MD - Mean Absolute Difference

Returns the mean absolute difference of the input data series.

Syntax

MD(X)

X

is the input data sample (one/two-dimensional array of cells (e.g., rows or columns)).

Remarks

1. The input time series data may include missing values (e.g., #N/A, #VALUE! #NUM! empty cell), but those will not be included in the calculations.
2. The sample mean difference (MD) is computed as follows: $$\Delta = {\rm{MD}} = \frac{{\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n ||{{x_i} - {x_j}}|| } }}{{n \times \left( {n - 1} \right)}}$$ Where:
   • $x_i$ is the value of the i-th non-missing observation.
   • $n$ is the number of non-missing observations in the sample.
3. The mean absolute difference is the product of the sample mean, and the relative mean difference (RMD) and so can also be expressed in terms of the Gini coefficient: \[{\rm{MD}} = 2 \times G \times \bar x\] Where:
   • $\bar x$ is the arithmetic sample mean.
   • $G$ is the Gini Coefficient.
4. Because it is related to the Gini coefficient, the mean difference is called the "Gini mean difference." It is also known as the "Absolute mean difference."
5. The sample mean difference is not dependent on a specific measure of central tendency like the standard deviation.
6. The mean difference of a sample is an unbiased and consistent estimator of the population mean difference.

Related Links

• Wikipedia - Mean difference.

References

• Hamilton, J.D.; Time Series Analysis, Princeton University Press (1994), ISBN 0-691-04289-6.
• Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0-471-69074.