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\displaystyle\sum_{i-1}^{n} X_i = Normal \bigg(\displaystyle\sum_{i-1}^{n} \mu_i,\sqrt {\displaystyle\sum_{i-1}^{n} \displaystyle\sum_{j-1}^{n}\sigma_{ij}}\bigg)

The formula says that the Normal distribution has a mean equal to the sum of the means for the individual distributions being added together. It also says that the variance (the square of the standard deviation in the formula) of the Normal distribution is equal to the square of the covariance terms between each variable. The covariance terms s_ij are calculated as follows:

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If we have data sets for the variables being modeled, EXCEL can calculate the covariance and correlation coefficients using the functions COVAR( ) and CORREL( ) respectively. If we were thinking of using a rank order correlation matrix, each element corresponds reasonably accurately to r_ij for roughly Normal distributions (at least, not very heavily skewed distributions), so the standard deviation of the Normally distributed sum could be calculated directly from the correlation matrix, as shown in model Image RemovedCovariance _ and _ correlation. This model actually uses some pretty skewed distributions and comes up with a good approximation to the sum one gets from running a simulation of each variable.

The links to the Covariance and correlation software specific models are provided here: