In this paper we analyze numerically the  effect of spurious multifractality caused entirely by the presence of fat-tailed symmetric  and asymmetric probability distributions of fluctuations in time series. In the presented approach different kinds of symmetric, asymmetric, thin- and fat-tailed probability distributions of synthetic data are examined starting from Levy regime up to those with finite variance, lying outside Levy regime.
We use nonextensive Tsallis statistics to construct all considered data. The semi-analytical compact formula are then provided to express the level of spurious multifractality generated by fat tails in terms of Tsallis parameter $q$. The results are presented in Hurst and Holder languages - more often used in study of multifractal phenomena. It turns out that fat tails and, in particular, the level of symmetry (asymmetry) of probalility distributions have a non-trivial influence on the measured multifractal properties of the time series. According to semianalitycal relations we provide we argue that it is possible to make a clear quantitative distinction between true multifractality caused by the presence of nonlinear correlations in data and spurious multifractality  generated by effects not connected with nonlinear correlations like fat-tailed shape of distributions, their asymmetry, linear autocorrelations or finite length of analysed time series.

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