{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 } {PSTYLE "" 4 256 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 257 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 258 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 259 "" 0 "" {TEXT -1 30 "Scores, Grades, and Deviations " }}{PARA 256 "" 0 "" {TEXT -1 9 "Jim Herod" }}{PARA 257 "" 0 "" {TEXT -1 21 "School of Mathematics" }}{PARA 258 "" 0 "" {TEXT -1 12 "G eorgia Tech" }}{PARA 260 "" 0 "" {TEXT 260 21 "herod@math.gatech.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 352 " S tudents aways want to know how their grades are determined and how wel l other students are doing in their class. This worksheet provides a w ay to think of this question. It recalls how to find the mean and the \+ standard deviation of a collection of numbers. Histograms and normal d istributions are drawn. Scores from a previous class are provided." }} {PARA 0 "" 0 "" {TEXT -1 50 " First we need to have the statistics package." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "with(stats): wi th(describe): with(statplots):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 38 "M ean, variance, and standard deviation" }}{PARA 0 "" 0 "" {TEXT -1 17 " We find the " }{TEXT 256 7 "average" }{TEXT -1 65 " of a collecti on of numbers. There is a standard way to find the " }{TEXT 257 4 "mea n" }{TEXT -1 47 ", or average, for a set of, say, nine numbers:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 " \+ mean(S) = " }{XPPEDIT 18 0 "(S[1]+S[2]+S[3]+S[4]+S[5]+S[6]+S[7] +S[8]+S[9])/9" "6#*&,4&%\"SG6#\"\"\"F(&F&6#\"\"#F(&F&6#\"\"$F(&F&6#\" \"%F(&F&6#\"\"&F(&F&6#\"\"'F(&F&6#\"\"(F(&F&6#\"\")F(&F&6#\"\"*F(F(F@! \"\"" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "sum(S[n],n=1..9)/9" "6#*&-%$s umG6$&%\"SG6#%\"nG/F*;\"\"\"\"\"*F-F.!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 " We do a comp utational example. Note that there is a one-word Maple command to find the mean." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "S1:=[1,2,2,2,2 ,3,4,4,4];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "sum(S1[n],n=1 ..9)/9; evalf(%); \nmean(S1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 187 " This average says that the numbers \+ are clustered between 2 and 3 -- a little closer to 3 than to 2. Compa re the following set that has the same mean but a wider spread, or dev iation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "S2:=[-1,1,1,1,4,4,4,5,5]; mean(S2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 " There is a stand ard measure of spread away from the mean. The measure of spread away f rom the mean is called " }{TEXT 258 8 "variance" }{TEXT -1 39 ". If th e mean is computed to be \265, then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 33 " variance = " } {XPPEDIT 18 0 "sum((S[i]-mu)^2,i=1..n)" "6#-%$sumG6$*$,&&%\"SG6#%\"iG \"\"\"%#muG!\"\"\"\"#/F+;F,%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 " Thus it is expected that the variance for the set S1 above will be smaller than the varia nce for the set S2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "variance(S1); evalf(%);\nvariance(S2); eval f(%);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 55 " An alternate, more common, measure of spread is the " }{TEXT 259 18 "standard deviation" }{TEXT -1 57 ", which is defined to be the square root of the variance." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "standarddeviation(S1); evalf (%);\nstandarddeviation(S2); evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "Histograms." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 484 " Histograms provide a geometric way \+ to see the distribution of a collection of numbers. In the next two hi stograms, rectangles are drawn with base of length one. The histogram \+ for set S1 has each rectangle centered at one of the numbers in the se t S1 and the height is the number of times that number appears in the \+ set. The mean and standard deviation can be calculated from histograms . Recognize, while looking at the histogram, that the rectangles are c lustered about the mean." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "HS1:=[Weight(0..3/2,3/2*1),Weight(3 /2..5/2,4),\n Weight(5/2..7/2,1),Weight(7/2..9/2,3)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "histogram(HS1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "mean(HS1);standarddeviation(HS1);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 156 "The hist ogram for the second set of numbers, S2, should have a different appea rance. While it is also clustered about the same mean, it has a larger spread." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "HS2:=[Weight(-3/ 2..-1/2,1),Weight(1/2..3/2,3),\n Weight(7/2..9/2,3),Weight(9/2..11/2 ,2)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "histogram(HS2);\nm ean(HS2);standarddeviation(HS2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 "Normal Distributions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 " Neither of the sets fall into what is called a " }{TEXT 261 19 "normal distribution" }{TEXT -1 162 " about the mean. A normal distribution has a technical meaning. The n ormal distribution is defined in terms of the exponential function, a \+ number \265, and a number " }{XPPEDIT 18 0 "sigma" "6#%&sigmaG" } {TEXT -1 37 ": the normal distribution is given as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 " \+ nd(x) = " }{XPPEDIT 18 0 "exp(-((x-mu)/sigma)^2/2)/(sigma*sqrt(2)*Pi) " "6#*&-%$expG6#,$*&*&,&%\"xG\"\"\"%#muG!\"\"F,%&sigmaGF.\"\"#F0F.F.F, *(F/F,-%%sqrtG6#F0F,%#PiGF,F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 52 " For the purposes of drawing the graph, we take " } {XPPEDIT 18 0 "mu" "6#%#muG" }{TEXT -1 15 " to be 8/3 and " }{XPPEDIT 18 0 "sigma" "6#%&sigmaG" }{TEXT -1 18 " to be sqrt(10)/3." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "mu:=8/3; sigma:=sqrt(10)/3;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "nd:=x->exp(-((x-mu)/sigma)^2 /2)/(sigma*sqrt(2)*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " plot(nd(x),x=0..5);" }}}{PARA 0 "" 0 "" {TEXT -1 123 "Note that the pe ak of the curve actually occurs at the mean, 8/3. We see this by askin g where we have a horizontal tangent." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(diff(nd(x),x)=0,x);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 199 "We will wait for \+ integral calculus to define the mean and standard deviation of a gener al distribution. At that time we will verify that the normal-distribut ion above has standard deviation given by " }{XPPEDIT 18 0 "sigma" "6# %&sigmaG" }{TEXT -1 1 "." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 "Actual Distributions" }}{PARA 0 "" 0 "" {TEXT -1 139 " Perhaps we can now talk intelligently about the dis tribution of scores in a mathematics class. Here are scores from a Mat h 2507 class." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 340 "Scores:=[W eight(90..95,20),Weight(95..100,15),Weight(80..85,27),\n Weight(85..9 0,16),Weight(75..80,21),Weight(70..75,19),\n Weight(65..70,18),Weight (60..65,12),Weight(55..60,5),\n Weight(50..55,6),Weight(45..50,2),Wei ght(40..45,2),\n Weight(35..40,2),Weight(30..35,1),Weight(25..30,2), \n Weight(20..25,2),Weight(15..20,0),Weight(10..15,2)];" }}}{PARA 0 " " 0 "" {TEXT -1 50 " We count how many students were in the class. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "x:=n->op(2,op(n,Scores)) ;Tot:=sum('x(n)','n'=1..18);" }}}{PARA 0 "" 0 "" {TEXT -1 51 " We \+ draw a histogram of the grade distribution." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "histogram(Scores);" }}}{PARA 0 "" 0 "" {TEXT -1 17 " We find the " }{TEXT 262 4 "mean" }{TEXT -1 6 ", the " }{TEXT 263 18 "standard deviation" }{TEXT -1 9 ", and the" }{TEXT 264 6 " ran ge" }{TEXT -1 15 " of the grades." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "mean(Scores);evalf(%);standarddeviation(Scores);evalf (%); spread:=range(Scores);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "mu:=mean(Scores); sigma:=standarddeviation(Scores);" }}}{PARA 0 " " 0 "" {TEXT -1 79 " We define a normal distribution with the same mean and standard deviation." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "npd:=x->exp(-((x-mu)/sigma)^2/2)/(sigma*sqrt(2)*Pi);" }}}{PARA 0 "" 0 "" {TEXT -1 189 " We multiply the normal distribution by a \+ factor so that it approximates the grade distribution. This gives a se nse for the extent to which the scores fell into a \"normal distributi on.\"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "spread:=evalf(int(n pd(x),x=range(Scores)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "J:=histogram(Scores): K:=plot(Tot/spread*npd(x),x=0..100):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plots[display](\{J,K\});" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "Grade Distribution" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 " The question inevitably arises, what grade will be associated with the sc ores. A common choice is the following." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 300 "grade:=proc(score)\n if score >= evalf(mu+sigma ) then print(`Grade is A`)\n else if score >= evalf(mu) then print(` Grade is B`)\n else if score >= evalf(mu-sigma) then print(`Grade is \+ C`)\nelse if score >= evalf(mu-2*sigma) then print(`Grade is D`)\n e lse print(`Grade is F`)\nfi; fi; fi; fi;end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "grade(40);grade(45);\ngrade(50);grade(55);\ngra de(60);grade(65);\ngrade(70);grade(75);\ngrade(80);grade(85);\ngrade(9 0);grade(95);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {MARK "0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }