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)/2Arithmetic 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

ab

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

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) = a m(1,b/a)

Examples

ba

babaL

loglog),(

ba

babaC

22

),(

2),(

22 babaRMS

A, G, H

e

ba

baI

ba

b

a )/(1

),(

Algebraic approach 1: Powers

ppp

p

babaM

/1

2),(

Algebraic approach 2: Gini

11),(

ss

ss

s ba

babaG

Graphical approach: Moskovitz

a b

Mf

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

x

A 1

G

H x

C 1/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

b

adxxfabcf )())((

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

Numerical analysis 1: compounding

11 ),1(),1( bnAnnHa

211112 ),(),( bbaAnbaHa

),(),(),( baAbaGbaH

221

5.121.3333

4166.121.412

414.12

Numerical analysis 24 2 00 ba

12

122112

2

ba

babbaa

01

011001

2

ba

babbaa

23

233223

2

ba

babbaa

a0 = 2 b0 = 4

a1 = 2.8284 b1 = 3.3137

a2 = 3.06 b2 = 3.1825

a3 = 3.12 b3 = 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.