5.

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		1.5.15— Biological Implications of the Statistical Properties of Fractals

		1— Not Gaussian or Asymptotically Gaussian

		The statistical knowledge of most scientists is limited to the statistical properties of Gaussian distributions. Fractals do not have the properties of Gaussian distributions. In order to understand the many fractal objects and processes in the natural world, it is required to learn about the properties of stable distributions. Stable distributions are more general than Gaussian distributions.

		2— The Average and Variance Do Not Exist

		The moments of a fractal, such as the mean and variance, do not exist. As more data are included, the measurements of these moments do not approach finite, limiting values.

		3— Large Variations

		The variance of a fractal increases as more data are analyzed. The average values measured for the properties of the data will have wide variation from one time to another and among repetitions of the same experiment.

		4— When are these large variations significant?

		When the variance of a fractal increases as more data are analyzed, we do not know how to perform statistical tests to determine if the parameters of the mechanism that generated the data have changed from one time to another or between experiments run under different conditions.

		The statistical tests taught in the usual statistics courses are based on the assumption that the variance is finite. These tests are not valid to analyze fractal data where the variance is infinite. It would be very worthwhile for mathematicians to formulate statistical tests for fractal distributions where the variance is infinite.

 

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