- The mean error and the percentage error
- The mean absolute deviation
- The mean squared error
- The tracking signal

**Mean error**

The mean error is the average of the forecast error for the number of data points considered. Its equation is:

Where S is the actual sales, F is the forecast and N the number of data points.

This indicator assesses any systematic deviation in the forecast, positive or negative. Note that the positive deviations can be balanced by the negative deviations, and as a result the mean error could be low even though big variations (positive and negative) occur.

So we better use this indicator as an evaluation of the model centering, meaning if globally the forecasts are centered on actual sales over a period of time.

The percentage error is the similar to the mean error except that we compare the sum of (S-F)/S instead of (S-F) to get a mean percentage error.

**Mean absolute deviation**

The mean absolute deviation is the mean of all the individual forecast errors:

This indicator checks all the deviations from the actual sales whenever positive or negative, and assesses the model spread.

Note as well that a model could have a low spread although the forecast are always higher (or lower) than actual sales, meaning that the model is accurate but not centered.

We commonly use the MAD indicator for the standard deviation (see below) using a good proxy: Standard error = 4/3 * MAD

Mean squared error

The mean squared error is given by:

This indicator is more appropriate if the errors are small and less frequent but larger. We commonly use the standard deviation, which is the square root of the mean squared error.

Also since the standard deviation could be estimated trough the mean absolute deviation, we better compute the first two indicators to make a first check.

**Tracking signal**

Once a forecast model is developed it should indicate if the actual demand is following the forecast, and indicate if any deviation in order to correct the model.

The tracking signal formula is the following at a given period (T):

Tracking signal (T) = Cumulative forecast error (period T) / Mean absolute deviation (period T)

We decide on an upper limit and a lower limit to track and control the forecast deviation over time.**Example**

Let’s take the following dataset:

to compute the 4 indicators:

Now let’s add the tracking signal, and let fix the upper limit at 4 and the lower limit at -4:

We can notice that the forecast is going out of boundaries at the end, meaning that the forecast model needs to be adjusted.

The last 2 forecast periods are getting out of track as the error is about 10% per period in the two consecutive periods, meaning that there is consequent change in the demand.