Topics

Moving averages and finding the trend

Why you need a moving average

The trend is the long-term direction, but you cannot see it clearly through the seasonal zigzag.
A moving average removes the seasonal pattern by averaging over exactly one full cycle, so the highs and lows cancel out. What remains is the trend.

Choosing the order

The order of the moving average must equal the length of the seasonal cycle.

Data	Cycle	Moving average
Quarterly	4 quarters	4-point
Monthly	12 months	12-point
Daily with a weekly pattern	7 days	7-point

Using the wrong order leaves seasonal pattern in the trend, which is the commonest technical error in this internal. If your fitted trend still zigzags, check the order.

Odd and even orders

With an odd order (5, 7), each average falls exactly on a data point, and no further work is needed.
With an even order (4, 12), each average falls between two data points. This is why quarterly and monthly data need a centred moving average.

The centred moving average

The procedure for quarterly data:
- Average quarters 1–4. This value sits between Q2 and Q3.
- Average quarters 2–5. This value sits between Q3 and Q4.
- Average those two averages. The result sits exactly on Q3.
Equivalently, a centred 4-point moving average is a weighted average of five values with weights $\tfrac{1}{8}, \tfrac{1}{4}, \tfrac{1}{4}, \tfrac{1}{4}, \tfrac{1}{8}$ .
The software does this for you. You need to understand what it produces and why, not compute it by hand — but being able to explain the centring is worth marks when asked how the trend was found.

What you lose at the ends

A moving average cannot be calculated for the first and last few points, because there are not enough values on both sides.
For a centred 4-point average you lose two points at each end.
This matters for forecasting, because the most recent trend values — the ones a forecast depends on most — are the ones you do not have. Saying so is a genuine Excellence-level reflection on the method.

Describing the trend once you have it

Plot the trend line on the same axes as the original series. This single graph is the most useful display in the whole report.
Then describe it, with values:
- Direction and size: "The trend rises from about 410 units in early 2019 to about 520 in late 2023, an increase of roughly 27% over five years."
- Rate: "The rise averages about 22 units per year, and is close to linear."
- Changes: "The trend flattens between 2021 and 2022 before resuming its rise."
Then explain each feature from the context. A flattening in the trend is a fact; a flattening caused by a supply disruption or a competitor entering the market is an explanation.

Common problems

Problem	What it means	What to do
Trend still zigzags	Wrong moving average order	Match the order to the cycle length
Trend is very jagged	Order too short, or a noisy series	Check the order; comment on the variability
Trend is over-smoothed	Order longer than the cycle	Use exactly one cycle
A large jump in the trend	A level shift in the data	Investigate the cause and consider excluding that period

Tips and Tricks

State the order you used and why. "A centred 4-point moving average was used because the data are quarterly, so one full seasonal cycle is four quarters" is one sentence and directly evidences a required component.

Plot the trend over the raw data. A marker should be able to see at a glance that the trend follows the series without the zigzag.

Quote trend values, not just the shape. Give the start value, the end value, and the change — ideally as both an absolute amount and a percentage.

Say what happens at the ends. Noting that the moving average cannot reach the most recent points, and that this limits the forecast, is exactly the kind of reflection Excellence rewards.

If the trend has more than one phase, describe each phase separately with its own values and its own contextual explanation. Averaging across a structural break describes neither period accurately.

Worked ExampleExplaining the trend section of a report

A student investigating quarterly sales of outdoor furniture at a New Zealand retailer has fitted a centred 4-point moving average. The trend rises from about $140 000 per quarter in 2018 to about $205 000 in 2021, is flat at around $205 000 through 2022, then rises again to about $240 000 by 2024.

Write the trend section of her report.

Step 1 — Say what was done and why

Step 2 — Describe the trend, with values

Step 3 — Explain each phase from the context

The 2018–2021 growth. Several factors plausibly contributed. Outdoor furniture is a discretionary household purchase, so demand tends to follow housing activity and household spending confidence. The strong growth over this period coincides with a period of rising house prices and high levels of home renovation, and — during 2020 and 2021 — with restrictions on international travel, which redirected discretionary household spending from overseas holidays towards the home. Lockdown periods in particular increased interest in outdoor living space.

The 2022 flattening. This is the most interesting feature and worth explaining carefully. Two mechanisms are plausible and may both apply:

Demand was pulled forward. Households that upgraded their outdoor furniture during 2020–21 do not need to do so again for several years, so the earlier surge borrowed from later demand.
Conditions reversed. Rising interest rates and cost-of-living pressure through 2022 reduced discretionary spending, and the reopening of international borders redirected household spending back towards travel.

The 2023–24 recovery. Growth resumes but at a slower rate than 2018–21, which is consistent with a return to underlying demand growth without the exceptional boost of the earlier period.

Step 4 — Reflect on the method

What the smoothing cost. Because a centred 4-point moving average requires two quarters either side, no trend value exists for the final two quarters of the data. The most recent trend value available is therefore from mid-2024, and any forecast must extrapolate beyond that point. This is a limitation worth stating explicitly, since the most recent behaviour of the series is precisely what a forecast depends on most.

Whether one trend describes the series. Because the trend has three clearly distinct phases, describing it with a single average growth rate would be misleading. A forecast based on the overall 2018–2024 average growth would be too high, because it incorporates the exceptional 2018–21 period. The 2023–24 growth rate is the more appropriate basis for forecasting, since it reflects current conditions rather than a period whose drivers no longer apply.

Why this reaches Excellence

Moving averages and finding the trend

Covers AS91580

Why you need a moving average

The trend is the long-term direction, but you cannot see it clearly through the seasonal zigzag.
A moving average removes the seasonal pattern by averaging over exactly one full cycle, so the highs and lows cancel out. What remains is the trend.

Choosing the order

The order of the moving average must equal the length of the seasonal cycle.

Data	Cycle	Moving average
Quarterly	4 quarters	4-point
Monthly	12 months	12-point
Daily with a weekly pattern	7 days	7-point

Using the wrong order leaves seasonal pattern in the trend, which is the commonest technical error in this internal. If your fitted trend still zigzags, check the order.

Odd and even orders

With an odd order (5, 7), each average falls exactly on a data point, and no further work is needed.
With an even order (4, 12), each average falls between two data points. This is why quarterly and monthly data need a centred moving average.

The centred moving average

The procedure for quarterly data:
- Average quarters 1–4. This value sits between Q2 and Q3.
- Average quarters 2–5. This value sits between Q3 and Q4.
- Average those two averages. The result sits exactly on Q3.
Equivalently, a centred 4-point moving average is a weighted average of five values with weights $\tfrac{1}{8}, \tfrac{1}{4}, \tfrac{1}{4}, \tfrac{1}{4}, \tfrac{1}{8}$ .
The software does this for you. You need to understand what it produces and why, not compute it by hand — but being able to explain the centring is worth marks when asked how the trend was found.

What you lose at the ends

A moving average cannot be calculated for the first and last few points, because there are not enough values on both sides.
For a centred 4-point average you lose two points at each end.
This matters for forecasting, because the most recent trend values — the ones a forecast depends on most — are the ones you do not have. Saying so is a genuine Excellence-level reflection on the method.

Describing the trend once you have it

Plot the trend line on the same axes as the original series. This single graph is the most useful display in the whole report.
Then describe it, with values:
- Direction and size: "The trend rises from about 410 units in early 2019 to about 520 in late 2023, an increase of roughly 27% over five years."
- Rate: "The rise averages about 22 units per year, and is close to linear."
- Changes: "The trend flattens between 2021 and 2022 before resuming its rise."
Then explain each feature from the context. A flattening in the trend is a fact; a flattening caused by a supply disruption or a competitor entering the market is an explanation.

Common problems

Problem	What it means	What to do
Trend still zigzags	Wrong moving average order	Match the order to the cycle length
Trend is very jagged	Order too short, or a noisy series	Check the order; comment on the variability
Trend is over-smoothed	Order longer than the cycle	Use exactly one cycle
A large jump in the trend	A level shift in the data	Investigate the cause and consider excluding that period

Tips and Tricks

Plot the trend over the raw data. A marker should be able to see at a glance that the trend follows the series without the zigzag.

Quote trend values, not just the shape. Give the start value, the end value, and the change — ideally as both an absolute amount and a percentage.

Say what happens at the ends. Noting that the moving average cannot reach the most recent points, and that this limits the forecast, is exactly the kind of reflection Excellence rewards.

Worked ExampleExplaining the trend section of a report

Write the trend section of her report.

Step 1 — Say what was done and why

Step 2 — Describe the trend, with values

Step 3 — Explain each phase from the context

The 2022 flattening. This is the most interesting feature and worth explaining carefully. Two mechanisms are plausible and may both apply:

Demand was pulled forward. Households that upgraded their outdoor furniture during 2020–21 do not need to do so again for several years, so the earlier surge borrowed from later demand.
Conditions reversed. Rising interest rates and cost-of-living pressure through 2022 reduced discretionary spending, and the reopening of international borders redirected household spending back towards travel.

The 2023–24 recovery. Growth resumes but at a slower rate than 2018–21, which is consistent with a return to underlying demand growth without the exceptional boost of the earlier period.