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{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 12 "P O Box 1038" }}{PARA 258 "" 0 "" {TEXT -1 25 "Grove Hill
, Alabama 36451" }}{PARA 260 "" 0 "" {TEXT 260 21 "herod@math.gatech.e
du" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 353 " \+
Students always want to know how their grades are determined and ho
w well other students are doing in their class. This worksheet provide
s a way to think of this question. It recalls how to find the mean and
the standard deviation of a collection of numbers. Histograms and nor
mal distributions are drawn. Scores from a previous class are provided
." }}{PARA 0 "" 0 "" {TEXT -1 50 " First we need to have the stati
stics package." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "with(stats
): with(describe): with(statplots):" }}}{PARA 0 "" 0 "" {TEXT -1 0 ""
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 38
"Mean, variance, and standard deviation" }}{PARA 0 "" 0 "" {TEXT -1
17 " We find the " }{TEXT 256 7 "average" }{TEXT -1 65 " of a coll
ection of numbers. There is a standard way to find the " }{TEXT 257 4
"mean" }{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#*&-%
$sumG6$&%\"SG6#%\"nG/F*;\"\"\"\"\"*F-F.!\"\"" }{TEXT -1 1 "." }}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 " We do a co
mputational example. Note that there is a one-word Maple command to fi
nd 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 "" {MPLTEXT 1 0 0 "
" }}}{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. Compare the following set that has the
same mean but a wider spread, or deviation." }}{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 standard measure of spread away from the
mean. The measure of spread away from the mean is called " }{TEXT
258 8 "variance" }{TEXT -1 39 ". If the mean is computed to be \265, t
hen" }}{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 S
1 above will be smaller than the variance for the set S2." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "varia
nce(S1); evalf(%);\nvariance(S2); evalf(%);" }}}{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 " His
tograms provide a geometric way to see the distribution of a collectio
n of numbers. In the next two histograms, rectangles are drawn with ba
se of length one. The histogram for set S1 has each rectangle centered
at one of the numbers in the set S1 and the height is the number of t
imes that number appears in the set. The mean and standard deviation c
an be calculated from histograms. Recognize, while looking at the hist
ogram, that the rectangles are clustered about the mean." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "HS1:=
[Weight(0..3/2,3/2*1),Weight(3/2..5/2,4),Weight(5/2..7/2,1),\n Weig
ht(7/2..9/2,3)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "histogr
am(HS1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "mean(HS1);stand
arddeviation(HS1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 156 "The histogram for the second set of numbers, S2, shoul
d have a different appearance. While it is also clustered about the sa
me mean, it has a larger spread." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 91 "HS2:=[Weight(-3/2..-1/2,1),Weight(1/2..3/2,3),Weight(7/2..9/2,
3),\n Weight(9/2..11/2,2)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 49 "histogram(HS2);\nmean(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 distribut
ion" }{TEXT -1 162 " about the mean. A normal distribution has a techn
ical meaning. The normal distribution is defined in terms of the expon
ential 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)/(s
igma*sqrt(2)*Pi)" "6#*&-%$expG6#,$*&*&,&%\"xG\"\"\"%#muG!\"\"F,%&sigma
GF.\"\"#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 grap
h, we take " }{XPPEDIT 18 0 "mu" "6#%#muG" }{TEXT -1 15 " to be 8/3 an
d " }{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(1
0)/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 peak of the curve actually occurs at the mean, 8/3. We \+
see this by asking 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 stan
dard deviation of a general distribution. At that time we will verify \+
that the normal-distribution 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 Distributi
ons" }}{PARA 0 "" 0 "" {TEXT -1 139 " Perhaps we can now talk inte
lligently about the distribution of scores in a mathematics class. Her
e are scores from a Math 2507 class." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 340 "Scores:=[Weight(90..95,20),Weight(95..100,15),Weight
(80..85,27),\n Weight(85..90,16),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),Weight(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),Weigh
t(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 " range" }{TEXT -1 15 " of the grades." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "mean(Scores);evalf(%);\nstan
darddeviation(Scores);evalf(%); \nspread:=range(Scores);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "mu:=mean(Scores); sigma:=standardde
viation(Scores);" }}}{PARA 0 "" 0 "" {TEXT -1 79 " We define a nor
mal distribution with the same mean and standard deviation." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "npd:=x->exp(-((x-mu)/sigma)^2/2)/(s
igma*sqrt(2)*Pi);" }}}{PARA 0 "" 0 "" {TEXT -1 189 " We multiply t
he normal distribution by a factor so that it approximates the grade d
istribution. This gives a sense for the extent to which the scores fel
l into a \"normal distribution.\"" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 43 "spread:=evalf(int(npd(x),x=range(Scores)));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "J:=histogram(Scores): K:=plo
t(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 scores. A common choice is th
e 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 else print(`Grade is F`)\nfi; \+
fi; fi; fi;end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "grade(4
0);grade(45);\ngrade(50);grade(55);\ngrade(60);grade(65);\ngrade(70);g
rade(75);\ngrade(80);grade(85);\ngrade(90);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 }