# 3. In the third step, we center the 4-quarter moving average. The moving averages in column 5 fall in the middle of the quarters. Perhaps it would be better to have moving averages associated with each quarter. In order to focus our moving averages, we associate to each quarter the average of the two 4-quarter moving averages that fall just above and below it. For the 1991-III quarter, the resulting 4-quarter centered moving average is 2,104.25, that is (2,096.75 2,111.75) / 2. The other items in column 6 are calculated in the same way. Figure 15-7 illustrates how the moving average smooths the peaks and troughs in the original time series. The seasonal and irregular components were smoothed, and the resulting dotted line represents the cyclical and trend components of the series.

Step 4: Percentage of the real value with respect to the moving average (3) (7) 100 (6)

Suppose we are working with admission data from a hospital emergency room, and we want to calculate daily rates. In steps 1 and 2, we calculate the moving totals and the 7-day moving averages, and the moving averages are already centered (since the midpoint of a 7-day period is the fourth day). In this case, step 3 is not necessary. As long as the number of periods for which we want to obtain indices is odd (7 days in a week, 3 shifts in a day), we can skip step 3. However, when the number of periods is even (4 quarters, 12 months , 24 hours), then we must follow step 3 to center the moving averages obtained in step 2. 4. Next, we calculate the percentage of the real value with respect to the value of the moving average for each quarter of the time series that has a 4-quarter moving average element. This step allows us to recover the seasonal component for the quarters. We determine this percentage by dividing each of the actual quarterly values in column 3 of Table 15-10 by the corresponding 4-quarter centered moving average values found in column 6, and then multiplying the result by 100. For example , we find that the percentage corresponding to 1991-III is:

2,415 Real 100 100 Moving Average 2,104.2