In the analysis of neural data, measures of non-Gaussianity are generally applied in two ways: as tests of normality for validating model assumptions and as Independent Component Analysis (ICA) contrast functions for separating non-Gaussian signals. Consequently, there is a wide range of methods for both applications, but they all have trade-offs. We propose a new strategy that, in contrast to previous methods, directly approximates the shape of a distribution via Hermite functions. Applicability as a normality test was evaluated via its sensitivity to non-Gaussianity for three families of distributions that deviate from a Gaussian distribution in different ways (modes, tails, and asymmetry). Applicability as an ICA contrast function was evaluated through its ability to extract non-Gaussian signals in simple multi-dimensional distributions, and to remove artifacts from simulated electroencephalographic datasets. The measure has advantages as a normality test and, for ICA, for heavy-tailed and asymmetric distributions with small sample sizes. For other distributions and large datasets, it performs comparably to existing methods. Compared to standard normality tests, the new method performs better for certain types of distributions. Compared to contrast functions of a standard ICA package, the new method has advantages but its utility for ICA is more limited. This highlights that even though both applications -- normality tests and ICA -- require a measure of deviation from normality, strategies that are advantageous in one application may not be advantageous in the other. Here, the new method has broad merits as a normality test but only limited advantages for ICA.