[This article was first published on Statforbiology, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)Want to share your content on R-bloggers? click here if you have a blog, or here if you don't. When we have a multi-trait table of data, we might be interested in investigating variability structure and the reciprocal relationships (whether causal or non-causal) among variables. For example, let’s consider the ‘mtcars’ dataset, shown below:data(mtcars)head(mtcars)## mpg cyl disp hp drat wt qsec vs am gear carb## Mazda RX4 21.0 6 160 110 3.90 2.620 16.46 0 1 4 4## Mazda RX4 Wag 21.0 6 160 110 3.90 2.875 17.02 0 1 4 4## Datsun 710 22.8 4 108 93 3.85 2.320 18.61 1 1 4 1## Hornet 4 Drive 21.4 6 258 110 3.08 3.215 19.44 1 0 3 1## Hornet Sportabout 18.7 8 360 175 3.15 3.440 17.02 0 0 3 2## Valiant 18.1 6 225 105 2.76 3.460 20.22 1 0 3 1Considering the different cars (along the rows), we might be interested in investigating the structure of variability for fuel consumption (mpg), number of cylinders (cyl), displacement (disp) and horsepower (hp). How much does each variable change across the different cars? How much does each variable change when another variable changes? Is there any kind of joint variation, either positive or negative?One suitable statistic to answer these question is the covariance, that is:where and are the means for each variable and is the number of individuals. In words, covariance is the sum of products of the ith deviations from the mean for both variables. In this setting, the variance is just the covariance of a variable with itself.For a data table, variances and covariances can be calculated by using the cov() function, that returns a matrix (the variance-covariance matrix, indeed):matr