§-Syllabus dot point

TASMathematics ApplicationsSyllabus dot point

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.

Generated by Claude Opus 4.88 min answerUpdated 2026-06-02

Reviewed by: AI editorial process; not yet individually human-reviewed

Have a quick question? Jump to the Q&A page

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 $1.20$ means that season runs $20\%$ above the average; $0.85$ means $15\%$ below. For data with $n$ seasons, the indices must add to $n$ (for example, four quarterly indices sum to $4$ , and average to $1$ ).

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 $t$ 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.

Calculating seasonal indices from data

To find seasonal indices yourself, average each season's values, divide each seasonal average by the overall average of all the data, and the results are the indices. They must sum to the number of seasons (4 for quarterly, 12 for monthly), so if they do not, scale them: multiply each by the number of seasons divided by their current sum. This forces the average index to 1, which is the property that makes deseasonalising and reseasonalising consistent.

Computing quarterly seasonal indices

Average sales per quarter are Q1 $80$ , Q2 $120$ , Q3 $140$ , Q4 $60$ . We need to find the seasonal index for each quarter, which tells us how much each quarter departs from the typical quarterly level.

Step 1: Find the overall average across all quarters

We need a reference level to compare each quarter against, and the overall average across all four quarters serves this role. Dividing the total of the quarterly averages by the number of quarters gives a single "typical quarter" value:

\text{Overall average} = \frac{80 + 120 + 140 + 60}{4} = \frac{400}{4} = 100

Step 2: Compute each seasonal index by dividing by the overall average

A seasonal index is simply the ratio of each quarter's average to the overall average. An index greater than 1 means that quarter is above average; an index less than 1 means it is below. Dividing each quarterly average by 100 gives the four indices directly:

\text{Q1 index} = \frac{80}{100} = 0.80, \quad \text{Q2 index} = \frac{120}{100} = 1.20

\text{Q3 index} = \frac{140}{100} = 1.40, \quad \text{Q4 index} = \frac{60}{100} = 0.60

Step 3: Check that the indices sum to the number of seasons

For quarterly data the four indices must sum to exactly 4, which ensures the average index is 1 and that deseasonalising then reseasonalising returns the original value without distortion. If the sum is not 4, each index must be rescaled. Here:

0.80 + 1.20 + 1.40 + 0.60 = 4.00

No scaling is needed. Q3 runs $40\%$ above the quarterly average and Q4 runs $40\%$ below, reflecting a strong high season in Q3 and a quiet Q4.

Final answer: Q1 index $= 0.80$ , Q2 $= 1.20$ , Q3 $= 1.40$ , Q4 $= 0.60$ .

A complete forecast answer states the trend equation, shows the deseasonalising and reseasonalising steps where seasons apply, checks that seasonal indices sum to the number of seasons, 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.

TCE 20243 marksA graph shows actual and deseasonalised power costs for a household, with a regression line for the deseasonalised data. a) What long-term trend does the regression line suggest? b) i. To convert a deseasonalised prediction to actual, what is the final step? ii. When 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 (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 $\times$ index. This puts the seasonal variation back in (reseasonalise using $D = \frac{A}{I}$ rearranged to $A = D \times I$ ).

b) ii. (1 mark) 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.

TCE 20243 marksA table shows rainfall on Tasmania's east coast in 2023. a) Complete the 3-point moving-average row (two values missing). b) Add your values to the graph. c) The line of best fit for the smoothed data is

R = 0.006M + 30.1

, where

M

is months since December 2022. Add

(2, 30.1)

and

(11, 30.2)

and draw 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 January value cannot be found (no value before it), so the first computed average is for February: $\frac{3 + 54 + 19}{3} = 25$ (given). The missing entries are found the same way, for example the November average $= \frac{62 + 22 + 29}{3} = 38$ (round to whole numbers to match the table).

b) (1 mark) Plot the two computed averages at their correct months, joining the smoothed series.

c) (1 mark) Plot $(2, 30.1)$ and $(11, 30.2)$ , rule a straight line through them and extend it. The almost flat line shows a very slight upward trend in smoothed rainfall.

TCE 20233 marksA table shows quarterly power costs, 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 2024 (Spring index

0.952

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 $x = 16$ .
Substitute into the trend line for the deseasonalised prediction: $D = 1.0459(16) + 516.33 = 16.73 + 516.33 = 533.07$ .
Reseasonalise by multiplying by the Spring index: actual $= 533.07 \times 0.952 = \$507.48$ (about $\$507$ ). The final answer must be the actual cost in dollars, not the deseasonalised value.