A scatter plot shows the relationship between two quantitative variables 𝒙 and 𝒚. Sometimes, we can predict one variable as a linear function of the other using the least squares regression lines of 𝒚 on 𝒙 or 𝒙 on 𝒚. These two regression lines together suffice to identify the mean vector, the coefficient of determination, the correlation coefficient, and the ratio of the standard deviations (SD). So, do our proposed summary ellipses. Additionally, the inner ellipse reveals the SDs and the outer ellipse flags potential outliers.
Sarkar, J., and Rashid, M. (2022), “Visualizing Bivariate Statistics Using Ellipses Over a Scatter Plot," Journal of Probability and Statistical Science 20(1), 150-165, August 2022.