Practice Maths

Seasonal Adjustment

Key Terms

Seasonal index (SI)
Measures how a season’s typical value compares to the overall mean; SI > 1 = above average, SI < 1 = below average.
Calculating SI
Average value for that season across all years ÷ overall mean of all data.
Check
All seasonal indices for one full cycle must sum to the number of seasons (4 for quarterly, 12 for monthly).
Deseasonalising
Actual value ÷ SI; removes the predictable seasonal effect to expose the trend.
Re-seasonalising
Trend value × SI; multiplies the seasonal effect back in when making a final forecast.
Seasonal adjustment
Another term for deseasonalising; produces data comparable across different seasons.

Seasonal Indices

A seasonal index (SI) measures how a particular season (quarter or month) compares to the annual average. An SI > 1 means that season is above average; an SI < 1 means it is below average; SI = 1 means exactly average.

Seasonal Index for a season = (average value for that season across all years) ÷ (overall mean of all data)

Check: the sum of all seasonal indices for one full cycle must equal the number of seasons.
(e.g. for quarterly data: SIQ1 + SIQ2 + SIQ3 + SIQ4 = 4)

Deseasonalising Data

To remove the seasonal effect from a data value (to “seasonally adjust” it):

Deseasonalised value = Actual value ÷ Seasonal Index

Deseasonalised values show the underlying trend and irregular variation, with the predictable seasonal effect removed. They allow fair comparisons between seasons.

Hot Tip: Always check that your seasonal indices sum to the number of seasons (4 for quarterly, 12 for monthly). If they don’t, the indices are wrong. Also remember: to deseasonalise, divide by the seasonal index — do not multiply. Multiplying is how you re-introduce the seasonal effect when forecasting.

Worked Example

Quarterly sales ($000) for a surf shop over 2 years:

Q1 Q2 Q3 Q4
Year 1 180 120 90 150
Year 2 200 130 100 170

Step 1 — Average for each quarter: Q1: (180+200)÷2 = 190; Q2: (120+130)÷2 = 125; Q3: (90+100)÷2 = 95; Q4: (150+170)÷2 = 160.

Step 2 — Overall mean: (190+125+95+160)÷4 = 570÷4 = 142.5.

Step 3 — Seasonal indices: SIQ1 = 190÷142.5 = 1.33; SIQ2 = 125÷142.5 = 0.88; SIQ3 = 95÷142.5 = 0.67; SIQ4 = 160÷142.5 = 1.12. Sum = 1.33+0.88+0.67+1.12 = 4.00 ✓

Step 4 — Deseasonalise Q1 Y1: 180 ÷ 1.33 = 135.3. This means Q1 Year 1 sales, after removing the seasonal boost, correspond to a trend level of about $135,300.

Full Lesson: Seasonal Adjustment

1. The Need for Seasonal Adjustment

Seasonal variation can mask the true trend. If retail sales in December are always higher than in July, a simple comparison of the two months tells us nothing about whether the business is growing. We need to remove the seasonal effect before we can make valid comparisons and detect genuine trend changes.

Australian Bureau of Statistics (ABS) uses seasonal adjustment routinely — when the government reports “seasonally adjusted unemployment” or “seasonally adjusted GDP growth”, they have removed the predictable seasonal patterns so month-to-month comparisons are meaningful.

2. Computing Seasonal Indices — Step by Step

Step 1: For each season, calculate the average value of that season across all years in the dataset. (If you have 3 years of quarterly data, average the three Q1 values, average the three Q2 values, etc.)

Step 2: Calculate the overall mean of all data values (or equivalently, the mean of the seasonal averages from Step 1).

Step 3: Divide each seasonal average by the overall mean. The result is the seasonal index for that season.

Step 4: Check: the sum of all seasonal indices must equal the number of seasons (4 for quarterly, 12 for monthly). If not, you have made an arithmetic error.

3. Interpreting Seasonal Indices

SI = 1.25 for Q1 means Q1 is typically 25% above the annual average.

SI = 0.72 for Q3 means Q3 is typically 28% below the annual average.

SI = 1.00 for Q2 means Q2 is exactly at the annual average.

Seasonal indices are dimensionless ratios — they have no units.

4. Deseasonalising

To deseasonalise a specific value, divide it by the seasonal index for that season:

Deseasonalised = Actual ÷ SI

This removes the predictable seasonal boost or dip, leaving the trend and irregular components. You can then plot deseasonalised values over time to see the underlying trend much more clearly than from the raw data.

5. Re-seasonalising (Going Back)

If you know the trend value for a future period and want to add back the seasonal effect to make a realistic forecast:

Forecast = Trend value × SI

This is the bridge between trend analysis and practical forecasting (covered in the Forecasting lesson).

6. Limitations

Seasonal indices assume the seasonal pattern is stable across all years in the dataset. If the seasonal pattern is changing (e.g. summer sales are becoming relatively larger each year due to climate change), the indices computed from historical data become less reliable over time. Always inspect the raw data for any evidence of changing seasonality before using indices for long-range forecasts.

Mastery Practice

  1. Fluency

    The quarterly visitor numbers (thousands) to a theme park over two years are: Q1: 85, 72; Q2: 120, 132; Q3: 60, 65; Q4: 95, 103. Calculate the seasonal index for each quarter. Show that the indices sum to 4.

  2. Fluency

    A business has quarterly seasonal indices of SIQ1 = 1.18, SIQ2 = 0.94, SIQ3 = 0.72, SIQ4 = 1.16. The actual Q3 sales are $84,000.

    1. Interpret the Q3 seasonal index.
    2. Calculate the deseasonalised Q3 sales.
  3. Fluency

    A seasonal index for Q2 is 1.35. Explain in plain language what this tells you about Q2 sales relative to the annual average.

  4. Fluency

    Three seasonal indices for a quarterly dataset are: SIQ1 = 0.85, SIQ2 = 1.10, SIQ4 = 0.95. Calculate SIQ3 without additional data. Justify your method.

  5. Understanding

    Monthly electricity consumption (MWh) for a Queensland school is recorded over 3 years. The average for each month across the 3 years is given below. The overall monthly mean is 145 MWh.

    Month Jan Feb Mar Apr May Jun
    Avg (MWh) 190 175 150 130 115 110
    Month Jul Aug Sep Oct Nov Dec
    Avg (MWh) 120 130 140 155 175 145
    1. Calculate the seasonal index for each month (round to 2 decimal places).
    2. Which months have SI > 1? Suggest a reason for the high consumption in those months.
    3. The actual June consumption in Year 4 is 118 MWh. Calculate the deseasonalised value. Is this above or below the overall average?
  6. Understanding

    The quarterly sales data ($000) for a garden centre over 3 years is:

    Year Q1 Q2 Q3 Q4
    195185210130
    2100198225142
    3108210238154
    1. Calculate the seasonal index for each quarter (to 2 d.p.).
    2. Deseasonalise all 12 data values.
    3. Compare the deseasonalised Q1 values across the three years. What does this reveal?
  7. Understanding

    A media company reports seasonally adjusted monthly app downloads. In January the actual downloads were 280,000 and the January SI = 1.40. In July actual downloads were 175,000 with SIJuly = 0.88.

    1. Calculate the deseasonalised (seasonally adjusted) downloads for January and July.
    2. Which month shows stronger underlying performance? Explain.
    3. The media company’s manager says “January had 280,000 downloads vs July’s 175,000 — we should focus all marketing on January.” Evaluate this reasoning statistically.
  8. Understanding

    Four quarterly seasonal indices are computed from 4 years of data: SIQ1 = 1.22, SIQ2 = 0.96, SIQ3 = 0.78, SIQ4 = 1.04.

    1. Verify that these indices are consistent (i.e. sum to the correct total).
    2. If the overall quarterly average revenue is $250,000, what is the expected revenue for Q3 based on the seasonal index alone?
    3. Actual Q3 revenue is $185,000. Calculate the deseasonalised value and interpret it.
  9. Problem Solving

    A tourism board records quarterly international visitor arrivals (thousands) over 3 years. The deseasonalised values for Q1 are 42.3, 44.8, 47.5 across Years 1–3. The actual Q1 arrivals were 55.0, 58.2, 61.8 thousand.

    1. Calculate the Q1 seasonal index (use Year 1 data). Verify it is consistent with Years 2 and 3.
    2. Using the deseasonalised values, describe and quantify the underlying trend in visitor arrivals.
    3. If the trend continues, estimate the deseasonalised Q1 arrivals in Year 4. Hence estimate the actual Q1 arrivals in Year 4.
  10. Problem Solving

    A financial analyst notices that the deseasonalised monthly revenue figures for a business form an approximately linear trend: the revenue appears to grow by about $3,200 per month on a seasonally adjusted basis. The seasonal index for December is 1.85.

    1. If the seasonally adjusted revenue for October (Month 10) is $82,000, what is the expected seasonally adjusted revenue for December (Month 12)?
    2. Use the seasonal index to forecast the actual December revenue.
    3. The actual December revenue turns out to be $166,000. Calculate and interpret the “residual” between the forecast and the actual value. What might explain this difference?
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