For a normal distribution is there a relation between the FWHM of the bell curve and the standard deviation of the numbers in the normal distribution?

Question

suppose we are measuring a variable repeatedly, and due to random measurement errors, we get different values each time. Since the error is random, we would expect the measured values to follow a normal distribution.

So if we plot the number of occurrences of each value wrt the value, we will get a bell curve of the shape $y = \frac{1}{\sigma\sqrt{2\pi}} e^\frac{-(x-\mu)^2}{2\sigma^2}$, where $\sigma$ is the standard deviation, say SD1.

Now, say the standard deviation of the measured values is SD2, given by $\sqrt{\frac{\Sigma(x-\mu)^2}{N}}$.

I would have expected SD1 = SD2, but it's not. Can someone explain why?

Thanks.