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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?. 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

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  1. Graphical representations of mean values Mike Mays Institute for Math Learning West Virginia University

  2. Why means?

  3. 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

  4. 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

  5. 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

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

  7. 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

  8. Fancier interactive version http://jacobi.math.wvu.edu/~mays/AVdemo/Labs/AGH.htm

  9. 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)

  10. Examples A, G, H

  11. Algebraic approach 1: Powers

  12. Algebraic approach 2: Gini

  13. Graphical approach: Moskovitz Mf a b

  14. Fancier interactive version http://math.wvu.edu/~mays/AVdemo/deployed/Moskovitz.html

  15. Homogeneous Moskovitz means Mf is homogeneous, f (1)=1 iff f is multiplicative A1 G H x C 1/x

  16. 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

  17. Special caseVs(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

  18. Numerical analysis 1: compounding

  19. Numerical analysis 2

  20. a0 = 2 b0 = 4 a1 = 2.8284 b1 = 3.3137 a2 = 3.06 b2 = 3.1825 a3 = 3.12 b3 = 3.1510 •  

  21. 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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