These days the interest of quality leads to the necessity of control charts for monitoring the process in various fields of practical applications. The $overline{X}$ chart is one of the most widely used tools for quality control that also performs well under the normality of quality characteristics. However, quality characteristics tend to have nonnormal properties in real applications. Numerous recent studies have tried to find and explore the performance of $overline{X}$ chart due to non-normality; however previous studies numerically examined the effects of non-normality and did not provide any theoretical justification. Moreover, numerical studies are restricted to specific type of distributions such as Burr or gamma distribution that are known to be flexible but can hardly replace other general distributions. In this paper, we approximate the false alarm rate(FAR) of the $overline{X}$ chart using the Edgeworth expansion up to 1/n-order with the fourth cumulant. This allows us to examine the theoretical effects of nonnormality, as measured by the skewness and kurtosis, on $overline{X}$ chart. In addition, we investigate the effect of skewness and kurtosis on $overline{X}$ chart in numerical studies. We use a skewed-normal distribution with a skew parameter to comprehensively investigate the effect of skewness.