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Bivariate skewness, kurtosis and surface plot
Journal of the Korean Data & Information Science Society 2017;28:959-70
Published online September 30, 2017
© 2017 Korean Data & Information Science Society.

Chong Sun Hong1 · Jae Hyun Sung2

12Department of Statistics, Sungkyunkwan University
Correspondence to: Chong Sun Hong
Professor, Department of Statistics, Sungkyunkwan University, Seoul 03063, Korea. E-mail:
Received August 10, 2017; Revised September 15, 2017; Accepted September 18, 2017.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License ( which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this study, we propose bivariate skewness and kurtosis statistics and suggest a surface plot that can visually implement bivariate data containing the correlation coefficient. The skewness statistic is expressed in the form of a paired real values because this represents the skewed directions and degrees of the bivariate random sample. The kurtosis has a positive value which can determine how thick the tail part of the data is compared to the bivariate normal distribution. Moreover, the surface plot implements bivariate data based on the quantile vectors. Skewness and kurtosis are obtained and surface plots are explored for various types of bivariate data. With these results, it has been found that the values of the skewness and kurtosis reflect the characteristics of the bivariate data implemented by the surface plots. Therefore, the skewness, kurtosis and surface plot proposed in this paper could be used as one of valuable descriptive statistical methods for analyzing bivariate distributions.
Keywords : Box plot, mahalanobis distance, mixture, quantile vector, surface plot