Does the value of a standard deviation have any meaning

Question

I understand that the standard deviation corresponds deviance from the mean but does the actual value of the standard deviation have any direct meaning? For example if you have a $\sigma$ of 10, does 10 mean anything by itself? (relative to the problem)

Answer 1

A little hard to understand what you mean by the words "mean anything by itself"... Anyway, relative to some real life problem one shall note that the units of measurement of the standard deviation are the same as of the random variable itself. So in some very crude meaning it can be understood as the range of the random outcomes that you'll get on average during some long enough set of observations.
For example if the random outcome is the voltage of some device (let's assume normally distributed with mean $220$ Volts and variance $100$ Volts), then applying the three-sigma rule one can say that almost all the observable realizations ($99.7\%$) will (or shall) lie inside the $220\pm 30$ Volts. Or rephrasing it (if you have no outliers) one can expect that with probability $99.7\%$ all the measurements will be in the range of $220\pm 30$ Volts.