What is Mean or Average?
Phillips (2000) defined the mean of two real positive numbers, and as a mapping or function of the two numbers into a real positive number . The mapping must satisfy three properties:
- If then . The mean value always lies between the two numbers.
- . Mean is symmetric in and
- If then
Other properties of Mean:
4. Homogeneous mean will satisfy one additional property . Examples of homogeneous mean are arithmetic mean, geometric mean and harmonic mean.
5. If , then . 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:
- If we have numbers , then
- The order of input numbers is not important. Mean is symmetric in dimension. Example:
- 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 then
- Mean is defined only for real positive number input. No negative numbers is allowed as input
- The number of input is finite up to (not infinite)
- 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.
- Actually, the symmetric property of mean is not a necessary condition. Others may define unsymmetrical mean.
Preferable reference for this tutorial is
Teknomo, Kardi (2015) Mean and Average. https:\\people.revoledu.com\kardi\tutorial\BasicMath\Average\