Find the four-yearly centred moving averages for the following data.

| Year | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 |
|---|---|---|---|---|---|---|---|---|
| Value | 2 | 6 | 1 | 5 | 3 | 7 | 2 | 6 |

Question

Find the four-yearly centred moving averages for the following data.

| Year | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 |
|---|---|---|---|---|---|---|---|---|
| Value | 2 | 6 | 1 | 5 | 3 | 7 | 2 | 6 |

Accepted Answer

The period $k = 4$ is **even**, so a 4-yearly average falls *between* two years. We therefore add a **centering** step: pair consecutive 4-yearly totals and divide by $8\ (= 4 	imes 2)$.

| Year | Value | 4-yearly moving total | 2-item (centered) total | Centred average $(\div 8)$ |
|---|---|---|---|---|
| 2010 | 2 | — | — | — |
| 2011 | 6 | — | — | — |
| | | $2+6+1+5 = 14$ | | |
| 2012 | 1 | | $14+15 = 29$ | $29/8 = 3.625$ |
| | | $6+1+5+3 = 15$ | | |
| 2013 | 5 | | $15+16 = 31$ | $31/8 = 3.875$ |
| | | $1+5+3+7 = 16$ | | |
| 2014 | 3 | | $16+17 = 33$ | $33/8 = 4.125$ |
| | | $5+3+7+2 = 17$ | | |
| 2015 | 7 | | $17+18 = 35$ | $35/8 = 4.375$ |
| | | $3+7+2+6 = 18$ | | |
| 2016 | 2 | — | — | — |
| 2017 | 6 | — | — | — | …

Years	1974	1975	1976	1977	1978	1979	1980	1981	1982
Production	0	4	9	9	8	5	4	8	10
Find trend values for the above data by using 5 yearly moving averages.

Year	Value	4-yearly moving total	2-item (centered) total	Centred average $(\div 8)$
2010	2	—	—	—
2011	6	—	—	—
		$2+6+1+5 = 14$
2012	1		$14+15 = 29$	$29/8 = 3.625$
		$6+1+5+3 = 15$
2013	5		$15+16 = 31$	$31/8 = 3.875$
		$1+5+3+7 = 16$
2014	3		$16+17 = 33$	$33/8 = 4.125$
		$5+3+7+2 = 17$
2015	7		$17+18 = 35$	$35/8 = 4.375$
		$3+7+2+6 = 18$
2016	2	—	—	—
2017	6	—	—	—

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