Question

Obtain the trend value for the following data using 4-yearly centered moving averages:

Years	1976	1977	1978	1979	1980	1981	1982	1983	1984	1985
Index	0	2	3	3	2	4	5	6	7	10

Answer 1

Step 1: Understanding the 4-Yearly Centered Moving Average

A 4-yearly moving average is calculated by taking the average of four consecutive years’ index values. Since we have an even number of years, the average is centered between the second and third years of each group.

Step 2: Calculate 4-Year Moving Averages

Years	Index	4-Year Moving Sum	4-Year Moving Average
1976	0
1977	2
1978	3	0 + 2 + 3 + 3 = 8	`8/4 = 2.0`
1979	3	2 + 3 + 3 + 2 = 10	`10/4 = 2.5`
1980	2	3 + 3 + 2 + 4 = 12	`12/4 = 3.0`
1981	4	3 + 2 + 4 + 5 = 14	`14/4 = 3.5`
1982	5	2 + 4 + 5 + 6 = 17	`17/4 = 4.25`
1983	6	4 + 5 + 6 + 7 = 22	`22/4 = 5.5`
1984	7	5 + 6 + 7 + 10 = 28	`28/4 = 7.0`
1985	10	-	-

Step 3: Final Trend Values

The trend values, calculated as 4-yearly moving averages, are:

• 1978 → 2.0
• 1979 → 2.5
• 1980 → 3.0
• 1981 → 3.5
• 1982 → 4.25
• 1983 → 5.5
• 1984 → 7.0

Conclusion

The 4-yearly centered moving average smooths short-term fluctuations in the data and provides a clear trend pattern.