MAD/Mean Ratio - advantages/disadvantages of using average or sum

Question

The calculation the MAD (Mean Absolute Deviation)/Mean ratio is this, according to the title: $$ \frac{\overline{\left | Forecast - Demand \right |}}{\overline{Demand}}$$ However, the calculation is often shown as this, because of the derivation from wMAPE (weighted Mean Absolute Percentage Error), with sum not average: $$ \frac{1}{\sum Demand}\sum_{}^{}Demand\frac{\left | Forecast - Demand \right |}{Demand}=\frac{\sum_{}^{}\left | Forecast - Demand \right |}{\sum_{}^{}Demand} $$ I understand why it's shown as that, because of the derivation, but that means that the title (mean) and the calculation (sum) don't match up. Are there pros and cons to using mean or sum? The key ones that I can think for using average are:

• Fair comparison between numerators and denominators of different populations, because sum is affected by number of observations
• Average is less affected by missing data than sum is, when you apply that aggregation across a column

Answer 1

If you are a little more careful and explicit in your formulas, it might well get a lot clearer.

Specifically, one typically calculates means or sums over forecasts and actuals for the same periods (as opposed, say, to calculating the MAD over the forecast horizon, but the mean of actuals over the training sample). And then both the numerator and the denominator have the same number of summands, say $N$. But then the sums you have yield the exact same ratio as dividing the means instead - because going from the sums to averages only means that we divide both the numerator and the denominator by $N$, and this cancels.

Some more on this can be found in Kolassa & Schütz (2007).