Seasonal indices and deseasonalising in WACE Mathematics Applications Unit 4

Syllabus dot point

“Calculate seasonal indices, deseasonalise and reseasonalise a time series, and interpret seasonal indices in context.”

WACEMathematics ApplicationsUnit 48 min readUpdated 2 Jun 2026

Quick answer

A seasonal index measures how a particular season typically compares with the average: an index of $1.20$ means that season runs $20\%$ above average, $0.80$ means $20\%$ below. Indices for one cycle sum to the number of seasons (so $4$ for quarters, $12$ for months). Deseasonalise by dividing the actual value by its index; reseasonalise by multiplying. Deseasonalised data lets you compare periods fairly and fit a trend line.

Jump to a section

What this dot point is asking
What a seasonal index is
Interpreting an index
Deseasonalising and reseasonalising
Why deseasonalise
Calculating seasonal indices from data
Exam-style questions

What this dot point is asking

You must calculate seasonal indices, deseasonalise and reseasonalise, and interpret indices in context.

What a seasonal index is

A seasonal index captures the repeating effect of a season (a quarter, month or day) relative to the average.

Seasonal index

A seasonal index of $1$ means a season is exactly average. Above $1$ means above average (a peak season); below $1$ means below average (a quiet season). The indices for one full cycle add to the number of seasons in the cycle.

So four quarterly indices sum to $4$ (averaging $1$ ), and twelve monthly indices sum to $12$ . This summing rule lets you find a missing index.

Interpreting an index

An index minus $1$ , as a percentage, is the seasonal effect. An index of $1.30$ means $30\%$ above average; $0.85$ means $15\%$ below average. Always interpret in the context of the data: "December sales are typically $30\%$ above the monthly average".

Index above or below 1

An index greater than $1$ is a high (above-average) season; less than $1$ is a low season. The further from $1$ , the stronger the seasonal effect. An index of exactly $1$ would mean that season is unaffected by seasonality.

Deseasonalising and reseasonalising

Removing the seasonal effect lets you see the underlying trend; restoring it converts a trend forecast back to an actual figure.

Deseasonalise and reseasonalise

Deseasonalised value $= \dfrac{\text{actual value}}{\text{seasonal index}}$ .

Reseasonalised value $=$ deseasonalised value $\times$ seasonal index.

Dividing strips out the seasonal bump or dip; multiplying puts it back. Deseasonalised data is the right input for fitting a trend line, because the trend line should not be distorted by the seasonal pattern.

Worked example

Working with seasonal indices

A cafe's quarterly sales have indices Q1 $= 1.25$ , Q2 $= 1.05$ , Q3 $= 0.90$ and an unknown Q4. The actual Q3 sales were $\$36\,000$ .

Step 1: Find the missing seasonal index using the summing rule.

Seasonal indices for a full cycle must sum to the number of seasons in that cycle. For quarterly data the four indices sum to $4$ , because their average is $1$ (meaning the average quarter is exactly average). Setting up the equation: $1.25 + 1.05 + 0.90 + x = 4$ , so $x = 4 - 3.20 = 0.80$ . The Q4 index is $0.80$ .

Step 2: Interpret the Q1 and Q4 indices in context.

An index minus $1$ , expressed as a percentage, gives the seasonal effect relative to the average quarter. Q1 index $1.25$ means Q1 sales are typically $25\%$ above the quarterly average, marking Q1 as a peak season for this cafe. Q4 index $0.80$ means Q4 sales are typically $20\%$ below the quarterly average, so Q4 is the cafe's quietest quarter.

Step 3: Deseasonalise the Q3 actual figure.

Deseasonalising removes the seasonal effect so we can compare the underlying level across different quarters. We divide the actual value by the seasonal index for that quarter: deseasonalised Q3 $= \dfrac{36000}{0.90} = 40000$ , that is $\$40\,000$ . This figure represents what Q3 would have been if it were a perfectly average quarter, free from the seasonal dip.

Step 4: Reseasonalise a deseasonalised forecast.

A trend line fitted to deseasonalised data gives forecasts without the seasonal pattern, so we must restore that pattern before quoting an actual prediction. If the trend line gives a deseasonalised Q1 forecast of $\$42\,000$ , we multiply by the Q1 index to put the seasonal peak back in: $42000 \times 1.25 = \$52\,500$ . This is the actual expected sales figure for Q1.

Why deseasonalise

Comparing raw seasonal figures is misleading: a December peak is not "growth", it is the season. Deseasonalising removes that effect so you can judge whether the business is genuinely trending up or down. After fitting a trend line to deseasonalised data and forecasting, you reseasonalise the forecast to predict the actual figure for a specific season.

Calculating seasonal indices from data

When indices are not given, compute them. The standard method: find each value as a proportion of the average for its cycle (or of a centred moving average), then average those proportions for each season across all cycles, and finally scale the averaged figures so they sum to the number of seasons. For instance, if four raw quarterly averages came to $0.78, 1.08, 0.92$ and $1.18$ , summing to $3.96$ , multiply each by $\dfrac{4}{3.96} = 1.0101$ so the adjusted indices sum to exactly $4$ . This final scaling step is where marks are commonly lost, so always confirm the indices total the number of seasons before using them.

Common errors

Multiplying when you should divide: Deseasonalise by dividing by the index; reseasonalise by multiplying. Mixing these reverses the seasonal effect.
Indices not summing to the number of seasons: Four quarterly indices sum to $4$ , twelve monthly to $12$ ; use this to find a missing index.
Misreading an index as a percentage directly: An index of $1.20$ is $20\%$ above average, not $120\%$ above; subtract $1$ first.
Fitting a trend line to raw seasonal data: Deseasonalise first, then fit the trend, then reseasonalise the forecast.

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 2022-style6 marks

A shop's quarterly sales have seasonal indices: Quarter 1

= 0.80

, Quarter 2

= 1.10

, Quarter 3

= 0.90

, Quarter 4

= ?

. (a) Find the missing seasonal index. (b) Interpret the Quarter 2 index. (c) The actual Quarter 1 sales were

\$48\,000

; deseasonalise this figure.

Show worked answer →

Seasonal indices for four quarters sum to $4$ .

(a) $0.80 + 1.10 + 0.90 + x = 4$ , so $x = 4 - 2.80 = 1.20$ . Quarter 4 index $= 1.20$ . (2 marks)

(b) A Quarter 2 index of $1.10$ means Quarter 2 sales are typically $10\%$ above the quarterly average. (2 marks)

Markers reward indices summing to $4$ , the "10% above average" interpretation, and dividing by the index to deseasonalise.

WACE 2024-style5 marks

Monthly visitor numbers have a December seasonal index of

1.45

. (a) Interpret this index. (b) The deseasonalised forecast for next December is

8200

visitors. Reseasonalise it to give the actual forecast.

Show worked answer →

An index above $1$ marks an above-average season.

(a) A December index of $1.45$ means December visitor numbers are typically $45\%$ above the monthly average, a strong peak season. (2 marks)

(b) Reseasonalise by multiplying the deseasonalised value by the index: $8200 \times 1.45 = 11890$ visitors. (3 marks)

Markers reward the "45% above average" interpretation and multiplying by the index to reseasonalise.