 In particular if all of the "X""i" have the same variance, then this simplifies to

The reason not to subtract off 3 is that the bare fourth moment better generalizes to multivariate distributions, especially when independence is not assumed.The cokurtosis between pairs of variables is an order four tensor.For a bivariate normal distribution, the cokurtosis tensor has off-diagonal terms that are neither 0 nor 3 in general, so attempting to "correct" for an excess becomes confusing.