Numerical Biases on Initial Mass Function Determinations Created by Binning

Abstract

We detect and quantify significant numerical biases in the determination of the slope of power laws with Salpeter (or similar) indices from uniformly-binned data using chi-square minimization. The biases are caused by the correlation between the number of stars per bin and the assigned weights and are especially important when the number of stars per bin is small. This result implies the existence of systematic errors in the values of IMFs calculated in this way. We propose as an alternative using variable-size bins and dividing the stars evenly among them. Such variable-size bins yield very small biases that are only weakly dependent on the number of stars per bin. Furthermore, we show that they allow for the calculation of reliable IMFs with only a small total number of stars. Therefore, they are a preferred alternative to the standard uniform-size binning.