A 2022 paper arXiv:2009.10305v4 introduced the notion of true positive and negative skewness for continuous random variables via Fréchet $p$-means. In this work, we find novel criteria for true skewness, establish true skewness for the Weibull, Lévy, skew-normal, and chi-squared distributions, and discuss the extension of true skewness to discrete and multivariate settings. Furthermore, some relevant properties of the $p$-means of random variables are established.