  The fractal properties of models of randomly placed $n$-dimensional spheres
($n$=1,2,3) are studied using standard techniques for calculating fractal
dimensions in empirical data (the box counting and Minkowski-sausage
techniques). Using analytical and numerical calculations it is shown that in
the regime of low volume fraction occupied by the spheres, apparent fractal
behavior is observed for a range of scales between physically relevant
cut-offs. The width of this range, typically spanning between one and two
orders of magnitude, is in very good agreement with the typical range observed
in experimental measurements of fractals. The dimensions are not universal and
depend on density. These observations are applicable to spatial, temporal and
spectral random structures. Polydispersivity in sphere radii and
impenetrability of the spheres (resulting in short range correlations) are also
introduced and are found to have little effect on the scaling properties. We
thus propose that apparent fractal behavior observed experimentally over a
limited range may often have its origin in underlying randomness.
