You misunderstood. The formula is most probably correct. My question was that why do you think that standard deviation is more interesting/useful than the extrema of the deviation?

For example I have a machine which manufactures screws.

In general I want to know when the tool in this machine gets worn out so I measure the screws at an appropriate point and I calculate the standard deviation from the size given on the technical drawing.

But if for some unknown reason the tool breaks (the standard deviation is still in acceptable range) I would probably start looking at the min/max deviation which may or may not shed some light one what circumstances caused the problem.