graphical representations of mean values mike mays institute for math learning west virginia...
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Graphical representations of mean values
Mike Mays
Institute for Math Learning
West Virginia University
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Why means?
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Suppose you have a 79 on one test and an 87 on another, towards a midterm grade. B cutoff is 82. Do you have a B?
A(a,b) = (a+b)/2Arithmetic mean
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Suppose you earn 6% interest on a fund the first year, and 8% on the fund the second year. What is the average interest over the two year period?
G(a,b) =Geometric mean
ab
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Theorem: For a and b ≥ 0, G(a,b) ≤ A(a,b), with equality iff a=b.
a b
h
h/a=b/h
h2=a b
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Interactive version
http://jacobi.math.wvu.edu/~mays/AVdemo/Labs/AG.htm
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Morgantown is 120 miles from Slippery Rock. Suppose I drive 60mph on the way up and 40mph on the way back. What is my average speed for the trip?
H(a,b) = 2ab/(a+b)
Harmonic mean
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Fancier interactive version
http://jacobi.math.wvu.edu/~mays/AVdemo/Labs/AGH.htm
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A mean is a symmetric function m(a,b) of two positive variables a and b satisfying the intermediacy property
min(a,b) ≤ m(a,b) ≤ max(a,b)
Homogeneity: m(a,b) = a m(1,b/a)
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Examples
ba
babaL
loglog),(
ba
babaC
22
),(
2),(
22 babaRMS
A, G, H
e
ba
baI
ba
b
a )/(1
),(
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Algebraic approach 1: Powers
ppp
p
babaM
/1
2),(
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Algebraic approach 2: Gini
11),(
ss
ss
s ba
babaG
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Graphical approach: Moskovitz
a b
Mf
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Fancier interactive version
http://math.wvu.edu/~mays/AVdemo/deployed/Moskovitz.html
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Homogeneous Moskovitz means
Mf is homogeneous, f (1)=1 iff f is multiplicative
x
A 1
G
H x
C 1/x
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Calculus: means and the MVT
Mean Value Theorem for Integrals (special case): Suppose f(x) is continuous and strictly monotone on [a,b]. Then there is a unique c in (a,b) such that
b
adxxfabcf )())((
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Special case Vs(a,b) from f(x) = xs
• s → ∞ max• s = 1 A• s → 0 I• -1/2 (A+G)/2• -1 L• -2 G• -3 (HG2)1/3
• s → -∞ min
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Numerical analysis 1: compounding
11 ),1(),1( bnAnnHa
211112 ),(),( bbaAnbaHa
),(),(),( baAbaGbaH
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221
5.121.3333
4166.121.412
414.12
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Numerical analysis 24 2 00 ba
12
122112
2
ba
babbaa
01
011001
2
ba
babbaa
23
233223
2
ba
babbaa
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a0 = 2 b0 = 4
a1 = 2.8284 b1 = 3.3137
a2 = 3.06 b2 = 3.1825
a3 = 3.12 b3 = 3.1510
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Thank you
• math.wvu.edu/~mays/
• Beckenbach, E. F. and Bellman, R. Inequalities. New York: Springer-Verlag, 1983
• Bullen, P. S.; Mitrinovic, D. S.; and Vasic, P. M. Means and Their Inequalities. Dordrecht, Netherlands: Reidel, 1988.