The answer will depend on what the question actually is!
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Yes. Normal (or Gaussian) distribution are parametric distributions and they are defined by two parameters: the mean and the variance (square of standard deviation). Each pair of these parameters gives rise to a different normal distribution. However, they can all be "re-parametrised" to the standard normal distribution using z-transformations. The standard normal distribution has mean 0 and variance 1.
It means independent of the underlying distribution.
A variable is a measure that can take different values. How often it can take these different values defines its distribution. Mode, median and mean are three common measures of central tendency of distributions.
When comparing the spread or variability rather than the location or mean. For example, men's heights and women's heights. You "know" that, on average, men will be taller but you may want to see if the variability within the two sets is the same or different.
Easy. The mean deviation about the mean, for any distribution, MUST be 0.