In my humble opinion, what people mean with a statistical variable is the same as a random variable, which is explained well in wikipedia: http://en.wikipedia.org/wiki/Random_variable.
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Because its the group for which the idependent variable is help constand in a statistical study.
No, it is a statistical measure of the spread of a distribution of some variable.
A dummy variable is a variable that takes on the values 1 and 0; 1 means something is true. These are used in statistical analyses.
Standard deviation of 0 can only be attained if all observations are identical. That is, the variable in question has just one possible value so statistical considerations are irrelevant.