Clustering, as a traditional machine learning method, is still playing a significant role in data analysis. The most of clustering algorithms depend on a predetermined exact number of clusters, whereas, in practice, clusters are usually unpredictable. Although elbow method is one of the most commonly used methods to discriminate the optimal cluster number, the discriminant of the number of clusters depends on manual identification of the elbow points on the visualization curve, which will lead to the experienced analysts not being able to clearly identify the elbow point from the plotted curve when the plotted curve being fairly smooth. To solve this problem, a new elbow point discriminant method is proposed to work out a statistical metric estimating an optimal cluster number when clustering on a dataset. Firstly, the average degree of distortion obtained by Elbow method is normalized to the range of 0 to10; Secondly, the normalized results are used to calculate Cosine of intersection angles between elbow points; Thirdly, the above calculated Cosine of intersection angles and Arccosine theorem are used to compute the intersection angles between elbow points; Finally, the index of the above computed minimal intersection angles between elbow points is used as the estimated potential optimal cluster number. The experimental results based on simulated datasets and a public well-known dataset (Iris Dataset) demonstrated that the estimated optimal cluster number output by our newly proposed method is better than widely used Silhouette method.