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The mean deviation of a set of data is defined as the arithmetic mean of deviations measured either from the mean or from the median, all deviation being counted as positive. The reason to count the deviations as positive i.e. to disregard the algebraic signs is to avoid the difficulty arising from the property that the sum of deviations of   the observations from their mean is zero. The symbolic definition of the mean deviation from the mean is

M. D =∑ |Xi – X| /n             for sample data,

M.D =∑| Xi – X|/N              for population data

Where and (pronounced “mod. Deviations”) indicate the absolute deviations of the observations from the mean of a sample and population respectively. It is more appropriate to call it the mean absolute deviation (M.A.D). For the data organized in to grouped frequency distribution having k classes with midpoints x1, x2,….. xk  and the corresponding frequencies     fi,f2,….fk(∑fi=n), the mean deviation of the sample is given by

M.D=∑fi|xi-x|/n

The mean deviation is also defined in terms of absolute deviations from the median in a similar way. Theory tells us that the mean deviation is least when the deviations are measured from the median. But in practice, it is generally calculated from the arithmetic mean. The mean deviation gives more information than the range or the quartile deviation as it is based on all observed values. It is easily calculated and readily understood as it is not amenable to mathematical treatment, its usefulness is limited. Mean deviation does not give undue weight to occasional large deviations, so it is used in situations where such deviations are likely to occur. It is unsatisfactory for statistical inference.

Mean deviation is an absolute measure of dispersion. Its relative measure, known as the co-efficient of mean deviation, is obtained by dividing the mean deviation by the average used in the calculation of deviations. Thus,

Coefficient of Mean Deviation . M.D = M.D/ Mean     OR      M.D/Median

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