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SD assumes the distribution of observations is symmetrical around the mean; i.e., that the normal distribution is the correct statistical model to use.

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Whether or not the assumption of normality is justified, calculating the standard deviation requires no knowledge of the shape of the distribution. It would be a valid statistic, whatever the distribution. Or am I crazy?

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Not crazy. Inferential statistics using the mean and standard deviation assume a normal distribution, but as descriptive statistics the standard deviation and mean require no distributional assumptions.