# Graphical representations of mean values - PowerPoint PPT Presentation

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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

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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

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 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

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

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

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)

A, G, H

Mf

a

b

### Fancier interactive version

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

### Homogeneous Moskovitz means

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

A1

G

Hx

C1/x

### 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

• s → ∞max

• s = 1A

• s → 0I

• -1/2(A+G)/2

• -1L

• -2G

• -3(HG2)1/3

• s → -∞min

### Numerical analysis 2

a0 = 2b0 = 4

a1 = 2.8284b1 = 3.3137

a2 = 3.06b2 = 3.1825

a3 = 3.12b3 = 3.1510

### 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.