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		1.5.1— The Statistical Properties of Fractals

		Fractals may seem to have strange statistical properties. These properties really aren't strange. They only appear to be strange because they are not covered in the usual statistics courses. Those courses only cover the properties of statistical distributions that are called Gaussian or asymptotically Gaussian distributions. The statistical properties of fractals belong to a more general class of distributions which are called stable distributions. Although few scientists seem to be aware of the properties of stable distributions, mathematicians have studied them for over 250 years.

		For example, the average of a fractal may not exist. How can the average not exist? If we have 5 measurements, can't we just punch those 5 values into a hand calculator, determine their sum, divide the sum by 5, and get the average?

		That calculation on the hand calculator alone is not enough to demonstrate that the average exists. The average from one set of data that we determine with our hand calculator is called the sample mean. To show the average exists we must show that as more data are analyzed, these sample means reach a limiting value. We then consider that limiting value as the ''real" average of the thing that we measured. This "real" average is called the population mean.

		1— Non-Fractal

		For non-fractal objects, as more data is included, the averages of the data reach a limiting value which we therefore consider to be the "real" average.

		(That is, the sample means converge to a finite, nonzero, limiting value that we identify as the population mean.)

		2— Fractal

		However, for a fractal, as more data are included, the averages of the data will continue to increase or continue to decrease. Thus there is no limiting value, we can consider to be the "real" average. The average does not exist.

		(That is, the sample means do not converge to a finite, nonzero, limiting value, and thus there is no value that we can identify as the population mean.)

 

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