Peak fitting is frequently performed in X-ray photoelectron spectroscopy (XPS). However, recent reports suggest that the current quality of this peak fitting is often inadequate in the scientific literature. Various statistical methods may be advantageously applied to an XPS peak fit to help determine the quality and validity of a fit. In this paper we describe a new statistical tool, which we believe will be helpful for determining the quality of protocols for fitting XPS data. This tool, box plots of random starting conditions, helps identify multiple local minima in a fit space. That is, ideally, different, reasonable starting conditions for a fit should lead to the same result, i.e., ideally, there should be a single global minimum for a fitting protocol. To determine whether a fit space contains multiple local minima, a series of reasonable, random starting conditions are chosen for the fit. If the boxes in the box plot of the peak areas of these fits are narrow, the different possibilities converge to a single global minimum. Conversely, if the boxes are wide, multiple local minima are present. Our approach is similar to the mathematical concept of 'disproof by contradiction'. It is demonstrated herein in four-and tencomponent fits to a moderately complex C 1s narrow scan. The resulting box plots compare favorably to traditional Monte Carlo analyses and uniqueness plots, although each of these statistical tools performs a different function/probes the fit space differently.