Measurement of Trend: Method of Moving Averages

The method of moving averages smooths a series by replacing each value with the average of itself and its neighbours, so that short-term fluctuations cancel out and the trend stands clear. A moving average of period $k$ is a series of successive arithmetic means, each taken over $k$ consecutive values; as we step one period forward in time, the oldest value is dropped and the next one added — the 'window' of $k$ values moves along the series.

Odd period (e.g. 3-yearly, 5-yearly). When $k$ is odd the average falls naturally against the middle year of the group, so no extra step is needed. For a 3-yearly moving average:

• form the 3-yearly moving totals $y_1+y_2+y_3,\ y_2+y_3+y_4,\ \ldots$;
• divide each total by $3$;
• place each result against the middle year of its group.

The first and last years get no moving-average value.

Even period (e.g. 4-yearly). When $k$ is even, the average of a group falls between two years, not against any actual year, so a second smoothing — centering — is needed:

• form the 4-yearly moving totals and write each between the middle two years;
• add each pair of consecutive 4-yearly totals to get 2-item moving totals (this is the centering step);
• divide these centered totals by 8 ($= 4 \times 2$) to get the centered 4-yearly moving average, which now falls against an actual year.