To discern how scattered observations are about some central value, choose either the median or the mean for the central value. If you should choose the mean, average deviation from the mean value is one measure of dispersion. The average deviation (also known as the mean deviation) is the average absolute difference between the observed values and the arithmetic mean (average) for all values in the data set. Sometimes, the calculation is performed using distance from the median instead of the mean (See AVEDEVMEDIAN). The term average deviation is something of a misnomer, since by definition of the mean the sum of all deviations about the mean are zero except for possible rounding errors. The true average deviation cannot be used since that sum is always zero, which says nothing about how far the average observation is from the mean. Use the absolute value of the difference between each observation and the mean to find a correct answer. While the mean deviation is sometimes called the mean absolute deviation, this usage is not strictly correct unless the data is categorized into bins first. For estimating population standard deviation in a normal population, the mean deviation is not as efficient as the sample standard deviation.
In plain English, we take the sum of the absolute value of all observations minus the mean and divide that sum by the number of observations (N).
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Flaws exist in using this calculation. If a sample is taken and the
accuracy of a process using our sample is estimated, a different estimate
if the result if the sample is cut into two smaller samples and the calculation
is performed on the subsamples. The amount of underestimation is not only
a function of the sample size, but also a function of the probability
of the distribution of the errors in measurement.
This strongly indicates against the use of small samples for performing
this calculation.
There are also some special merits in this calculation. It is not unheard
of to be dealing with a distribution whose variance does not exist. In
this case, all higher moments and derivative measures such as the standard
deviation are useless as a measure of the data's width around its mean.
Attempted calculations of the statistics using higher moments produce
random results. The average deviation does not suffer from this defect
but is a good measure for estimation for broad distributions with a significant
number of outlier points. Higher order moments or statistics involving
higher powers of the input data are less robust than lower moments or
statistics that involve only linear sums or counting.