Graphical representations of mean values
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Graphical representations of mean values. Mike Mays Institute for Math Learning West Virginia University. Why means?. 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 )/2. Arithmetic mean.

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Graphical representations of mean values

Graphical representations of mean values

Mike Mays

Institute for Math Learning

West Virginia University


Why means

Why means?


Graphical representations of mean values

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)/2

Arithmetic mean


Graphical representations of mean values

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


Graphical representations of mean values

Theorem: For a and b≥ 0, G(a,b) ≤ A(a,b), with equality iff a=b.

h/a=b/h

h2=a b

h

b

a


Interactive version

Interactive version

http://jacobi.math.wvu.edu/~mays/AVdemo/Labs/AG.htm


Graphical representations of mean values

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


Fancier interactive version

Fancier interactive version

http://jacobi.math.wvu.edu/~mays/AVdemo/Labs/AGH.htm


Graphical representations of mean values

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) = am(1,b/a)


Examples

Examples

A, G, H


Algebraic approach 1 powers

Algebraic approach 1: Powers


Algebraic approach 2 gini

Algebraic approach 2: Gini


Graphical approach moskovitz

Graphical approach: Moskovitz

Mf

a

b


Fancier interactive version1

Fancier interactive version

http://math.wvu.edu/~mays/AVdemo/deployed/Moskovitz.html


Homogeneous moskovitz means

Homogeneous Moskovitz means

Mf is homogeneous, f (1)=1 iff f is multiplicative

A1

G

Hx

C1/x


Calculus means and the mvt

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


Special case v s a b from f x x s

Special caseVs(a,b) from f(x) = xs

  • s → ∞max

  • s = 1A

  • s → 0I

  • -1/2(A+G)/2

  • -1L

  • -2G

  • -3(HG2)1/3

  • s → -∞min


Numerical analysis 1 compounding

Numerical analysis 1: compounding


Numerical analysis 2

Numerical analysis 2


Graphical representations of mean values

a0 = 2b0 = 4

a1 = 2.8284b1 = 3.3137

a2 = 3.06b2 = 3.1825

a3 = 3.12b3 = 3.1510


Thank you

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.


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