By Kardi Teknomo, PhD .

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What is Mean or Average?

Phillips (2000) defined the mean of two real positive numbers, Mean or Average and Mean or Average as a mapping or function of the two numbers into a real positive number Mean or Average . The mapping must satisfy three properties:

Mean or Average

  1. If Mean or Average then Mean or Average . The mean value always lies between the two numbers.
  2. Mean or Average . Mean is symmetric in Mean or Average and Mean or Average
  3. If Mean or Average then Mean or Average

Other properties of Mean:

4. Homogeneous mean will satisfy one additional property Mean or Average . Examples of homogeneous mean are arithmetic mean, geometric mean and harmonic mean.

5. If Mean or Average , then Mean or Average . This property happens because of the first property. Note that the property number 3 is the exact contrary of this property number 5.

Expanding for more than two numbers, I may generalize the three properties of mean and the definition of mean. Mean is defined as a mapping or function of the real positive numbers into a real positive number that satisfy property that the mean value always lies between the input numbers .

Three Properties that defined Mean:

  1. If we have numbers Mean or Average , then Mean or Average
  2. The order of input numbers is not important. Mean is symmetric in Mean or Average dimension. Example: Mean or Average
  3. If all inputs are equal, then the mean value is equal to any one of its input. From property number 1 we can deduce that if Mean or Average then Mean or Average

Note:

  1. Mean is defined only for real positive number input. No negative numbers is allowed as input
  2. The number of input is finite up to Mean or Average (not infinite)
  3. The mean value is somewhere in between the lowest and the highest input numbers. The mean value will not go outside this range of input.
  4. Actually, the symmetric property of mean is not a necessary condition. Others may define unsymmetrical mean.

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See also: Kolmogorov's Generalized Mean

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Preferable reference for this tutorial is

Teknomo, Kardi (2015) Mean and Average. http:\\people.revoledu.com\kardi\tutorial\BasicMath\Average\