Q.Explain why a 3-yearly moving-average trend series always has fewer values than the original time series it was calculated from.

A moving average needs a full block of neighbouring years on both sides of the year it is centred on, and the first and last years in any series simply do not have that.

A $k$-yearly moving average places its computed value against the middle year of a block of $k$ consecutive years. For a 3-yearly average, every trend value needs one year before it and one year after it in the original data to form its block of 3.

The very first year of the series has no year before it, so no 3-year block can be centred on it; symmetrically, the very last year has no year after it. Every other (interior) year does have a complete block on both sides, so it does get a trend value.

This means exactly one value is lost at the start and one at the end — 2 values lost in total for a 3-yearly average out of, say, 7 original years, leaving 5 trend values (as seen in Worked Example 1, where 2015 and 2021 have no trend value but 2016-2020 do). In general, a $k$-yearly moving average loses $(k-1)/2$ years at each end, i.e. $(k-1)$ values lost in total — which is precisely why an even wider moving average (5-yearly, 7-yearly) smooths the series more but sacrifices more years at the boundaries.

Assuming the lost values can simply be estimated by extending the pattern (the moving-average method genuinely has no trend value there, and reporting one would misrepresent the method); and forgetting that a wider moving average (5-yearly, 7-yearly) loses even more boundary years than a 3-yearly one.