We present a general overview of multiscaling properties in finite time series with broad probability distribution of data. It is indicated how important are finite size effects (FSE) and broad data distribution in proper analysis of multifractality in such a case and what the relative strength of false multifractal effects generated by FSE, the broad data distribution  and the strength of real multiscaling property of time series could be. After presentation of the general scheme to calculate these three independent ingredients of the observed multifractal effects, we turn to examination of such properties for particular real financial time series from world stock indices. It is shown that within MFDFA (multifractal detrended fluctuation analysis) the hierarchy of real unbiased multiscaling effects formed after subtraction of multifractal bias generated by FSE or fat tails in probability distributions of returns can be very much different from the corresponding hierarchy based entirely only on the "naked" observed multifractal features. It proves an importantance of a deep comparative analysis of these three ingredients before any final conclusions about real multiscaling property of given data is drawn.

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