Pavel Kroupa discusses the variation of the initial mass function (IMF) in star-forming environments. While a universal IMF is not intuitive, there is currently no convincing evidence for a variable IMF. The detection of systematic variations in the IMF would be a key insight into star formation. The paper defines an average or Galactic-field IMF, noting that there is evidence for a change in the power-law index at two masses: near 0.5 M☉ and 0.08 M☉. Using this IMF, the uncertainty in observational estimates of the IMF is investigated, considering Poisson noise and dynamical evolution of star clusters. The apparent scatter in the power-law index is found to reproduce the observed scatter, defining the fundamental limit within which any true variation becomes undetectable. The absence of evidence for a variable IMF suggests that any true variation in well-studied populations must be smaller than this scatter. Determinations of the power-law indices α are subject to systematic errors, primarily from unresolved binaries. The systematic bias is quantified, showing that the single-star IMF for young star clusters is systematically steeper by Δα ≈ 0.5 between 0.1 and 1 M☉ than the Galactic-field IMF. Globular clusters appear to have systematically flatter IMFs than the Galactic-field IMF. The detection of ancient white-dwarf candidates in the Galactic halo and absence of associated low-mass stars suggests a radically different IMF for this ancient population. Star-formation in higher-metallicity environments appears to produce relatively more low-mass stars. This is an interesting trend, consistent with a systematic variation of the IMF as expected from theoretical arguments. The paper presents an analysis of the alpha-plot, which shows the power-law index α against the mean log10m of the mass range over which the index is measured. The alpha-plot clearly shows the flattening of the IMF for m ≲ 0.5 M☉. It also shows no systematic difference between MW and LMC populations. The models discussed show that unresolved binary systems mostly affect the region m ≲ 1 M☉. The data for which are listed in Table 1. The paper also discusses the effects of stellar evolution and the application of incorrect pre-main sequence and main-sequence evolutionary tracks, which can corrupt the masses inferred from observed quantities as the stars evolve to or along the main sequence. The paper concludes that the observed scatter in the alpha-plot is due to a combination of Poisson noise and dynamical evolution of star clusters, and that the fundamental limit within which any true variation becomes undetectable is approximately 0.5. The paper also discusses the implications of these findings for the IMF in different star-forming environments.Pavel Kroupa discusses the variation of the initial mass function (IMF) in star-forming environments. While a universal IMF is not intuitive, there is currently no convincing evidence for a variable IMF. The detection of systematic variations in the IMF would be a key insight into star formation. The paper defines an average or Galactic-field IMF, noting that there is evidence for a change in the power-law index at two masses: near 0.5 M☉ and 0.08 M☉. Using this IMF, the uncertainty in observational estimates of the IMF is investigated, considering Poisson noise and dynamical evolution of star clusters. The apparent scatter in the power-law index is found to reproduce the observed scatter, defining the fundamental limit within which any true variation becomes undetectable. The absence of evidence for a variable IMF suggests that any true variation in well-studied populations must be smaller than this scatter. Determinations of the power-law indices α are subject to systematic errors, primarily from unresolved binaries. The systematic bias is quantified, showing that the single-star IMF for young star clusters is systematically steeper by Δα ≈ 0.5 between 0.1 and 1 M☉ than the Galactic-field IMF. Globular clusters appear to have systematically flatter IMFs than the Galactic-field IMF. The detection of ancient white-dwarf candidates in the Galactic halo and absence of associated low-mass stars suggests a radically different IMF for this ancient population. Star-formation in higher-metallicity environments appears to produce relatively more low-mass stars. This is an interesting trend, consistent with a systematic variation of the IMF as expected from theoretical arguments. The paper presents an analysis of the alpha-plot, which shows the power-law index α against the mean log10m of the mass range over which the index is measured. The alpha-plot clearly shows the flattening of the IMF for m ≲ 0.5 M☉. It also shows no systematic difference between MW and LMC populations. The models discussed show that unresolved binary systems mostly affect the region m ≲ 1 M☉. The data for which are listed in Table 1. The paper also discusses the effects of stellar evolution and the application of incorrect pre-main sequence and main-sequence evolutionary tracks, which can corrupt the masses inferred from observed quantities as the stars evolve to or along the main sequence. The paper concludes that the observed scatter in the alpha-plot is due to a combination of Poisson noise and dynamical evolution of star clusters, and that the fundamental limit within which any true variation becomes undetectable is approximately 0.5. The paper also discusses the implications of these findings for the IMF in different star-forming environments.