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The answer depends on absolute deviation from what: the mean, median or some other measure.
Suppose you have n observations, x1, x2, ... xn and you wish to calculate the sum of the absolute deviation of these observations from some fixed number c.
The deviation of x1 from c is (x1 - c).
The absolute deviation of x1 from c is |x1 - c|. This is the non-negative value of (x1 - c). That is,
if (x1 - c) ≤ 0 then |x1 - c| = (x1 - c)
while
if (x1 - c) < 0 then |(x1 - c)| = - (x1 - c).
Then the sum of absolute deviations is the above values, summed over x1, x2, ... xn.
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Can the standard deviation or variance be negative?
No, a standard deviation or variance does not have a negative sign. The reason for this is that the deviations from the mean are squared in the formula. Deviations are squared to get rid of signs. In Absolute mean deviation, sum of the deviations is taken ignoring the signs, but there is no justification for doing so. (deviations are not squared here)
What is the arithmetic mean of the absolute values of the deviations of the individual values from the mean in a data set?
It is the mean absolute deviation.
For which measure of central tendency will the sum of the deviations always be zero?
Mean
What is the sum of the squared deviations from the mean divided by the count minus one?
variation
What is the Sum of deviation from the mean is?
The sum of deviations from the mean, for any set of numbers, is always zero. For this reason it is quite useless.
