The 4-year centred moving averages for the following data are
| Year | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 |
| Sales (in thousand units) | 30 | 34 | 38 | 42 | 40 | 44 |
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 37.25, 39.75
For an even period, each 4-year moving average falls between two years, so it is not yet a trend value. The 4-year moving totals and averages are
\( \displaystyle 30 + 34 + 38 + 42 = 144,\quad \frac{144}{4} = 36 \) (between 2019 and 2020)
\( \displaystyle 34 + 38 + 42 + 40 = 154,\quad \frac{154}{4} = 38.5 \) (between 2020 and 2021)
\( \displaystyle 38 + 42 + 40 + 44 = 164,\quad \frac{164}{4} = 41 \) (between 2021 and 2022)
Centring takes the mean of each pair of neighbouring averages and places it against an actual year:
\( \displaystyle 2020: \frac{36 + 38.5}{2} = 37.25 \)
\( \displaystyle 2021: \frac{38.5 + 41}{2} = 39.75 \)
So the 4-year centred moving averages are 37.25 and 39.75.
Why the other options are incorrect
Option (1) 36, 38.5, 41 — Incorrect. These are the 4-year moving averages before centring; they lie between two years.
Option (2) 37.5, 40 — Incorrect. The centred values are 37.25 (the mean of 36 and 38.5) and 39.75 (the mean of 38.5 and 41), not 37.5 and 40.
Option (4) 36.5, 39.75 — Incorrect. The first centred value is 37.25, not 36.5.