# Concordance correlation always less than or equal to pearson correlation

Can someone explain intuitively (or even mathematically) why Lin's concordance correlation coefficient (CCC) is always less than or equal to pearson's correlation coefficient (PCC). A related query exists as How to compare concordance correlation coefficient to Pearson's r? but there is no quantitative comparison between CCC and PCC. The reason why I say $$CCC<=PCC$$ is based on formula of CCC as $$CCC = 2\times PCC\times \sigma_x \times \sigma_y /(\sigma_x^2+\sigma_y^2+(\mu_x-\mu_y)^2)$$, where x and y are the 2 variables on which we evaluate correlation. Clearly, if $$\mu_x = \mu_y$$ and $$\sigma_x = \sigma_y$$, then $$CCC=PCC$$. For all other scenarios, $$CCC, based on arithmetic and geometric mean relationships.