No, not always. It depends on the type of data you collect. If it is quantitative data, you will be able to calculate a mean. If it is qualitative data, a mean can't be calculated but you can describe the data in terms of a mode.
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In parametric statistical analysis we always have some probability distributions such as Normal, Binomial, Poisson uniform etc.In statistics we always work with data. So Probability distribution means "from which distribution the data are?
The variance is always positive. The variance is not directly related to the sign (nor magnitude) of the mean.
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The importance is that the sum of a large number of independent random variables is always approximately normally distributed as long as each random variable has the same distribution and that distribution has a finite mean and variance. The point is that it DOES NOT matter what the particular distribution is. So whatever distribution you start with, you always end up with normal.
The geometric mean, if it exists, is always less than or equal to the arithmetic mean. The two are equal only if all the numbers are the same.