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Graphical representations of mean values - PowerPoint PPT Presentation

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

Mike Mays

Institute for Math Learning

West Virginia University

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

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

Theorem: For 8% on the fund the second year. What is the average interest over the two year period? 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 8% on the fund the second year. What is the average interest over the two year period?

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

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 60mph on the way up and 40mph on the way back. What is my average speed for the trip?

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

A mean is a symmetric function 60mph on the way up and 40mph on the way back. What is my average speed for the trip? 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 60mph on the way up and 40mph on the way back. What is my average speed for the trip?

A, G, H

Algebraic approach 1: Powers 60mph on the way up and 40mph on the way back. What is my average speed for the trip?

Algebraic approach 2: Gini 60mph on the way up and 40mph on the way back. What is my average speed for the trip?

Graphical approach: Moskovitz 60mph on the way up and 40mph on the way back. What is my average speed for the trip?

Mf

a

b

Fancier interactive version 60mph on the way up and 40mph on the way back. What is my average speed for the trip?

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

Homogeneous Moskovitz means 60mph on the way up and 40mph on the way back. What is my average speed for the trip?

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

A1

G

H x

C 1/x

Calculus: means and the MVT 60mph on the way up and 40mph on the way back. What is my average speed for the trip?

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 60mph on the way up and 40mph on the way back. What is my average speed for the trip? 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

Numerical analysis 1: compounding 60mph on the way up and 40mph on the way back. What is my average speed for the trip?

Numerical analysis 2 60mph on the way up and 40mph on the way back. What is my average speed for the trip?

a 60mph on the way up and 40mph on the way back. What is my average speed for the trip? 0 = 2 b0 = 4

a1 = 2.8284 b1 = 3.3137

a2 = 3.06 b2 = 3.1825

a3 = 3.12 b3 = 3.1510

Thank you 60mph on the way up and 40mph on the way back. What is my average speed for the trip?

• 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.