Mean M(x 1 ,x 2 ,…x N ) (no weighted):

1. Mean M(x1,x2,…xN) (no weighted): 1. M is a function: 2. Simmetry: for all permutation 3. Fixed point: 4. Homogenity: 5. Monotony: 6. Continuity:

2. Geometric Mean Arithmetic Mean So for the point 2, to make a mean, we have to use some operation with commutativeproperty; the simplest are: Moreover for the fixed pointproperty (3), if all number are iguals the mean is the same value, so we have to add something:

3. Traditional Block ………… ………… Harmonic Mean First method to build anothers means: Media For example, if we choose f(x)=1/x and arithmetic mean as traditional block, we discover the Harmonic Mean:

4. Power Mean if we choose f(x)=xs and arithmetic mean as traditional block, we find the Power Mean:

5. Second method to build anothers means: for two numbers, but is possible to generalize

6. Third form, to build anothers means: Lehmer mean

7. First group Harmonic Mean Power Mean Power Mean of orden 2 (Root Mean Squares) Quadratic Mean

8. Strange connection geometric-quadratic Power mean orden 1/2 Looks Heronian Second group Identric Mean Logarithmic Mean Stolarsky Mean

9. All means with negative p have the the coniugate mean respect the arithmentic mean, with positive p. Clic there is a demostration Third group Contraharmonic Mean

10. Equivalent Form of Lehmer Mean q=-p We can write the Lehmer Mean in this form too: Like weighted mean….

11. Heronian Mean Heinz Mean 1 0 Logarithmic Mean Average of Means like Heronian and Heinz. Another type

12. Mixed Means Arithmetic-Geometric Mean Arithmetic-Harmonic Mean Geometric-Harmonic Mean These two sequences converge to the same number.

13. For the Arithmetic-Geometric Mean there is a closed for expression: Where K(x) is the complete elliptic integral of first kind.

14. Demostration of an expression:

15. An observation: Another observation:

16. Moreover: • Elementary Symmetric Mean • And…. • 1. Median • 2. Mode