The standard deviation is the square root of the variance, as you might know. The variance is calculated by summing up the deviation from the mean and square it, and dividing it by $n$.
$$\sigma^2 = \frac{\sum_{i}^{n} (x_i-\mu)^2 }{n}$$
Every difference $(x-\mu)$ is squared. When you take the square root of the variance, it is not the same as taking the square root of every $(x-\mu)^2$ and sum it up afterwards...
Hope this helps,
Basil