# Mean absolute percentage error

The mean absolute percentage error (MAPE), also known as mean absolute percentage deviation (MAPD), is a measure of accuracy of a method for constructing fitted time series values in statistics, specifically in trend estimation. It usually expresses accuracy as a percentage, and is defined by the formula:

$\mbox{M} = \frac{1}{n}\sum_{t=1}^n \left|\frac{A_t-F_t}{A_t}\right|,$

where At is the actual value and Ft is the forecast value.

The difference between At and Ft is divided by the Actual value At again. The absolute value in this calculation is summed for every fitted or forecasted point in time and divided again by the number of fitted points n. Multiplying by 100 makes it a percentage error.

Although the concept of MAPE sounds very simple and convincing, it has two major drawbacks in practical application:[citation needed]

• If there are zero values (which sometimes happens for example in demand series) there will be a division by zero
• When having a perfect fit, MAPE is zero. But in regard to its upper level the MAPE has no restriction.

When calculating the average MAPE for a number of time series there might be a problem: a few of the series that have a very high MAPE might distort a comparison between the average MAPE of time series fitted with one method compared to the average MAPE when using another method. In order to avoid this problem other measures have been defined, for example the sMAPE (symmetrical MAPE), weighted absolute percentage error (WAPE), real aggregated percentage error (RAPE),or a relative measure of accuracy (ROMA).[citation needed]

## Alternative MAPE definitions

Problems can occur when calculating the MAPE value with a series of small denominators. A singularity problem of the form 'one divided by zero' and/or the creation of very large changes in the Absolute Percentage Error, caused by a small deviation in error, can occur.

The difference with the original formula is that each Actual Value (At) of the series is replaced by the average Actual Value (Āt) of that series. Hence, the distortions are smoothed out. This alternative is still being used for measuring the performance of models that forecast spot electricity prices.[1]

Another variant is the sum of absolute differences is divided by sum of actual values. Sometimes, it is referred as WAPE.