why it is said that mean deviation does not possess mathematical properties
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Both answer how far your values are spread around the mean of the observations.
An observation that is 1 under the mean is equally "far" from the mean as a value that is 1 above the mean. Hence you should neglect the sign of the deviation. This can be done in two ways:
Calculate the absolute value of the deviations and sum these.
Square the deviations and sum these squares. Due to the square, you give more weight to high deviations, and hence the sum of these squares will be different from the sum of the means.
After calculating the "sum of absolute deviations" or the "square root of the sum of squared deviations", you average them to get the "mean deviation" and the "standard deviation" respectively.
The mean deviation is rarely used.
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