{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 }