How do we describe patterns in data over time and forecast future values?
Analyse time series using smoothing and trend lines, then forecast future values.
Trend, seasonal and irregular components, moving-average smoothing, seasonal indices, and trend-line forecasting in TCE Mathematics Applications.
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What this dot point is asking
A time series is data recorded in time order, such as monthly sales or quarterly rainfall. The goal is to see through the wobble and describe the underlying movement, then project it forward.
Smoothing with moving averages
A moving average replaces each value with the average of itself and its neighbours, which smooths out short-term fluctuations and reveals the trend. The number of points averaged is the order. For an odd order (e.g. 3 or 5), the smoothed value lines up directly with a data point. For an even order (e.g. 4 for quarterly data), you must centre the average so it lines up with a time point.
Seasonal indices
A seasonal index measures how a particular season compares with the average. An index of means that season runs above the average; means below. For data with seasons, the indices must add to (for example, four quarterly indices sum to , and average to ).
Deseasonalising strips out the seasonal pattern so the underlying trend is clear. You fit a trend line to the deseasonalised data, forecast along it, then reseasonalise by multiplying back by the relevant index.
Trend-line forecasting
Once you have a trend line (often a least-squares line fitted to time on the horizontal axis), substitute the future time value to forecast. As with regression, forecasting far beyond the data is extrapolation and grows less reliable the further out you go.
A complete forecast answer states the trend equation, shows the deseasonalising and reseasonalising steps where seasons apply, and comments on reliability if the forecast reaches well beyond the data.
Exam-style practice questions
Practice questions written in the style of TASC exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
2024 TASC General Mathematics3 marksFigure 1 is a graph of actual and deseasonalised power costs for a Tasmanian household, with a regression line for the deseasonalised data. a) What long term trend does the regression line of the deseasonalised data suggest? b) i. When using deseasonalised data to make predictions, the final step is to convert the deseasonalised prediction to actual. How is this achieved? ii. In what situation would a smoothing process, such as 3 or 5 point averaging, be used instead of deseasonalising?Show worked answer →
a) (1 mark) The regression line of the deseasonalised data has a slight positive slope, so the long term trend is a small steady increase in power costs over time (accept a comment that there is little or no increase, since the rise is very gradual).
b) i. (1 mark) Multiply the deseasonalised prediction by the seasonal index for that season: actual = deseasonalised x index. This puts the seasonal variation back in. (Re-seasonalise using D = A / I rearranged to A = D x I.)
b) ii. (1 mark) A moving-average smoothing is used when the data has no clear repeating seasonal pattern (or the period is unknown or irregular), so seasonal indices cannot be calculated. Smoothing just removes random fluctuation to reveal the underlying trend.
2024 TASC General Mathematics3 marksTable 4 shows rainfall data recorded on Tasmania's east coast during 2023. a) Complete the 3-point moving average row of Table 4 (there are two values missing). b) Add your values to the graph in Figure 4. c) The line of best fit for the smoothed data is R = 0.006M + 30.1, where R is the 3-point moving average of rainfall and M is the number of months since December 2022. Add the points (2, 30.1) and (11, 30.2) to Figure 4 and draw in the line.Show worked answer →
a) (1 mark) A 3-point moving average replaces each value with the mean of it and its two neighbours. The missing January value cannot be found (no value before it), so the first computed average is for February: (3 + 54 + 19) / 3 = 25 (given). The two missing entries are found the same way, for example the November average = (62 + 22 + 29) / 3 = 38 (round to whole numbers to match the table).
b) (1 mark) Plot the two computed averages on the graph at their correct months, joining the smoothed series.
c) (1 mark) Plot (2, 30.1) and (11, 30.2), then rule a straight line through them and extend it across the plot. The almost flat line shows a very slight upward trend in smoothed rainfall over the year.
2024 TASC General Mathematics3 marksTable 8 shows quarterly power costs for a Tasmanian household, with deseasonalising almost complete. The regression line for the deseasonalised data is D = 1.0459x + 516.33 where x is the season number. Predict the actual power cost for Spring of 2024.Show worked answer →
(3 marks) Work in three steps.
Find the season number x for Spring 2024. The data runs from Summer 2021 (x = 1) so Spring 2024 is season x = 16.
Substitute into the trend line to get the deseasonalised prediction: D = 1.0459(16) + 516.33 = 16.73 + 516.33 = 533.07.
Re-seasonalise by multiplying by the Spring seasonal index (0.952): actual = 533.07 x 0.952 = 507). The final answer must be the actual cost in dollars, not the deseasonalised value.