At present, data with sharp peaks, thick tails and skew appear in medicine, sociology, biology and other fields. For such data, adopting the Skew Laplace normal data to fit will get more accurate results. At the same time, in statistics, abnormal points or strong influence points will have a great impact on the results of statistical diagnosis, and hence the diagnosis of abnormal points or strong influence points is particularly important. Common methods such as Likelihood distance, Cook distance, etc., study the impact of deleting a point (group) on the regression analysis and predicted value. In the reasearch, the influence of Pena distance on the regression value and predicted value of a specific point after the deletion of each point in the sample is studied. Moreover, the influence of Pena distance on the Location regression model in the Skew Laplace normal data is studied, and the EM algorithm is applied to make a statistical diagnosis of the location regression model in the Skew Laplace normal distribution. The expression of the Pena distance and the discrimination method of high-leverage outliers under the location regression model with Skew Laplace normal data are obtained. The comparsion shows the Pena distance is compared with Cook distance and Likelihood distance, and the Pena distance is better than Cook distance and Likelihood distance in some cases. Through Monte Carlo simulation and a real example analysis, the proposed model and the proposed method are shown to be reasonable.