Question

Obtain the trend value for the following data using 4-yearly centered moving averages : \[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \mathbf{Years} & 1976 & 1977 & 1978 & 1979 & 1980 & 1981 & 1982 & 1983 & 1984 & 1985 \\ \hline \mathbf{Index} & 0 & 2 & 3 & 3 & 2 & 4 & 5 & 6 & 7 & 10 \\ \hline \end{array} \]

Hint:

The key to centered moving averages for an even period (like 4 years) is the two-step centering process. First, find the moving totals and place them *between* the years. Second, add adjacent pairs of these totals to center them *on* a year. Finally, divide by the total number of data points used (4 x 2 = 8).

Answer 1

\[ \begin{array}{|c|c|c|c|c|} \hline \mathbf{Year} & \mathbf{Index} & \mathbf{4\text{-}Yearly\ Moving\ Total} & \mathbf{2\text{-}Period\ Total\ (Centered)} & \mathbf{Trend\ Value\ (Centered\ Total / 8)} \\ \hline 1976 & 0 & & & \\ 1977 & 2 & 8 & & \\ 1978 & 3 & 10 & 18 & 2.25 \\ 1979 & 3 & 12 & 22 & 2.75 \\ 1980 & 2 & 14 & 26 & 3.25 \\ 1981 & 4 & 17 & 31 & 3.875 \\ 1982 & 5 & 22 & 39 & 4.875 \\ 1983 & 6 & 28 & 50 & 6.25 \\ 1984 & 7 & & & \\ 1985 & 10 & & & \\ \hline \end{array} \] The trend values are: \[ 2.25,\; 2.75,\; 3.25,\; 3.875,\; 4.875,\; 6.25 \] for the years 1978 to 1983, respectively.