The Jarque-Bera test [JAR1] is a two-sided goodness-of-fit test for Normality suitable when a fully-specified null distribution is unknown and its parameters must be estimated. It is based on the sample skewness and kurtosis and was developed for use in connection with regression analysis. The test statistic is

where n is the sample size, s is the sample skewness, and k is the sample kurtosis. For (very) large sample sizes, the test statistic has a chi-square distribution with two degrees of freedom, but more generally its distribution is obtained via Monte Carlo simulation. The test is, by definition, particularly well suited to evaluating the departure of the sample skewness and kurtosis from that expected under the assumption of a Normal distribution.

Reference

[JAR1] Jarque C M, Bera A K (1987) A test for normality of observations and regression residuals. Intl Statistical Review, 55(2), 163–172