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““Don’t worry about your difficulties in Don’t worry about your difficulties in Mathematics,Mathematics,
I can assure you that mine are still I can assure you that mine are still greater”greater”Albert EinsteinAlbert Einstein
Math ReviewMath Review
NASANASANational Space Grant NetworkNational Space Grant Network
““The greatest sin committed by formal education The greatest sin committed by formal education is the continued and relentless extermination of is the continued and relentless extermination of the natural pleasure of learning; and the greatest the natural pleasure of learning; and the greatest sin committed by "progressive" education is the sin committed by "progressive" education is the delusion that such pleasure need not be delusion that such pleasure need not be accompanied by a certain measure of pain”.accompanied by a certain measure of pain”.
Sydney HarrisSydney Harris,, Thoughts at Large Thoughts at Large
A)A) IntroductionIntroductiona.a. SymbolsSymbolsb.b. OperationsOperationsc.c. Central TendenciesCentral Tendencies
B)B) Linear AlgebraLinear AlgebraC)C) Correlation/Regression AnalysisCorrelation/Regression AnalysisD)D) System of Equations: Linear/QuadraticSystem of Equations: Linear/QuadraticE)E) Applied CalculusApplied Calculus
Math ReviewMath ReviewThursday-Monday, June 3-7 2003Thursday-Monday, June 3-7 2003
A)A) IntroductionIntroductiona.a. SymbolsSymbolsb.b. OperationsOperationsc.c. Central TendenciesCentral Tendencies
B)B) Linear AlgebraLinear Algebra
Math Review #1Math Review #1Thursday, June 3 2003Thursday, June 3 2003
http://www.columbia.edu/itc/sipa/envp/louchouarn/courses/mpa.htm
a. SymbolsBasic Math ReviewBasic Math Review
We need symbols to simplify expressions and develop abstract arguments
E = mc2
particularly for quantitative analysis.
F = ma
QuickTime™ and aCinepak decompressor
are needed to see this picture.N = ike
A Better Math CurriculumA Better Math CurriculumDr. Tom Davis (B.Sc. Math at Caltech, Ph.D. in Math at Stanford, post-doc in electrical engineering at Stanford)Dr. Tom Davis (B.Sc. Math at Caltech, Ph.D. in Math at Stanford, post-doc in electrical engineering at Stanford)
The problem today: “math is generally taught by and aimed at mathematicians”
(In many universities, engineering Depts teach their own math courses since students are unable to solve engineering (applied) problems with the tools they learn from the math Dept!)
Basic Math:- Finances (household…)- Problem-solving skills!!!- Basic numeracy- Estimation (including probability & statistics)- Visualization, and- Bullshit detector!!!
What’s in a numberWhat’s in a number“Sex Drives”
vs.
Average driving distance of men's and women's tours (PGA and LPGA)
200
210
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1990 1992 1994 1996 1998 2000 2002 2004
Year
Average driving distance (Ft)
Men
Ladies
What’s in a numberWhat’s in a number“Sex Drives”
vs.
Average driving distance of men's and women's tours (PGA and LPGA)
y = 1.0082x - 36.144
R2 = 0.9392
210
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255 260 265 270 275 280 285 290
Men's driving distance (Ft)
Women's driving distance (Ft)
Average driving distance of men's and women's tours (PGA and LPGA)
210
215
220
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230
235
240
245
250
255
260
255 260 265 270 275 280 285 290
Men's driving distance (Ft)
Women's driving distance (Ft)
What’s in a numberWhat’s in a number“Bullshit detector”
Tour de France
y = 0.173x - 305.41
R2 = 0.8601
20
25
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45
1915 1925 1935 1945 1955 1965 1975 1985 1995 2005Year
Average Speed (km/hr)
Tour de France
50
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1915 1925 1935 1945 1955 1965 1975 1985 1995 2005Year
Total Time (hr)
What’s in a numberWhat’s in a number“Bullshit detector”
Tour de France
2020
2520
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1915 1925 1935 1945 1955 1965 1975 1985 1995 2005Year
Distance (km)
Tour de France
3000
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20 30 40Average Speed (km/hr)
Distance (km)
Summation: (sigma)xi = x1 + x2 + x3 + … + xn
Where:n, represents the sample sizex represents the variable, andxi represents the value of the ith observation
b. OperationsBasic Math ReviewBasic Math Review
€
x =x i∑
n=
1
nx i∑
xi+yi)2 = x1+y1)2 + x2+y2)2 + x3+y3)2 + … + xn+yn)2
b. OperationsBasic Math ReviewBasic Math Review
€
1
xaxx(-a)(-a) = =
Powers (Exponents):Powers (Exponents):xxaa xxbb = = xx(a + b)(a + b)
xxaayyaa = ( = (xyxy))aa
((xxaa))bb = = xx(ab)(ab)
€
xabxx(a/b)(a/b) = b = bthth root of ( root of (xxaa) = ) =
€
xa
xbxx(a - b)(a - b) = =
b. OperationsBasic Math ReviewBasic Math Review
Logarithms (base 10):Logarithms (base 10):loglogbb((xx) = ) = yy if and onlyif and only ifif bbyy = = xx
loglogbb(1) = 0(1) = 0
loglogbb(b) = 1 (b) = 1
loglogbb((xxyy) = log) = logbb((xx) + log) + logbb((yy))
loglogbb(x/y) = log(x/y) = logbb((xx) - log) - logbb((yy))
loglogbb((xxnn) = nlog) = nlogbb((xx))
€
xy = balso: also: Warning:
loglogbb((x)x)loglogbb((yy) ) log logbb((xxyy))
€
logb(x)
logb(y)≠ logb(
x
y)
b. OperationsBasic Math ReviewBasic Math Review
Logarithms (natural log):Logarithms (natural log):lnln ((xx) = ) = yy if and onlyif and only ifif eeyy = = xx
ln(1) = 0ln(1) = 0
ln(ln(ee) = 1 ) = 1
ln(ln(xxyy) = ln() = ln(xx) + ln() + ln(yy))
ln(x/y) = ln(ln(x/y) = ln(xx) - ln() - ln(yy))
ln(ln(xxyy) = ) = yyln(ln(xx))
ln(ln(eexx) = ) = xxln(ln(ee) = ) = x x 1 = 1 = xx
eeln(x)ln(x) = = xx
b. OperationsBasic Math ReviewBasic Math Review
a)a) Solve for Solve for xx: ln(: ln(eeaa) = b) = bxx
b)b) Solve for Solve for yy using common logarithms (base 10): using common logarithms (base 10):y = 175
c)c) Solve for Solve for yy using common logarithms (base 10): using common logarithms (base 10):y = 175 - 127
c. Central TendenciesBasic Math ReviewBasic Math Review
€
x =x i∑
n=
1
nx i∑
The most commonly used descriptive statistics are The most commonly used descriptive statistics are measures of central tendencymeasures of central tendencyThe The sample meansample mean (: pronounced “ (: pronounced “xx bar”) is: bar”) is:
Where Where xxii represents the sum of all values in the sample represents the sum of all values in the sample and and n n represents the sample sizerepresents the sample size
c. Central TendenciesBasic Math ReviewBasic Math Review
Let’s assume we have a student population (Let’s assume we have a student population (nn = 47) = 47)
But what happens if we have an outlier (skewed But what happens if we have an outlier (skewed distribution )?distribution )?
Frequency Distribution
0
2
4
6
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22 23 24 25 26 27 28 29 30 31 32 33 34
Age
Frequency
c. Central TendenciesBasic Math ReviewBasic Math Review
Let’s assume we have a real student population (Let’s assume we have a real student population (nn = = 47)47)
Frequency Distribution
0
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10
20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50
Age
Frequency
c. Central TendenciesBasic Math ReviewBasic Math Review
Mean: arithmetic averageMean: arithmetic average Median: middle value of a set of valuesMedian: middle value of a set of values Mode: the data value that occurs most oftenMode: the data value that occurs most often
FridayFriday Populations Lecture 1Populations Lecture 1 Math Review #2: Linear Algebra - Math Review #2: Linear Algebra -
Correlation/Regression AnalysisCorrelation/Regression Analysis Don’t forget the website AND the math sheets!Don’t forget the website AND the math sheets!
http://www.columbia.edu/itc/sipa/envp/louchouarn/courses/mpa.htm
MondayMonday Lamont orientation (LDEO Exec. Director and DEES Lamont orientation (LDEO Exec. Director and DEES
Chair)Chair) Math Review #3: System of Equations: Math Review #3: System of Equations:
Linear/Quadratic - Applied CalculusLinear/Quadratic - Applied Calculus