Probability and Statistics for Computer Science The statement that - PowerPoint PPT Presentation

Probability and Statistics ì for Computer Science “The statement that “The average US family has 2.6 children” invites mockery” – Prof. Forsyth reminds us about criAcal thinking Credit: wikipedia Hongye Liu, Teaching Assistant Prof, CS361, UIUC, 8.27.2020

Last lecture � Welcome/OrientaAon � Big picture of the contents � Lecture 1 - Data VisualizaAon & Summary (I) is � Some feedbacks

Warm up question: � What kind of data is a le[er grade? � What do you ask for usually about the stats of an exam with numerical scores?

Objectives � Grasp Summary StaAsAcs � Learn more Data VisualizaAon for Rela2onships

Summarizing 1D continuous data For a data set {x} or annotated as {x i }, we summarize with: N items � LocaAon Parameters Mode Mean tu ) Median , , � Scale parameters Inter quartile standard egg ' range ciqr ) deviation ( 62 ) variance

Summarizing 1D continuous data � Mean N mean ( x i ) = 1 � x i N i =1 It’s the centroid of the data geometrically, by idenAfying the data set at that point, you find the center of balance.

it [ 1,87 { Ki ) 7 , { Ki }=1 , 6 , 3 , 4 , 12 5 , 2 , E' ga - meancfxi } ,=o TIKI - -

Properties of the mean � Scaling data scales the mean meant a { Ki } t e ) = a means { xi } ) t C mean ( { k · x i } ) = k · mean ( { x i } ) � TranslaAng the data translates the mean mean ( { x i + c } ) = mean ( { x i } ) + c

Less obvious properties of the mean � The signed distances from the mean I sum to 0 N � ( x i − mean ( { x i } )) = 0 i =1 � The mean minimizes the sum of the squared distance from any real value WE N ( x i − µ ) 2 = mean ( { x i } ) � argmin µ i =1

N - meant { ki ) , )=o prove I l ki Ei ,cxi ) - II. E- I mean ' " :3 ) LHS : numen N E Xi meemflx :D = N ÷÷xi , - IN LHS : =o . -

argmuin Eic * u ) - = mean 4%3 , Prove 8=-24 - ie N df d ⇐ f ) - cxi.ee ) ' = ? - f - - u 't dm = ga die - E. et : - da 's :* ÷ - . = Erg -28=0 - e - t ) ddtg -_ 2g \ it = - EE ,ag=o dashed "I = - I

N Z g g = Ki - ee - - o ' . ( Xi - M ) =o - 2 E- I N N - I µ = o ( ki ) -2 F- I F- I N N . µ = 0 -2 Xi - N - I c- ' I I ^ = mean µ = N = mean Arginine . , . .

Q1: � What is the answer for mean ( mean ({x i })) ? A. mean ({x i }) B. unsure C. 0

Standard Deviation (σ) 1- = ex - at � The standard devia-on Arginine -2 f = mean u � N � � 1 � � std ( { x i } ) = ( x i − mean ( { x i } )) 2 N i =1 � = mean ( { ( x i − mean ( { x i } )) 2 } )

Q2. Can a standard deviation of a dataset be -1? A. YES B. NO

Properties of the standard deviation � Scaling data scales the standard deviaAon std ( { k · x i } ) = | k | · std ( { x i } ) � TranslaAng the data does NOT change the standard deviaAon std ( { x i + c } ) = std ( { x i } )

.¥l A mum y 3 2 I

Standard deviation: Chebyshev’s inequality (1 st look) A N � At most items are k standard k 2 devia-ons ( σ ) away from the mean � Rough jus-fica-on: Assume mean =0 # N − N k 2 0 . 5 N 0 . 5 N 0 k 2 k 2 k σ − k σ � 1 N [( N − N k )0 2 + N std = k 2 ( k σ ) 2 ] = σ ( o - O ) - O

Variance (σ 2 ) � Variance = (standard deviaAon) 2 } N var ( { x i } ) = 1 � ( x i − mean ( { x i } )) 2 N i =1 � Scaling and translaAng similar to standard I deviaAon var ( { k · x i } ) = k 2 · var ( { x i } ) var ( { x i + c } ) = var ( { x i } )

Q3: Standard deviation � What is the value of std ( mean ({x i }) ? D A. 0 B. 1 C. unsure

Standard Coordinates/normalized data � The mean tells where the data set is and the standard devia,on tells how spread out it is. If we are interested only in comparing the IT shape, we could → define: x i = x i − mean ( { x i } ) 0 � std ( { x i } ) for every i C � We say is in standard coordinates { � x i }

Q4: Mean of standard coordinates � μ of is: I { � x i } A. 1 B. 0 C. unsure i. mean x i = x i − mean ( { x i } ) � std ( { x i } )

Q5: Standard deviation (σ) of standard coordinates � σ of is: O { � x i } l A. 1 B. 0 C. unsure Std x i = x i − mean ( { x i } ) � std ( { x i } )

Q6: Variance of standard coordinates � Variance of is: { � x i } A. 1 B. 0 C. unsure x i = x i − mean ( { x i } ) � std ( { x i } )

Q7: Estimate the range of data in standard coordinates � EsEmate as close as possible, 90% data is within: A. [-10, 10] B. [-100, 100] C. [-1, 1] x i = x i − mean ( { x i } ) � std ( { x i } ) D. [-4, 4] I E. others

¥÷÷÷ " :* . = k t - ko . r .

Summary stats of standard Coordinates/normalized data

Standard Coordinates/normalized data to μ=0, σ=1, σ 2 =1 � Data in standard coordinates always has mean = 0; standard deviaAon =1; - - variance = 1. - � Such data is unit-less, plots based on this someAmes are more comparable � We see such normalizaAon very oren in staAsAcs

Additional References � Charles M. Grinstead and J. Laurie Snell "IntroducAon to Probability” � Morris H. Degroot and Mark J. Schervish "Probability and StaAsAcs”

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