Financial practitioners commonly use the average slope of a price chart over a given time interval to compute the fractal dimension of a stock price series. they approximate this average slope by simply taking the difference between the highest and lowest price during the interval and dividing through by the number of increments during the interval. They use this number to compute the scaling factor from which the fractal dimension is computed.
They typically provide no justification for (i) why high minus low over interval length should correspond to the average slope, or (ii) why the average slope should correspond to the number of boxes touched in the boxcount method. See, for example, this rather handwavy explanation [1].
Is there any mathematical justification for this "approximation" or are these people just too lazy to implement the boxcount method? Is there any mathematical argument by which the "average slope" (computed as described above) will be proportional to the number of boxes covered in the boxcount method?