**Measurement of Error**

Several of the common terms used to describe the degree of error associated with forecasting are standard error, mean squared error (or variance), and mean absolute deviation. In addition. tracking signals may be used to indicate the existence of any positive or negative bias in the forecast. Standard error is discussed in the section on linear regression later in the chapter. Since the standard error is the square root of a function, it is often more convenient to use the function itself. This is called the mean square error, or variance.

The mean absolute deviation (MAD) was at one time very popular but subsequently was ignored in favor of the standard deviation and standard error measures. In recent years however MAD has made a comeback because of its simplicity and usefulness in obtaining tracking signals. MAD is the average error in the forecasts. u ing absolute values. It is valuable because MAD, like the tan dard deviation, measures the dispersion (or variation) of observed values around some expected value. MAD is computed using the differences between the actual demand and the forecast demand without regard to whether it is negative or positive. It therefore is equal to the sum of the absolute deviations divided by the number of data points or stated in equation form:

When the errors that occur in the forecast are normally distributed (which is usually assumed to be the case), the mean absolute deviation relate to standard division as

**Conversely, I MAD 0.8 standard deviation** The standard deviation is the larger measure. If the MAD for a set of points was found to be 60 units. then the standard deviation would be 75 units. And. in the usual statistical manner, if control limits were set at ±3 standard deviations (or ±3.75 MADs). then 99.7 percent of the points would fall within these limits. (See Exhibit 9.9.)

A tracking signal is a measurement that indicates whether the forecast average is keeping pace with any genuine upward or downward changes in demand. As used in forecasting the tracking signal is the number of mean absolute deviations that the forecast value is above or below the actual occurrence. Exhibit 9.9 shows a normal distribution with a mean of zero and a MAD equal to one. Thus if we compute a tracking signal and find it equal to 2 we can conclude that the forecast model is providing forecasts that are quite a bit above the mean of the actual occurrences. A tracking signal can be calculated using the arithmetic sum of forecast deviations divided by the mean absolute deviation, or

It is important to note that while the MAD, being an absolute value, is always positive, the tracking’signal can take on positive and negative values. Exhibit 9.10 illustrates the procedure for computing MAD and the tracking signal for a six-month period where the forecast had been set at a constant 1.000 and the actual demands that occurred are as shown. In this example, the forecast, Rutherford average, was off by 66.7 units and the tracking signal was equal to 3.3 mean absolute deviations. . We can obtain a better interpretation of the MAD and tracking signal by plotting the points on a graph. While not completely legitimate from a sample size standpoint. we plotted each month in Exhibit 9.11 to show the drifting of the tracking signal. Note that it drifted from -1MAD to +3.3 MADs. This occurred because the actual demand was greater than the forecast in four of the six periods. If the actual demand doesn’t fall below the forecast to offset the continual positive RSFE, the tracking signal would continue to rise and we ‘ould conclude that the assumption that demand ~ 1,000 is a bad forecast. When the tracking signal exceeds a pre established limit (for example, ±2.0 or ±3.0), the manager should n eider changing the forecast model or the value of ex. Acceptable limits for the tracking signal depend on the size of the demand being forecast (high-volume or high-revenue items should be monitored frequently) and the amount