In turn, the covariance matrix for a pool of dice is just the sum of the covariance matrices of the individual dice; like the variance of the univariate case, the covariance matrix of a dice pool "stacks" additively.
The center of each ellipse is simply the mean result of the dice pool.
The center of each ellipse is simply the mean result of the dice pool.
Note that the [68–95–99.7 rule](https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule) only applies to univariate distributions. With bivariate distributions, the rule is closer to 39-86-99, following the [chi distribution](https://en.wikipedia.org/wiki/Chi_distribution). So about 39% of the results will fall within the ellipse, and 86% will fall within twice as far. However, if you only consider a single dimension at a time, then the original 68–95–99.7 rule applies.
Note that the [68–95–99.7 rule](https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule) only applies to univariate distributions. With bivariate distributions, the rule is closer to 39-86-99, following the [chi distribution](https://en.wikipedia.org/wiki/Chi_distribution). So about 39% of the results will fall within the ellipse, and 86% will fall within twice as far. However, if you only consider a single dimension at a time, then the original 68–95–99.7 rule applies.
