes. Simple to compute: range = largest observation – smallest observation. Examples: data = {- 1 , 1 } ; range=2. data = {- 2 , 2 } ; range=4. Danish fire insurance data: range(danishall) = [0 . 3134041 , 263 . 2503660] Miles per gallon of 19 cars: range(mpg) = [30 . 0 , 44 . 9] . Mile run data (ending in the 1980’s): range(mile) = [47 . 33 , 106 . 00] .
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Measures of Variability Probability Model Title Page JJ II J I Page 4 of 21 Go Back Full Screen Close Quit Variance and Standard Deviation Measure variability by quantifying spread about central point. For observations { x 1 , . . . , x n } define deviations from the mean of x i as 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.
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Measures of Variability Probability Model Title Page JJ II J I Page 5 of 21 Go Back Full Screen Close Quit Just right: p = 2. Definition: Sample variance : s 2 := 1 n - 1 n X i =1 ( x i - ¯ x ) 2 . Sample standard deviation : s = s 2 . Rules for calculating; critical 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 X i =1 x 2 i - n ¯ x 2 . .
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Measures of Variability Probability Model Title Page JJ II J I Page 6 of 21 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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