Time Series and Forecasting - TCE Mathematics Applications (Tasmania)

Syllabus dot point

“Analyse time series using smoothing and trend lines, then forecast future values.”

TCEMathematics ApplicationsUnit 48 min readUpdated 2 Jun 2026

Quick answer

You must identify the components of a time series (trend, seasonal, irregular), smooth out short-term noise with moving averages, calculate and apply seasonal indices, and fit a trend line to forecast future values.

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.

Components of a time series

Trend: the long-term upward or downward movement. Seasonal: a regular pattern that repeats over a fixed period (e.g. higher each summer). Irregular: random, unpredictable variation.

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.

Three-point moving average

Sales over five days: $10, 14, 11, 17, 15$ .

Compute the smoothed values

Day 2: $(10 + 14 + 11)/3 = 35/3 = 11.67$ .
Day 3: $(14 + 11 + 17)/3 = 42/3 = 14.00$ .
Day 4: $(11 + 17 + 15)/3 = 43/3 = 14.33$ .

The smoothed series $11.67, 14.00, 14.33$ rises steadily, showing an upward trend once the day-to-day noise is removed. There is no smoothed value for the first or last day.

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 and reseasonalising

\text{deseasonalised value} = \frac{\text{actual value}}{\text{seasonal index}}

\text{reseasonalised value} = \text{deseasonalised value} \times \text{seasonal index}

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.

Forecasting with a seasonal index

The deseasonalised trend line for quarterly sales is $\hat{y} = 200 + 15t$ , where $t$ is the quarter number. Forecast the actual sales for $t = 9$ , which falls in a quarter with seasonal index $1.30$ .

Project the trend, then reseasonalise

Deseasonalised forecast: $\hat{y} = 200 + 15(9) = 200 + 135 = 335$ .

Reseasonalise: $335 \times 1.30 = 435.5$ .

The forecast actual sales for that quarter are about $436$ units. The seasonal index lifts the trend value because this is a high quarter.

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.

Even-order smoothing needs centring

For quarterly data use a centred 4-point moving average; for monthly data use a centred 12-point average. Forgetting to centre an even-order average shifts every smoothed point half a period out of place.

Adding the seasonal index instead of multiplying

Seasonal indices are multipliers, not amounts to add. To deseasonalise you divide by the index, and to reseasonalise you multiply. Adding or subtracting the index value is a common and costly error.

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 questions

Questions in the style of TASC 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.

Practice3 marks

A 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.

Practice3 marks

A 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.

Practise this

Sources & how we know this

General Mathematics Level 3 (MTG315123) course document (2023 onwards) — TASC