math review - columbia.edu · a. symbols basic math review we need symbols to simplify expressions...

Math ReviewMath Review

“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 ReviewTuesdayTuesday--Friday, June 10Friday, June 10--13 200313 2003

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 Review #1Math Review #1Tuesday, June 10 2003Tuesday, June 10 2003

http://www.columbia.edu/~pl2065/courses/mpa.htm

A)A) IntroductionIntroductiona.a. SymbolsSymbolsb.b. OperationsOperationsc.c. Central TendenciesCentral Tendencies

B)B) Linear AlgebraLinear Algebra



a. SymbolsBasic Math ReviewBasic Math Review

We need symbols to simplify expressions and develop abstract arguments

E = mc2

particularly for quantitative analysis.

F = ma

A Better Math CurriculumA Better Math CurriculumDr. Tom Davis (B.Sc. Math at Caltech, Ph.D. in Math at Stanford,Dr. Tom Davis (B.Sc. Math at Caltech, Ph.D. in Math at Stanford, postpost--doc in electrical engineering at Stanford)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”

Or the“Gender Wars”

What’s in a numberWhat’s in a numberReally, shouldn’t we be serious?


Average driving distance of men's and women's tours (PGA and LPGA)

200

210

220

230

240

250

260

270

280

290

300

1990 1992 1994 1996 1998 2000 2002 2004

Year

Ave

rage

drivi

ng

dist

ance

(Ft

)

Men

Ladies

vs.



y = 1.0082x - 36.144

R2 = 0.9392

210

215

220

225

230

235

240

245

250

255

260

255 260 265 270 275 280 285 290

Men's driving distance (Ft)


210

215

220

225

230

235

240

245

250

255

260

255 260 265 270 275 280 285 290

Men's driving distance (Ft)

Wom

en's

drivi

ng

dist

ance

(Ft

)

vs.

What’s in a numberWhat’s in a number“Bullshit detector”

Tour de France

507090

110130150170190210230250

1915 1925 1935 1945 1955 1965 1975 1985 1995 2005Year

Tota

l Tim

e (h

r)

Tour de France

y = 0.173x - 305.41R2 = 0.8601

20

25

30

35

40

45

1915 1925 1935 1945 1955 1965 1975 1985 1995 2005Year

Ave

rage

Spe

ed (k

m/h

r)

What’s in a numberWhat’s in a number“Bullshit detector”

Tour de France

20202520

30203520

40204520

50205520

6020

1915 1925 1935 1945 1955 1965 1975 1985 1995 2005Year

Dis

tanc

e (k

m)

Tour de France

3000

3500

4000

4500

5000

5500

6000

20 30 40Average Speed (km/hr)

Dist

ance

(km

)

Summation: Σ (sigma)Σxi = 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 =xi∑

n=

1n

xi∑

Σ(xi+yi)2 = (x1+y1)2 + (x2+y2)2 + (x3+y3)2 + … + (xn+yn)2

Basic Math ReviewBasic Math Reviewb. OperationsFactorial: n!

n! = 1 x 2 x 3 x … x nWhere:0! = 11! = 12! = 1 x 2 = 23! = 1 x 2 x 3 = 65! = 1 x 2 x 3 x 4 x 5 = 12010! = 3,628,800 (try it out)

Orn! = (n-1)! x n

(for n > 0)


1xaxx((--a)a) = =

Powers (Exponents):Powers (Exponents):xxaa ×× xxbb = = xx(a + b)(a + b)

xxaayyaa = (= (xyxy))aa

((xxaa))bb = = xx((abab))

xabxx(a/b)(a/b) = = bbthth root of (root of (xxaa) = ) =

xa

xbxx(a (a -- b)b) = =


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((xx**yy) =) = loglogbb((xx) +) + loglogbb((yy))

loglogbb(x/y) =(x/y) = loglogbb((xx) ) -- loglogbb((yy))

loglogbb((xxnn) =) = nlognlogbb((xx))

xy = balso: also: Warning:loglogbb((x)x)××loglogbb((yy) ) ≠≠ loglogbb((xx××yy))logb(x)logb(y)

≠ logb(xy

)


Logarithms (natural log):Logarithms (natural log):lnln ((xx) = ) = yy if and onlyif and only ifif eeyy = = xx

ln ln (1) = 0(1) = 0ln ln ee = 1 = 1

ln ln ((xx××yy) = ) = lnln((xx) + ) + lnln((yy))ln ln (x/y) = (x/y) = lnln((xx) ) -- lnln((yy))ln ln ((xxyy) = ) = yylnln((xx))ln ln ((eexx) = ) = xxlnln((ee) = ) = x x ×× 1 = 1 = xxeelnln(x)(x) = = xx

Basic Math ReviewBasic Math Reviewb. Operationsa)a) Solve for Solve for xx:: lnln((eeaa) = ) = bbxx

b)b) Solve for Solve for yy using common logarithms (base 10):using common logarithms (base 10):y = 175

c. Central TendenciesBasic Math ReviewBasic Math Review

x =xi∑

n=

1n

xi∑

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:

WhereWhere ΣΣξξii represents the sum of all values in the sample and represents the sum of all values in the sample and n n represents the sample sizerepresents the sample size


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 distribution )?But what happens if we have an outlier (skewed distribution )?

Frequency Distribution

0

2

4

6

8

10

12

14

22 23 24 25 26 27 28 29 30 31 32 33 34

Age

Fre

qu

en

cy


Let’s assume we have a real student population (Let’s assume we have a real student population (nn = 47)= 47)Frequency Distribution

0

1

2

3

4

5

6

7

8

9

10

20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50

Age

Fre

qu

en

cy


Mean: arithmetic averageMean: arithmetic averageMedian: middle value of a set of valuesMedian: middle value of a set of valuesMode: the data value that occurs most oftenMode: the data value that occurs most often

ThursdayThursdayEnvironmental Chemistry IEnvironmental Chemistry IMath Review: Math Review: Linear Algebra Linear Algebra -- Correlation/Regression Correlation/Regression AnalysisAnalysisDon’t forget the website AND the math sheets!Don’t forget the website AND the math sheets!


math review - columbia.edu · a. symbols basic math review we need symbols to simplify expressions...

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