Time series and forecasting (WACE Year 12 Mathematics Applications)
“Plot and describe time series, smooth with moving averages, deseasonalise with seasonal indices, fit a trend line and forecast future values.”
Time series analysis describes data measured over time and uses it to forecast. The pipeline is: plot and describe the components (trend, seasonal, cyclic, irregular), smooth with moving averages, compute seasonal indices and deseasonalise (divide by the index), fit a least-squares trend line to the deseasonalised data, substitute a future time to forecast, then reseasonalise (multiply by the index) to get the actual forecast.
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What this dot point is asking
You must plot and describe a time series, smooth it, deseasonalise it, fit a trend line and forecast.
The components of a time series
A time series plots a variable against time. It blends up to four components.
Trend: the long-term upward or downward movement. Seasonal: a regular pattern repeating within a fixed period (quarters, months). Cyclic: longer swings not of fixed length (the business cycle). Irregular: random, unpredictable variation.
Describing the series means naming which components are present and the overall direction.
Smoothing and deseasonalising
To see the trend, first reduce the noise. Moving-average smoothing averages each value with its neighbours; for quarterly data a centred four-point average removes the seasonal cycle. Seasonal indices measure each season's typical deviation from average and let you remove it.
- Find the seasonal indices (they sum to the number of seasons).
- Deseasonalise: deseasonalised value .
- Fit a least-squares trend line to the deseasonalised data.
- Forecast: substitute the future into the trend line.
- Reseasonalise: actual forecast deseasonalised forecast seasonal index of the target season.
The order matters: deseasonalise before fitting the trend (so the seasonal swing does not distort the line), and reseasonalise after forecasting (to put the season back).
The whole point of deseasonalising is to fit a clean trend line. Forgetting to reseasonalise the final forecast is the most common error: it leaves the answer as a deseasonalised value rather than the actual predicted figure for that season.
A full forecast
Quarterly visitor numbers have seasonal indices Q1 , Q2 , Q3 , Q4 . The deseasonalised data gives a trend line , with the first quarter of data. Forecast the actual visitors for , which is a Quarter 4.
Check the indices
, correct for four quarters.
Forecast the deseasonalised value
At : (deseasonalised visitors).
Reseasonalise for Quarter 4
Multiply by the Q4 index: visitors.
Interpret with caution
The forecast for that Quarter 4 is about visitors. Because is beyond the data, it is an extrapolation: it assumes the linear trend and the seasonal pattern both continue, which may not hold.
Limitations of forecasts
A time-series forecast assumes the trend and seasonal pattern persist. Forecasting far beyond the data is extrapolation and is risky: trends can flatten, reverse, or be disrupted by one-off events. Always state that a forecast assumes the past pattern continues, and flag extrapolations.
Where each tool fits in the pipeline
It helps to see how the separate time-series tools connect. The time series plot identifies which components are present. Moving-average smoothing reduces noise to reveal the trend and is also used in computing seasonal indices. Seasonal indices quantify the repeating effect and let you deseasonalise. The least-squares trend line measures the genuine long-term movement of the deseasonalised data. Forecasting substitutes a future time into the trend line, and reseasonalising restores the season. Each tool is a step, and a full forecasting question moves through all of them in this order.
- Fitting the trend line to raw data
- Deseasonalise first, or the seasonal swing distorts the slope.
- Forgetting to reseasonalise
- The final forecast for a specific season must be multiplied by that season's index; otherwise it is only the deseasonalised value.
- Using the wrong season's index
- Match the index to the season of the forecast time, not the first quarter.
- Ignoring extrapolation risk
- Forecasts beyond the data assume the pattern continues; say so.
Exam-style questions
Questions in the style of SCSA exam questions on this dot point, each with a worked answer. They are written by ExamExplained unless tagged "Past paper"; the year shows the paper a question is modelled on.
WACE 2021-style8 marksDeseasonalised quarterly sales give a trend line , where is the first quarter of data. The Quarter 1 seasonal index is . (a) Forecast the deseasonalised sales for . (b) Reseasonalise the forecast if is a Quarter 4 with seasonal index . (c) State one limitation of this forecast.Show worked answer →
Use the trend line, then restore the season.
(a) Deseasonalised forecast at : . (3 marks)
(b) Reseasonalise by multiplying by the Quarter 4 index: , so about units. (3 marks)
(c) The forecast assumes the linear trend and seasonal pattern continue unchanged; as is beyond the data it is an extrapolation and may be unreliable. (2 marks)
Markers reward the trend substitution, multiplying by the correct season's index, and an extrapolation-based limitation.
WACE 2023-style5 marksOutline the steps to forecast a future seasonal value from a time series, from the raw data to the final forecast.Show worked answer →
The standard pipeline runs deseasonalise, trend, reseasonalise.
Step 1: calculate the seasonal indices from the data (using averages or smoothing). Step 2: deseasonalise the data by dividing each value by its seasonal index. Step 3: fit a least-squares trend line to the deseasonalised data against a time variable . Step 4: substitute the future to forecast the deseasonalised value. Step 5: reseasonalise by multiplying by the seasonal index of the target season. (5 marks)
Markers reward all five steps in order, especially deseasonalise before fitting and reseasonalise after forecasting.