Unformatted text preview: 4/11/11 Announcements Midterm 1 study guide available WebCT Exam Materials Study Guides Last Week Visual Displays of Data Measures of Central Tendency Measures of Variability 1 4/11/11 FR O FO BE M RE Frequency DistribuKon Graph 4 3 FR O FO BE M RE Grouped Frequency DistribuKon Graph 45 40 35 30 25 20 15 10 5 0 0  3 40 99 0 80 799 0 12 119 00 9  16 159 00 9  20 199 00 9  24 239 00 9  28 279 00 9  32 319 00 9  36 359 00 9  40 399 00 9 4 39 9 f 2 1 0 0 1 2 3 4 f Cups of Coffee PopulaKon of City (thousands) FR FO BE OM RE FR O FO BE M RE PopulaKon Sample PopulaKon Sample "mew" M
"M" X
"X bar" !Xi = N !Xi X= n 2 4/11/11 FR BE OM FO RE FR O BE M FO RE CalculaKng the Standard DeviaKon SS = !(Xi " )2 PopulaKon Sample Popula/on !
"sigma" s
"s" != SS N Sample SS = !(Xi " X)2 s= SS n !1 FR FO BE OM RE Parameters and StaKsKcs FR O FO BE M RE ProperKes of EsKmators PopulaKon Sample ? ? A ESTIM TION X s STATISTICS (ESTIMATES) PARAMETERS Consistency: "Does it get be`er with more data?" (RelaKve) Efficiency: "Does it err less than other esKmators?" Sufficiency: "Does it use all the data?" Bias: "Does it over or underesKmate the true value on average?" 3 4/11/11 E POPULATION OR N = 2BEF 0,000 OM FR = 100 = 15 FR O BE M FO RE CalculaKng the Standard DeviaKon SS = !(Xi " )2 Popula/on Standard deviaKons for 10 samples calculated using PopulaKon Formula (formula for ) Sample 01: = 13.43 Sample 02: = 11.86 Sample 03: = 11.87 Sample 04: = 13.12 Sample 05: = 13.96 Sample 06: = 11.97 Sample 07: = 13.05 Sample 08: = 9.65 Sample 09: = 15.25 Sample 10: = 12.34 Standard deviaKons for 10 samples calculated using corrected Formula (formula for s) Sample 01: s = 16.12 Sample 02: s = 14.23 Sample 03: s = 14.24 Sample 04: s = 15.75 Average s Sample 05: s = 16.75 = 151.83 / 10 Sample 06: s = 14.39 Sample 07: s = 15.66 = 15.18 Sample 08: s = 11.58 Sample 09: s = 18.30 Sample 10: s = 14.81 != SS N Sample SS = !(Xi " X)2 Average = 126.5 / 10 = 12.65 s= SS n !1 = 0 = 1.0 = 0 = 0.5 Describing the LocaKon of Scores This Week Probability The Normal DistribuKon 4 4/11/11 Describing the LocaKon of Scores Student Tom Quiz 1 12 Quiz 2 12 Quiz 1 TOM Quiz 1 TOM = 10.5 Quiz 2 TOM Quiz 2 TOM = 13.5 5 4/11/11 Quiz 1 FR O BE M FO RE DeviaKon +1.5 TOM = 10.5 deviationi = X i ! 1.5 Quiz 2 TOM = 13.5 Quiz 1 +1.5 Quiz 1 +1.5 TOM TOM = 10.5 = 10.5 Quiz 2 1.5 TOM Quiz 2 1.5 TOM = 13.5 = 13.5 6 4/11/11 Quiz 1 +1.5 Quiz 1 = 3 +1.5 TOM TOM = 10.5 = 10.5 Quiz 2 1.5 TOM Quiz 2 = 1.5 1.5 TOM = 13.5 = 13.5 Raw Score Student Tom Quiz 1 12 = 10.5 = 3 Raw DeviaKon Score Quiz 2 12 = 13.5 = 1.5 Student Tom Quiz 1 +1.5 = 10.5 = 3 Quiz 2 1.5 = 13.5 = 1.5 7 4/11/11 Raw DeviaKon Score Student Tom Quiz 1 +1.5/3 = 10.5 = 3 Number of Standard DeviaKons Student Tom Quiz 1 +.5 = 10.5 = 3 Quiz 2 1.5/1.5 = 13.5 = 1.5 Quiz 2 1 = 13.5 = 1.5 Quiz 1 = 3 +1.5 Number of Standard DeviaKons Student Tom Quiz 1 +.5 = 10.5 = 3 TOM = 10.5 Quiz 2 1 = 13.5 = 1.5 Quiz 2 = 1.5 1.5 TOM = 13.5 8 4/11/11 ZScore The number of standard deviaKons a parKcular score deviates from its corresponding mean ZScore zi = deviationi ! ZScore zi = Xi ! " Sign: whether score is above or below mean of the populaKon Value: number of standard deviaKons the score is from the mean 9 4/11/11 Raw Score Student Mike Quiz 1 10 = 10.5 = 3 Check Your Understanding Find Mike's zscore for quiz 1 Quiz 2 14 = 13.5 = 1.5 Student Mike Quiz 1 10 = 10.5 = 3 Quiz 2 14 = 13.5 = 1.5 A B C D E z = (0.5 3) = 0.166 z = (0.5 3) = 0.166 z = (10 3) = 3.333 z = (10 3) = 3.333 z = (10.5 3) = 3.5 Check Your Understanding Find Mike's zscore for quiz 1 Student Mike Quiz 1 10 = 10.5 = 3 Quiz 2 14 = 13.5 = 1.5 = 10.5 = 3 X = 10 CalculaKon for Quiz 1 zi = Xi ! " A B C D E z = (0.5 3) = 0.166 z = (0.5 3) = 0.166 z = (10 3) = 3.333 z = (10 3) = 3.333 z = (10.5 3) = 3.5 10 4/11/11 CalculaKon for Quiz 1 = 10.5 = 3 X = 10 CalculaKon for Quiz 1 = 10.5 = 3 X = 10 zi =
zi = Xi ! " zi =
zi = Xi ! " 10 !10.5 3 10 !10.5 3
!.5 3 zi = CalculaKon for Quiz 1 = 10.5 = 3 X = 10 Student Quiz 1 Mike z =  0.166 Quiz 2 14 = 13.5 = 1.5 zi = ! 1 6 = 10.5 = 3 11 4/11/11 CalculaKon for Quiz 2 = 13.5 = 1.5 X = 14 CalculaKon for Quiz 2 = 13.5 = 1.5 X = 14 zi = Xi ! " zi =
zi = Xi ! " 14 !13.5 1.5 CalculaKon for Quiz 2 = 13.5 = 1.5 X = 14 CalculaKon for Quiz 2 = 13.5 = 1.5 X = 14 zi =
zi = Xi ! " 14 !13.5 1.5
0.5 1.5 zi = 1 3 zi = 12 4/11/11 Z Scores Student Quiz 1 Quiz 2 Mike z =  0.166 z = 0.333 = 10.5 = 3 = 13.5 = 1.5 Raw Scores Student Tom Mike Quiz 1 12 10 Quiz 2 12 14 Mean 12 12 Quiz 1 Raw Scores Student Tom Mike Quiz 1 12 10 Quiz 2 12 14 Mean 12 12 = 3 .5 +1.5 MIKE TOM = 10.5 Z Scores Student Tom Mike Quiz 1 +.5 0.166 Quiz 2 1 +0.333 Mean Z 0.25 +.083 Quiz 2 = 1.5 +.5 1.5 TOM MIKE = 13.5 13 4/11/11 Raw Scores Student Tom Mike Quiz 1 12 10 Quiz 2 12 14 Mean 12 12 ConverKng between Raw and Zscores Raw score to ZScore zi =
ZScore to Raw Score Z Scores Student Tom Mike Quiz 1 +.5 0.166 Quiz 2 1 +0.333 Mean Z 0.25 +.083 Xi ! " X i = + zi! Check Your Understanding A populaKon of scores has = 50. In this populaKon, a score of X = 40 corresponds to z = 1.00. What is the populaKon standard deviaKon? Check Your Understanding A populaKon of scores has = 50. In this populaKon, a score of X = 40 corresponds to z = 1.00. What is the populaKon standard deviaKon? A B C D E 10 5 20 I don't know how to approach this I ran out of Kme A B C D E 10 5 20 I don't know how to approach this I ran out of Kme 14 4/11/11 Xi ! " 40 ! 50 !1 = ! !10 !1 = ! ! = 10 zi = Check Your Understanding For a distribuKon of scores, which of the following zscore values represents the most extreme value (furthest from the mean)? A B C D E z = 2.0 z = 1.5 z = 3.0 z = 2.5 I don't know how to approach this Check Your Understanding For a distribuKon of scores, which of the following zscore values represents the most extreme value (furthest from the mean)? The use of Zscores A B C D E z = 2.0 z = 1.5 z = 3.0 z = 2.5 I don't know how to approach this 15 4/11/11 The use of Zscores Describing scores in distribuKons... with a single number The use of Zscores Describing scores in distribuKons... with a single number Making scores from nonequivalent distribuKons comparable. 16 4/11/11 Which is Larger? Which is More ExcepKonal? 8oz 820lbs 8oz 820lbs Which is More ExcepKonal? Which is More ExcepKonal? 8oz = 5oz 820lbs = 600lbs 8oz = 5oz = 1oz 820lbs = 600lbs = 100lbs 17 4/11/11 Which is More ExcepKonal? Which is More ExcepKonal? 8oz = 5oz = 1oz z = (85) 1 820lbs = 600lbs = 100lbs z = (820600) 100 8oz = 5oz = 1oz z = +3 820lbs = 600lbs = 100lbs z = +2.20 The use of Zscores Describing scores in distribuKons... with a single number Making scores from nonequivalent distribuKons comparable. EquaKng and rescaling enKre distribuKons Standardized (Zscored) DistribuKon A distribuKon in which all scores have been zscored 18 4/11/11 Quiz 1 MIKE Quiz 1 = 10.5 = 3 = 10.5 = 3 TOM MIKE TOM 2.5 2.5 1.5 1 .5 0 .5 1 1.5 Quiz 2 TOM MIKE Quiz 2 = 13.5 = 1.5 = 13.5 = 1.5 TOM MIKE 5 4 3 2 1 0 1 WebCT Excel Materials Data Files 19 4/11/11 Cell LocaKons Cell LocaKons Column D Cell LocaKons Cell LocaKons Row 6 Cell D6 20 4/11/11 Cell Ranges Cell Ranges Range D3:D12 Cell Ranges Range B6:J6 21 4/11/11 Excel Formulas and FuncKons Formulas and funcKons always start with an equals sign and func/ons always have opening and closing parentheses =1+2 =B12*6 =B12/B14 =(B1210)/5 =sum() =average() =median() =mode() =stdev.p() =stdev() =round() =`est() FuncKons take arguments between the parentheses Tell Excel what data to operate on Tell Excel addiKonal informaKon about how you want the formula computed =Average() FuncKon computes the arithme/c mean =
!Xi N X= !Xi n Excel FuncKons Arguments: Cell locaKons Syntax: =average(D3,D4,D5,D6,D7,D8,D9,D11,D12) or =average(D3:D12) 22 4/11/11 Mean for Midterm 1 Mean for Midterm 2 Mean for Midterm 1 and Midterm 2 Mean for Midterm 1 and Midterm 2 =average(D3:D12) =average(H3:H12) 23 4/11/11 =stdev.p() FuncKon computes the standard devia/on for a popula/on Standard DeviaKon for Midterm 1 != SS N Arguments: Cell locaKons Syntax: =stdev.p(D3,D4,D5,D6,D7,D8,D9,D11,D12) or =stdev.p(D3:D12) Standard DeviaKon for Midterm 2 Standard DeviaKon for Midterm 2 =stdev.p(D3:D12) =stdev.p(H3:H12) 24 4/11/11 Standard DeviaKon for Midterm 2 Midterm 1 Midterm 2 ConverKng between Raw and Zscores Raw score to ZScore ConverKng between Raw and Zscores Raw score to ZScore zi =
ZScore to Raw Score Xi ! " zi =
ZScore to Raw Score Xi ! " X i = + zi! X i = + zi! 25 4/11/11 CreaKng ZScores CreaKng ZScores X ! =(.700.781)/0.098 zi = i " CreaKng ZScores CreaKng ZScores =(.700.781)/0.098 =(D3.781)/0.098 26 4/11/11 CreaKng ZScores "Filling" Formulas "Filling" Formulas "Filling" Formulas 27 4/11/11 "Filling" Formulas "Filling" Formulas =(D3.781)/0.098 =(D6.781)/0.098 =(D12.781)/0.098 CreaKng ZScores CreaKng ZScores X ! =(.700.781)/0.098 zi = i " =(H30.652)/0.132 28 4/11/11 CreaKng ZScores Effects of Standardizing Zscored Distribu/on Mean: Always 0 Standard DeviaKon: Shape: 29 4/11/11 Effects of Standardizing Zscored Distribu/on Mean: Always 0 Standard DeviaKon: Always 1 Shape: Midterm 1 Raw Scores Effects of Standardizing Zscored Distribu/on Mean: Always 0 Standard DeviaKon: Always 1 Shape: Same as original distribuKon Midterm 1 Z Scores 30 4/11/11 ConverKng between Raw and Zscores Raw score to ZScore zi =
ZScore to Raw Score Xi ! " X i = + zi! ConverKng between Raw and Zscores Raw score to ZScore Rescaling to a New DistribuKon zi =
ZScore to Raw Score Xi ! " X i = + zi! X i = + zi! 31 4/11/11 Rescaling to a New DistribuKon Rescaling to a New DistribuKon X i = + zi! = 0.80 = 0.10 =E3*0.10+0.80 = 0.80 =
0.10 Rescaling to a New DistribuKon Rescaling to a New DistribuKon =E3*0.10+0.80 = 0.80 = 0.10 =I3*0.10+0.80 = 0.80 = 0.10 32 4/11/11 Rescaling to a New DistribuKon Rescaling to a New DistribuKon =I3*0.10+0.80 = 0.80 = 0.10 = 0.80 = 0.10 Rescaling to a New DistribuKon Rescaling to a New DistribuKon = 0.80 = 0.10 = 0.80 = 0.10 33 4/11/11 Rescaling to a New DistribuKon Rescaling to a New DistribuKon = 0.80 = 0.10 = 0.80 = 0.10 Rescaling to a New DistribuKon Rescaling to a New DistribuKon = 0.80 = 0.10 = 0.80 = 0.10 34 4/11/11 Midterm 1 Original Midterm 1 Rescaled Midterm 2 Original Midterm 2 Rescaled This Week Describing the LocaKon of Scores Probability The Normal DistribuKon 35 4/11/11 For Next Time Read: Chapter 6 Do Review Homework 2 Answers Start Studying for Midterm 1 Work on notes for midterm (Two 8.5"x11" pages, both sides, any content desired) Exam covers Chapters 1 6 and lecture material through Friday Download Study Guide from WebCT 36 ...
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 Standard Deviation, zi, Tom Mike

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