A multifractal spectrum, based on an earlier paper and different from the Legendre multifractal spectrum (Padua et al. 2013) was examined in this paper. The examination yielded interesting results which enhanced the utility of the developed ?(s)-multifractal spectrum in analyzing real data. One of the results show that a mixture of several monofractal observations can be represented as a single monofractal distribution but whose spectrum is different from the spectrum of the original data. Thus, high fractal dimensional distributions can be infinitely decomposed into component monofractal dimensions. Further, we also show that given a multifractal set of observations, observations that fall on smaller scales obey a normal distribution. The study ends by providing possible avenues for future research particularly in the area of analytic number theory in relation to the Riemann hypothesis about the distribution of primes.
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