Variance of N variables There is yet another important lesson to be taken away from this discussion which bears saying again more explicitly. Suppose we have the same example as above with Bob's blades of grass and we know the variance of the length of each blade $x_i$ is $\sigma^2$, then what's the variance of the sum of their lengths $X$? We've just said that it's $N\sigma^2$. That means that the standard deviation of $X$ is $\sqrt{N}\sigma$. That's right, it's $\sqrt{N}$ times the standard deviation of a single blade. You might think that the $\sqrt{N}$ is wrong and it should be $N$. Nope, it's $\sqrt{N}$. We'll talk more about this because now we're going to talk about "random walks".