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The problem of measuring the disparity of a particular probability density function from a normal one has been addressed in several recent studies. The most used technique to deal with the problem has been exact expressions using information measures over particular distributions. In this paper, we consider a class of asymmetric distributions with a normal kernel, called Generalized Skew-Normal (GSN) distributions. We measure the degrees of disparity of these distributions from the normal distribution by using exact expressions for the GSN negentropy in terms of cumulants. Specifically, we focus on skew-normal and modified skew-normal distributions. Then, we establish the Kullback–Leibler divergences between each GSN distribution and the normal one in terms of their negentropies to develop hypothesis testing for normality. Finally, we apply this result to condition factor time series of anchovies off northern Chile.