This paper provides the statistical foundations for fractal analysis of real-life data. It begins by developing a fractal probability distribution based on a power-law formulation and proceeds by estimating the parameters through (a) maximum likelihood estimation method, (b) exponential parameter estimation, and (c) regression approach. Data analysis based on a fractal distribution hypothesis is heavily guided by the fact that for a random variable X with fractal distribution f(x;θ,λ), the random variable y=log(x/θ) has an exponential distribution with rate parameter β = λ-1. A new Q-Q plot is introduced for assessing the fractality of observations. Likewise, tests of hypotheses about the fractal dimension λ are introduced based on the pivotal statistic y=log(x/θ). Test statistics are constructed whose null distributions are shown to be chi-square, for testing H0: λ=λ0 or beta distributions, for testing H0: λ1=λ2.