Given the mean high temperature for my city for a particular month, I'd like to provide a sense of how much the high might typically vary during the month, e.g. "80°F (+/-10°F)" or alternately "70–90°F". Would I use the standard deviation or the variance?
The measure of variability is the standard deviation. The standard deviation is in the same units as the mean. The variance is not. Standard deviation is not the only measure that could be used. The mean absolute deviation is also possible. I can think of two reasons why standard deviation is favored.
The Chebyshev inequality tells you for all distributions with finite variance a lower bound on the percentage of the distribution that falls within k standard deviations of the mean.
For a Gaussian distribution the actual percentage of the distribution that falls within k standard deviations of the mean can readily be computed.