Normal mixture models provide the most popular framework for modelling heterogeneity in a population with continuous outcomes arising in a variety of subclasses. In the last two decades, the skew normal distribution has been shown beneficial in dealing with asymmetric data in various theoretic and applied problems. In this article, we address the problem of analyzing a mixture of skew normal distributions from the likelihood-based and Bayesian perspectives, respectively. Computational techniques using EM-type algorithms are employed for iteratively computing maximum likelihood estimates. Also, a fully Bayesian approach using the Markov chain Monte Carlo method is developed to carry out posterior analyses. Numerical results are illustrated through two examples.